On the Complexity of Scheduling in Wireless Networks

  • Changhee Joo1Email author,

    Affiliated with

    • Gaurav Sharma2,

      Affiliated with

      • Ness B. Shroff3 and

        Affiliated with

        • Ravi R. Mazumdar4

          Affiliated with

          EURASIP Journal on Wireless Communications and Networking20102010:418934

          DOI: 10.1155/2010/418934

          Received: 11 January 2010

          Accepted: 1 September 2010

          Published: 7 September 2010

          Abstract

          We consider the problem of throughput-optimal scheduling in wireless networks subject to interference constraints. We model the interference using a family of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq1_HTML.gif -hop interference models, under which no two links within a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq2_HTML.gif -hop distance can successfully transmit at the same time. For a given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq3_HTML.gif , we can obtain a throughput-optimal scheduling policy by solving the well-known maximum weighted matching problem. We show that for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq4_HTML.gif , the resulting problems are NP-Hard that cannot be approximated within a factor that grows polynomially with the number of nodes. Interestingly, for geometric unit-disk graphs that can be used to describe a wide range of wireless networks, the problems admit polynomial time approximation schemes within a factor arbitrarily close to 1. In these network settings, we also show that a simple greedy algorithm can provide a 49-approximation, and the maximal matching scheduling policy, which can be easily implemented in a distributed fashion, achieves a guaranteed fraction of the capacity region for "all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq5_HTML.gif ." The geometric constraints are crucial to obtain these throughput guarantees. These results are encouraging as they suggest that one can develop low-complexity distributed algorithms to achieve near-optimal throughput for a wide range of wireless networks.

          1. Introduction

          Scheduling link transmissions in a wireless network so as to optimize one or more of the performance objectives (e.g., throughput, delay, or energy) has been the topic of paramount interest over the past several decades. In their seminal work, Tassiulas and Ephremides [1] characterized the capacity region of constrained queuing systems, such as a wireless network. They developed a queue length-based scheduling scheme that is throughput-optimal, that is, it stabilizes the network if the user rates fall within the capacity region of the network. Unlike wireline networks, where all links have fixed capacities, the capacity of a wireless link can be influenced by channel variation due to fading, changes in power allocation or routing, changes in network topology, and so forth. Thus, the capacity region of a wireless network can vary due to changes in power allocation or routing. To efficiently utilize the wireless resources, one must therefore develop algorithms that can perform jointly routing, link scheduling, and power control under possibly varying channel conditions and network topology. This has spurred recent interest in developing cross-layer optimization algorithms (see, e.g., [25]).

          Motivated by the works on fair resource allocation in wireline networks [6, 7], researchers have also incorporated congestion control into the cross-layer optimization framework [810]. The congestion control component controls the rate at which users inject data into the network to ensure that the user rates fall within the capacity region.

          Most of the above cross-layer optimization problems have been shown to exhibit a mathematical decomposition [2, 8]. To elaborate, the cross-layer optimization problem can be decomposed into multiple subproblems, where each subproblem corresponds to optimization across a single layer. The subproblems are loosely coupled through parameters that correspond to congestion prices or queue lengths at the individual links.

          The main component of all these cross-layer optimization schemes is the optimal scheduler that solves a very difficult global optimization problem of the form:
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ1_HTML.gif
          (1)

          where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq6_HTML.gif denotes the set of wireless links; http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq7_HTML.gif is the vector of link rates http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq9_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq10_HTML.gif is the congestion price or possibly some function of queue length at link http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq11_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq12_HTML.gif is the capacity region of the network.

          The main difficulty in solving the above optimization problem is that the capacity region http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq13_HTML.gif depends on the network topology and, in general, has no easy representation in terms of the power constraints at the individual links or nodes. The above optimization problem is, in general, NP-Complete and Nonapproximable.

          In this paper, we consider a class of scheduling problems that we term Maximum Weighted http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq14_HTML.gif -Valid Matching Problems (MWKVMP s). These problems arise as simplifications to the scheduling problem specified by (1). The basic idea is to limit the interference to only http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq15_HTML.gif hops, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq16_HTML.gif is a positive integer. By varying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq17_HTML.gif , one can capture the interference characteristics of a broad range of wireless networks.

          The rest of the paper is organized as follows. The model, problem formulation, related works, motivation, and main contributions of this work are presented in the next section. Some hardness and approximability results for the class of scheduling problems that we consider are presented in Section 3. We then restrict our attention to geometric unit-disk graphs that naturally model the connectivity graph of wireless networks, and develop approximation schemes for our scheduling problems in Section 4. By focusing on the throughput performance in Section 5, we reduce the complexity of scheduling schemes further, and show that a distributed maximal matching algorithm achieves a provable throughput guarantee. The geometric constraints of graphs remain crucial to obtain the throughput guarantees. Finally, we provide concluding remarks in Section 6.

          2. System Model and Problem Formulation

          We consider a set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq18_HTML.gif of wireless nodes, each communicating over a single wireless interface. We assume that all transmissions are carried out over the same wireless channel, and therefore interfere with each other. We assume that all transmissions from a node are carried out at the same power level (which can be different for different nodes). We connect two nodes with an (undirected) edge if each of them can successfully receive from the other, provided no other node in the network transmits at the same time. The set of (undirected) edges so formed is denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq19_HTML.gif . Note that the existence of an edge between two nodes depends on the power allocated to the nodes, noise variances, as well as coding and modulation schemes. Our emphasis on bidirectional edges stems from the fact that most network and transport layer protocols assume bidirectional communications between the nodes. We also note that our main results can easily be extended to settings where directed edges are allowed between the nodes.

          We next introduce the class of scheduling problems we consider in this paper. We first introduce some notation. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq20_HTML.gif be an undirected graph (connectivity graph of a wireless network, in our case) having http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq21_HTML.gif as the set of vertices (nodes) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq22_HTML.gif as the set of edges (link). A matching is a set of edges no two of which share a common vertex. We now generalize this concept of matching to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq23_HTML.gif -valid matchings for an integer http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq24_HTML.gif .

          Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq25_HTML.gif denote the minimum number of hops between vertices http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq26_HTML.gif . Letting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq27_HTML.gif denote the set of nonnegative integers, we define a distance function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq28_HTML.gif as follows: for two edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq29_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq30_HTML.gif , let
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ2_HTML.gif
          (2)
          We call a set of edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq31_HTML.gif a " http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq32_HTML.gif -valid matching" if for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq33_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq34_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq35_HTML.gif . Observe that the concept of matching discussed before is equivalent to the concept of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq36_HTML.gif -Valid matching in this new terminology. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq37_HTML.gif denote the set of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq38_HTML.gif -Valid matchings of the graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq39_HTML.gif . We consider the following scheduling problems:
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ3_HTML.gif
          (3)
          where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq40_HTML.gif denotes the weight of edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq41_HTML.gif . Note that the weight of each edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq42_HTML.gif is a positive, but otherwise arbitrary, number that can possibly depend on many factors (e.g., congestion price, supported rate, queue length). The above class of problems will henceforth be referred to as Maximum Weighted http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq43_HTML.gif -Valid Matching Problems (MWKVMP s). When all edge weights are set to unity, we obtain the following class of problems:
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ4_HTML.gif
          (4)

          where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq44_HTML.gif denotes the cardinality of the set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq45_HTML.gif . In the sequel, we refer to these problems as Maximum http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq46_HTML.gif -Valid Matching Problems (MKVMP s).

          We note that the scheduling problems specified by (3) are natural simplifications of the complex scheduling problem specified by (1). This is because for a given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq47_HTML.gif , by satisfying the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq48_HTML.gif -hop interference constraints one can guarantee a certain fixed data rate at a given edge. The weight of each edge can then be determined as some function of the rate it supports and the congestion price at the edge. The scheduling problem specified by (1) then corresponds to MWKVMP for that particular value of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq49_HTML.gif . For simplicity of notation, we did not explicitly show the dependence of edge weights on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq50_HTML.gif in (3).

          From the above discussion, it is not surprising to see that MWKVMP s can represent the scheduling problem specified by (1) under a wide variety of interference models. Below we discuss two widely used interference models that can be obtained as special cases of the interference constraints in (3).

          Node-Exclusive (or Primary) Interference Model

          This is a commonly used model for Bluetooth and FH-CDMA networks [11, 12]. Under this model, the set of edges that transmit simultaneously must constitute a matching. Then the scheduling problems specified by (3) and (4) correspond to the classical Maximum Weighted Matching Problem (MWMP) and the Maximum Matching Problem (MMP), respectively. Both these problems can be solved in polynomial time [13].

          IEEE 802.11-Based Interference Model

          This is a commonly used model for IEEE 802.11-based wireless networks [9, 14], under which the chosen set of edges must constitute a 2-Valid matching. It models the communication under the RTS/CTS-based scheme of IEEE 802.11 DCF (see Figure 1). Note that the sender and the receiver exchange RTS and CTS messages preventing their neighboring nodes from participating in a communication, which is equivalent to saying that the chosen set of communicating node pairs must constitute a 2-Valid matching.

          In general, we use the term " http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq51_HTML.gif -hop interference model," under which a scheduler should provide a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq52_HTML.gif -valid matching. The node-exclusive and IEEE 802.11-based interference models correspond to the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq53_HTML.gif -hop interference model with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq54_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq55_HTML.gif , respectively.
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Fig1_HTML.jpg
          Figure 1

          The http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq56_HTML.gif -hop interference set of a given edge for RTS/CTS based communication model of IEEE 802. 11 DCF.

          2.1. Related Work

          The http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq57_HTML.gif -hop interference model has been studied in many different contexts due to its simplicity [1, 4, 8, 12, 1517]. A polynomial time link scheduling algorithm has been developed in [12], and distributed schemes that guarantee a throughput within a constant factor of the optimal have been developed in [8, 15]. Recently, a class of throughput-optimal scheduling policies, called pick-and-compare, has been proposed [16, 17]. Although they achieve the throughput-optimality with a low complexity, they result in causing significantly long queue lengths, which in turn results in high delays, and for practical buffer sizes, can result in low throughput performance [18].

          In [9], the performance of maximal scheduling schemes has been studied under the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq58_HTML.gif -hop interference model. It has been shown that the maximal scheduling schemes achieve a throughput within a factor of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq59_HTML.gif of the capacity region, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq60_HTML.gif denotes the maximum link degree. In [15], the maximal scheduling schemes are shown to achieve at least a factor of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq61_HTML.gif of the optimal throughput, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq62_HTML.gif is the interference degree of the connectivity graph (see Definition 11). It also has been shown in [19, 20] that random access scheduling policies can achieve comparable performance.

          The MKVMP for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq63_HTML.gif is often known as the induced matching problem, which has been shown to be NP-Hard [21]. The work of [14] is closest in spirit to our work. The authors consider the induced matching problem from the perspective of carrying out maximum number of simultaneous transmissions. They study the approximability of the induced matching problem for general as well as specific kinds of graphs, and develop a distributed constant factor Polynomial-Time Approximation Scheme (PTAS) for the induced matching problem under geometric unit-disk graphs.

          However, most previous studies are limited to the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq64_HTML.gif -hop or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq65_HTML.gif -hop interference model. It has been observed through simulations in [22] that, under different network settings, the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq66_HTML.gif -hop interference model with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq67_HTML.gif can better capture the network interference constraints. For the detailed results, we refer to our technical report [22].

          2.2. Main Contributions

          From a theoretical perspective, we provide several results on the hardness and approximability of MWKVMP and MKVMP for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq68_HTML.gif . Although some of these results have previously been obtained for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq69_HTML.gif , to the best of our knowledge no prior work has studied MWKVMP or MKVMP for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq70_HTML.gif . Since weighted matching problems arise in a variety of contexts, these results might find applications in other fields (e.g., VLSI) as well.

          From a wireless networking perspective, we provide a Polynomial-Time Approximation Scheme (PTAS) for MWKVMP restricted to geometric unit-disk graphs, which can be used to represent the connectivity graph of a wide range of wireless networks. We also characterize the performance of "natural" greedy scheme under the same class of graphs. Although it has been known that the greedy scheme yields a constant factor approximation to MWKVMP, we are more interested in specific performance bounds of the scheme for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq71_HTML.gif . We note that both PTAS and the greedy algorithm can be used to construct scheduling policies that achieve a constant fraction of the capacity region under http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq72_HTML.gif -hop interference models, but they can be implemented in a limited class of wireless networks (e.g., wireless mesh networks) due to high complexity and requirement for centralized control.

          We complement the results by showing that the maximal scheduling policy that can be implemented in a distributed manner with a low complexity achieves a guaranteed fraction of the capacity region. These results are encouraging as they indicate that one can develop distributed algorithms to achieve near optimal throughput in case of a wide range of wireless networks. Finally, we observe that the topological constraints of the underlying graphs play a critical role to guarantee the throughput performance, and that the maximal scheduling policy can achieve an arbitrarily small fraction of the capacity region in general network graphs.

          3. Hardness and Approximability Results

          We now formulate the decision problems KVMP and WKVMP corresponding to MKVMP and MWKVMP, respectively, and prove that they are NP-Complete. We also show that MKVMP and MWKVMP cannot be approximable within http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq73_HTML.gif in polynomial time while we can approximate them within http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq74_HTML.gif . We begin with the following definitions.

          Definition 1.

          For a given graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq75_HTML.gif and number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq76_HTML.gif , KVMP is a decision process that determines whether http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq77_HTML.gif has a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq78_HTML.gif -valid matching of size http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq79_HTML.gif .

          Definition 2.

          For a given graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq80_HTML.gif , number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq81_HTML.gif , and weight http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq82_HTML.gif , WKVMP is a decision process that determines whether http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq83_HTML.gif has a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq84_HTML.gif -valid matching of size http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq85_HTML.gif and total weight http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq86_HTML.gif .

          The following theorem shows that WKVMP http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq87_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq88_HTML.gif , which also implies that KVMP http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq89_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq90_HTML.gif .

          Theorem 3.

          WKVMP http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq91_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq92_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq93_HTML.gif .

          Proof.

          Given a certificate in the form of a list of edges, it can easily be verified in polynomial time whether that list corresponds to a set of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq94_HTML.gif edges that are at a distance of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq95_HTML.gif or more from each other and have a total weight of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq96_HTML.gif or not. Thus, whether the set of edges constitute a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq97_HTML.gif -valid matching of size http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq98_HTML.gif with a total weight of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq99_HTML.gif can be verified in polynomial time. Hence, WKVMP http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq100_HTML.gif NP.

          We next show that KVMP is NP-Hard, which implies that the decision problem WKVMP is NP-Hard as well.

          Theorem 4.

          KVMP is NP-Hard for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq101_HTML.gif .

          Proof.

          The proof uses a novel technique reducing 3-CNF-SAT problem to KVMP [23]. Since their result is stronger, MKVMP, and hence KWMVMP, are Nonapproximable for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq102_HTML.gif .

          We now analyze the approximability of MKVMP for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq103_HTML.gif . We have the following result.

          Theorem 5.

          Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq104_HTML.gif be a constant such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq105_HTML.gif . Then, MKVMP (and hence, MWKVMP) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq106_HTML.gif is not approximable within http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq107_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq108_HTML.gif , unless http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq109_HTML.gif . Further, it is not approximable within http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq110_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq111_HTML.gif , unless http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq112_HTML.gif is equivalent to Zero-error Probabilistic Polynomial time http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq113_HTML.gif problems [24].

          Before we prove Theorem 5, we introduce some terminology. Consider a graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq114_HTML.gif . A subset of vertices is termed "independent" provided that no two vertices in the set have an edge between them. The classical Maximum Independent Set Problem (MISP) is to find an independent subset of vertices of maximum possible cardinality. Note that we can easily convert MKVMP (4) to MISP by mapping an edge to a vertex and connecting two vertices when corresponding two edges are within http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq115_HTML.gif -hop distance. Hastad [25] has shown that MISP is not approximable within http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq116_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq117_HTML.gif unless NP http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq118_HTML.gif P, and it is not approximable within http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq119_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq120_HTML.gif unless NP http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq121_HTML.gif ZPP. We are now ready to prove Theorem 5.

          Proof.

          We show that given a graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq122_HTML.gif , we can construct a graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq123_HTML.gif in polynomial time such that a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq124_HTML.gif -valid matching of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq125_HTML.gif has cardinality no smaller than that of the maximum independent set of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq126_HTML.gif . Then we show that both http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq127_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq128_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq129_HTML.gif , which is equal to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq130_HTML.gif . Finally, we will show that given a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq131_HTML.gif -valid matching in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq132_HTML.gif , one can obtain an independent set of vertices in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq133_HTML.gif with the same cardinality in polynomial time.

          Suppose that MKVMP admits a polynomial time http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq134_HTML.gif -approximation scheme (PTAS). Given a graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq135_HTML.gif , one can construct the corresponding graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq136_HTML.gif in polynomial time, and use the PTAS for MKVMP to obtain a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq137_HTML.gif -valid matching of size at least http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq138_HTML.gif times the cardinality of any maximum independent set of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq139_HTML.gif . Then we can map it back to an independent set of vertices in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq140_HTML.gif with the same cardinality, in polynomial time. This would then result in a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq141_HTML.gif -approximation scheme for MISP of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq142_HTML.gif , which, in view of the results in [25], would imply Theorem 5.

          We next discuss how to construct the graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq143_HTML.gif from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq144_HTML.gif in polynomial time. We first consider even http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq145_HTML.gif .

          1. (1)

            For each vertex http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq146_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq147_HTML.gif , we place a pair of vertices http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq148_HTML.gif , and connect them with an edge.

             
          2. (2)

            For each edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq149_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq150_HTML.gif , we connect the vertices http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq151_HTML.gif through a sequence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq152_HTML.gif edges and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq153_HTML.gif vertices. Let the vertices be denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq154_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq155_HTML.gif being the vertex adjacent to vertex http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq156_HTML.gif .

             
          We denote the resultant graph as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq157_HTML.gif . Figure 2 illustrates an example of a graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq158_HTML.gif along with the constructed graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq159_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq160_HTML.gif . It is straightforward to see that the graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq161_HTML.gif can be constructed in polynomial (in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq162_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq163_HTML.gif ) time. Also, we have
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ5_HTML.gif
          (5)

          Now, suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq164_HTML.gif constitutes an independent set of vertices in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq165_HTML.gif . It is then clear that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq166_HTML.gif constitutes a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq167_HTML.gif -valid matching in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq168_HTML.gif . To see this, observe that since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq169_HTML.gif constitutes an independent set of vertices in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq170_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq171_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq172_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq173_HTML.gif . Hence, by the construction of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq174_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq175_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq176_HTML.gif . Then it follows that the graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq177_HTML.gif has a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq178_HTML.gif -valid matching of cardinality not smaller than the cardinality of the maximum independent set of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq179_HTML.gif .

          It remains to show that given a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq180_HTML.gif -valid matching http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq181_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq182_HTML.gif , one can, in polynomial time, obtain an independent set of vertices in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq183_HTML.gif with the same cardinality. To this end, we propose a systematic construction method in Algorithm 1.

          It is easy to see that the running time of Algorithm 1 is bounded above by a polynomial in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq184_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq185_HTML.gif . We check that the resulting set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq186_HTML.gif from Algorithm 1 is indeed an independent set in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq187_HTML.gif . It suffices to show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq188_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq189_HTML.gif . Suppose that there exist two vertices http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq190_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq191_HTML.gif . It then follows that there must exist edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq192_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq193_HTML.gif , which contradicts our assumption that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq194_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq195_HTML.gif -valid matching.

          Next, we discuss how to construct the graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq196_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq197_HTML.gif and odd. We make a minor change in the construction of the graph. In the first step, instead of placing a pair of vertices for each vertex http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq198_HTML.gif , we now place a triplet of vertices http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq199_HTML.gif (see Figure 3), and connect the pairs of vertices http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq200_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq201_HTML.gif with an edge. In the second step, for each edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq202_HTML.gif , we connect the vertices http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq203_HTML.gif through a sequence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq204_HTML.gif edges and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq205_HTML.gif vertices. We denote the resulting graph as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq206_HTML.gif . We now have
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ6_HTML.gif
          (6)

          Similarly, we can check that the graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq207_HTML.gif has a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq208_HTML.gif -valid matching of cardinality no smaller than the cardinality of the maximum independent set of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq209_HTML.gif . Suppose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq210_HTML.gif constitutes an independent set of vertices in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq211_HTML.gif . We have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq212_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq213_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq214_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq215_HTML.gif . Then by the construction of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq216_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq217_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq218_HTML.gif , and the result follows.

          We show that given a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq219_HTML.gif -valid matching in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq220_HTML.gif , we can obtain an independent set of vertices in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq221_HTML.gif with the same cardinality in polynomial time. The construction algorithm is the same as Algorithm 1, except for the following three lines:

          1. (i)

            Line 4: if   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq222_HTML.gif is of the form http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq223_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq224_HTML.gif   then

             
          2. (ii)

            Line 8: else if   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq225_HTML.gif is of the form http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq226_HTML.gif   then

             
          3. (iii)

            Line 11: if   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq227_HTML.gif   then

             

          We check that the resulting set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq228_HTML.gif is an independent set in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq229_HTML.gif as follows. Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq230_HTML.gif is not an independent set. Then there exist two vertices http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq231_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq232_HTML.gif . Then by the construction of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq233_HTML.gif , there must exist two edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq234_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq235_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq236_HTML.gif , which contradict our assumption that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq237_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq238_HTML.gif -valid matching. The running time of the algorithm is also bounded above by a polynomial in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq239_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq240_HTML.gif .

          For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq241_HTML.gif , we can construct the graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq242_HTML.gif as in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq243_HTML.gif case, and prove the corresponding results accordingly. We omit the details.

          Algorithm 1: Constructing an independent set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq244_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq245_HTML.gif from a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq246_HTML.gif -valid matching http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq247_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq248_HTML.gif , when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq249_HTML.gif is even ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq250_HTML.gif ).

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq251_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq252_HTML.gif

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq253_HTML.gif while   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq254_HTML.gif   do

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq255_HTML.gif   Pick an edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq256_HTML.gif

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq257_HTML.gif   if   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq258_HTML.gif is of the form http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq259_HTML.gif   then

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq260_HTML.gif     http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq261_HTML.gif

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq262_HTML.gif else if   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq263_HTML.gif is of the form http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq264_HTML.gif then

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq265_HTML.gif     http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq266_HTML.gif

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq267_HTML.gif   else if   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq268_HTML.gif is of the form http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq269_HTML.gif   then

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq270_HTML.gif     http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq271_HTML.gif

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq272_HTML.gif    else if   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq273_HTML.gif is of the form http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq274_HTML.gif   then

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq275_HTML.gif     if   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq276_HTML.gif   then

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq277_HTML.gif      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq278_HTML.gif

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq279_HTML.gif     else

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq280_HTML.gif      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq281_HTML.gif

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq282_HTML.gif     end if

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq283_HTML.gif   end if

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq284_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq285_HTML.gif

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq286_HTML.gif end while

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Fig2_HTML.jpg
          Figure 2

          A graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq287_HTML.gif along with the graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq288_HTML.gif constructed as specified in the proof of Theorem 5 for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq289_HTML.gif .

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Fig3_HTML.jpg
          Figure 3

          A graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq290_HTML.gif along with the graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq291_HTML.gif constructed as specified in the proof of Theorem 5 for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq292_HTML.gif .

          From http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq293_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq294_HTML.gif in the above proof, the following result follows from Theorem 5.

          Corollary 6.

          MKVMP (and hence, MWKVMP) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq295_HTML.gif is not approximable within http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq296_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq297_HTML.gif , unless http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq298_HTML.gif . Further, it is not approximable within http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq299_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq300_HTML.gif , unless http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq301_HTML.gif .

          Corollary 6 gives a lower bound on the approximation ratio of any polynomial time approximation algorithm for MWKVMP or MKVMP. The next result we have is opposite in flavor: it shows that there exists a polynomial time algorithm for MWKVMP whose approximation ratio is no worse than http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq302_HTML.gif .

          Theorem 7.

          MWKVMP can be approximated within a factor of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq303_HTML.gif .

          The following Corollary is an immediate consequence of Theorem 7.

          Corollary 8.

          MKVMP can be approximated within a factor of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq304_HTML.gif .

          We define the Vertex Weighted Maximum Independent Set Problem (VWMISP), which is the following variation of the Maximum Independent Set Problem (MISP). Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq305_HTML.gif denote the weight of vertex http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq306_HTML.gif . VWMISP is to find an independent set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq307_HTML.gif of vertices that maximizes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq308_HTML.gif . It is shown in [26] that VWMISP is approximable within http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq309_HTML.gif . We now proceed to the proof of Theorem 7.

          Proof.

          Given a network graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq310_HTML.gif , we construct a graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq311_HTML.gif from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq312_HTML.gif in polynomial time, and approximately solve VWMISP in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq313_HTML.gif using the results of [26]. We can then obtain the corresponding http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq314_HTML.gif -valid matching in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq315_HTML.gif from the independent set in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq316_HTML.gif .

          We first construct http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq317_HTML.gif from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq318_HTML.gif as follows. For each edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq319_HTML.gif , we generate a vertex http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq320_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq321_HTML.gif with weight http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq322_HTML.gif . If two edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq323_HTML.gif satisfy http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq324_HTML.gif , we connect the corresponding vertices http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq325_HTML.gif with an edge. The resulting graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq326_HTML.gif is often called the conflict graph of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq327_HTML.gif . Clearly, we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq328_HTML.gif , and we can construct the conflict graph in polynomial time. From the construction, it is clear that for a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq329_HTML.gif -valid matching in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq330_HTML.gif , there exists an independent set of vertices in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq331_HTML.gif with the same weighted sum, and vice versa.

          Now, using the results of [26], we can approximate VWMISP in polynomial time and obtain an independent set in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq332_HTML.gif with weight at least http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq333_HTML.gif times the weight of an optimal independent set. From the independent set in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq334_HTML.gif , we can reconstruct a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq335_HTML.gif -valid matching in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq336_HTML.gif with the same weight due to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq337_HTML.gif , in polynomial time.

          4. MWKVMP for Geometric Unit-Disk Graphs

          In this section, we focus on the MWKVMP problem for an important class of network graphs. In particular, we are interested in geometric unit-disk graphs, under which the connectivity and the interference constraints are determined by the location of vertices. Specifically, the vertices are placed on a plane, two vertices are connected if and only if their distance is no greater than http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq338_HTML.gif , and also interfere with each other if and only if their distance is no greater than http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq339_HTML.gif . Geometric graphs have been used extensively in the literature to model the connectivity of wireless networks [27, 28]. In this section, we show that MWKVMP can be approximated within a constant factor in case of unit-disk graphs. We also note that the results can also be extended to some other geometric graphs including the quasi-unit-disk graphs [29].

          We start with redefining http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq340_HTML.gif -valid matching in geometric graphs. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq341_HTML.gif denote the Euclidean distance between two nodes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq342_HTML.gif . We define the distance between edges and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq343_HTML.gif -valid matching accordingly as (2), for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq344_HTML.gif , we let
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ7_HTML.gif
          (7)

          A set of edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq345_HTML.gif is said to be a " http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq346_HTML.gif -valid matching" if for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq347_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq348_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq349_HTML.gif . We also denote the set of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq350_HTML.gif -valid matchings of the graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq351_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq352_HTML.gif .

          4.1. PTAS for MWKVMP

          Several NP-complete problems are known to admit PTAS when restricted to planar or geometric graphs. In [30], PTASs are developed for various NP-complete problems restricted to planar graphs. NC-approximation schemes for various NP-Hard and PSPACE-Hard problems restricted to geometric graphs are developed in [31]. Following the approach in [31], we now show that MWKVMP and, therefore, MKVMP admits a constant factor PTAS when restricted to geometric graphs. We present the polynomial time approximation algorithm for the completeness.

          Consider a geometric graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq353_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq354_HTML.gif , specified using the coordinates of its vertices in the plane. We now present an algorithm that yields a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq355_HTML.gif -valid matching with weight at least http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq356_HTML.gif times the weight of an optimal http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq357_HTML.gif -valid matching in polynomial time, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq358_HTML.gif is a constant, and can be chosen to be arbitrarily small.

          The basic technique is the following. Given any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq359_HTML.gif , we calculate the smallest possible http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq360_HTML.gif that satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq361_HTML.gif . We divide the plane into grids of width http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq362_HTML.gif and height http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq363_HTML.gif , and denote each grid by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq364_HTML.gif as shown in Figure 4. Each grid is left (top) closed and right (bottom) open. For each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq365_HTML.gif , we partition the set of edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq366_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq367_HTML.gif disjoint sets http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq368_HTML.gif by removing edges whose two end-vertices are within http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq369_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq370_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq371_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq372_HTML.gif be the smallest subset of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq373_HTML.gif such that all edges in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq374_HTML.gif are of the form http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq375_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq376_HTML.gif . Also, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq377_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq378_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq379_HTML.gif . For each subgraph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq380_HTML.gif , we find a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq381_HTML.gif -valid matching of size at least http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq382_HTML.gif times the size of the optimal http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq383_HTML.gif -valid matching in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq384_HTML.gif . Observe that since each subgraph has been separated by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq385_HTML.gif , the union of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq386_HTML.gif -valid matchings for subgraphs http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq387_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq388_HTML.gif -valid matching for the graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq389_HTML.gif . Using arguments similar to [31, 32], we then show that each iteration returns a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq390_HTML.gif -valid matching with weight at least http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq391_HTML.gif times the weight of an optimal http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq392_HTML.gif -valid matching in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq393_HTML.gif . Our algorithm is is described in detail in Table http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq394_HTML.gif , and achieves http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq395_HTML.gif of the optimal performance. For the detailed analysis, we refer to our technical report [22].
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Fig4_HTML.jpg
          Figure 4

          Graph partition at iteration http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq396_HTML.gif in Algorithm 2.

          Algorithm 2 has complexity of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq397_HTML.gif (see [22]). Hence, even for a small http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq398_HTML.gif , the complexity is likely to be a high-order polynomial of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq399_HTML.gif and becomes a major obstacle to its implementation in practice. In the next subsection, we show that a natural greedy algorithm with a lower complexity can approximate MWKVMP within a constant factor under geometric unit-disk graphs.

          Algorithm 2: A (1+ http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq400_HTML.gif )-approximation scheme for MWKVMP http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq401_HTML.gif

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq402_HTML.gif   Find the smallest http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq403_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq404_HTML.gif .

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq405_HTML.gif   Divide the plane into grids of width http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq406_HTML.gif and height http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq407_HTML.gif . Each grid is denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq408_HTML.gif .

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq409_HTML.gif for   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq410_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq411_HTML.gif   do

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq412_HTML.gif   Partition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq413_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq414_HTML.gif disjoint sets http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq415_HTML.gif by removing edges whose two end-vertices are within http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq416_HTML.gif such

               that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq417_HTML.gif .

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq418_HTML.gif  Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq419_HTML.gif denote the subgraph induced by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq420_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq421_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq422_HTML.gif .

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq423_HTML.gif for   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq424_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq425_HTML.gif   do

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq426_HTML.gif   for   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq427_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq428_HTML.gif   do

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq429_HTML.gif    Partition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq430_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq431_HTML.gif disjoint sets http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq432_HTML.gif by removing edges whose two end-vertices are within

                http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq433_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq434_HTML.gif .

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq435_HTML.gif    Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq436_HTML.gif denote the subgraph induced by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq437_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq438_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq439_HTML.gif .

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq440_HTML.gif      for   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq441_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq442_HTML.gif   do

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq443_HTML.gif     Obtain an optimal http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq444_HTML.gif -valid matching http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq445_HTML.gif .

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq446_HTML.gif      end for

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq447_HTML.gif      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq448_HTML.gif

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq449_HTML.gif   end for

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq450_HTML.gif     http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq451_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq452_HTML.gif

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq453_HTML.gif end for

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq454_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq455_HTML.gif

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq456_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq457_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq458_HTML.gif

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq459_HTML.gif end for

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq460_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq461_HTML.gif

          4.2. Greedy Approach for MWKVMP

          We study the performance of the greedy scheduling scheme illustrated in Algorithm 3. Note that the algorithm is greedy in the sense that it schedules links in decreasing order of the weight. Some other works uses the term "greedy'' for a simpler scheme that schedules a set of links that no other links can be added to without violating the interference constraints. In this paper, we denote such a scheme by "maximal scheduling'', and differentiate from our greedy algorithm. It is well known that this greedy approach yields a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq462_HTML.gif -approximation algorithm for MWMP in general network graphs under the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq463_HTML.gif -hop interference model [33], and a constant approximation algorithm in planar graphs under the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq464_HTML.gif -hop interference model [34]. However, the performance can be much worse for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq465_HTML.gif . In this section, we characterize the performance of the greedy approach in geometric unit-disk graphs by providing a lower bound for "all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq466_HTML.gif ." We begin with some definitions.

          Algorithm 3: Greedy weighted http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq467_HTML.gif -valid matching algorithm http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq468_HTML.gif

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq469_HTML.gif   Arrange edges of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq470_HTML.gif in descending order of weight as

             http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq471_HTML.gif

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq472_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq473_HTML.gif

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq474_HTML.gif for   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq475_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq476_HTML.gif   do

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq477_HTML.gif   if   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq478_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq479_HTML.gif -valid matching  then

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq480_HTML.gif     http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq481_HTML.gif

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq482_HTML.gif   end if

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq483_HTML.gif end for

          Definition 9.

          The http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq484_HTML.gif -hop interference set of an edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq485_HTML.gif , denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq486_HTML.gif , is the set of edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq487_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq488_HTML.gif .

          Definition 10.

          The http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq489_HTML.gif -hop interference degree of an edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq490_HTML.gif , denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq491_HTML.gif , is defined as
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ8_HTML.gif
          (8)

          Definition 11.

          The http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq492_HTML.gif -hop interference degree of the graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq493_HTML.gif , denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq494_HTML.gif , is defined as
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ9_HTML.gif
          (9)

          The following is the main result of this subsection.

          Theorem 12.

          The weight of the matching returned by Algorithm 3 is always within a factor http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq495_HTML.gif of the weight of an optimal matching. Further, there exists a graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq496_HTML.gif for which the above ratio is tight.

          Proof.

          Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq497_HTML.gif be the edge added to the matching during the first step by the greedy algorithm. Then, we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq498_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq499_HTML.gif . Now, the optimal matching can contain at most http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq500_HTML.gif edges belonging to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq501_HTML.gif , each with a weight no larger than http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq502_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq503_HTML.gif be the edge added to the matching during the second step by the greedy algorithm. Then, we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq504_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq505_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq506_HTML.gif denotes the set consisting of elements of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq507_HTML.gif that are not in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq508_HTML.gif . Moreover, the optimal matching can contain at most http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq509_HTML.gif edges belonging to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq510_HTML.gif , each with a weight no larger than http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq511_HTML.gif .

          For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq512_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq513_HTML.gif . Arguing as above, it can be shown that during the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq514_HTML.gif th step, the greedy algorithm adds an edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq515_HTML.gif to the matching that satisfies
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ10_HTML.gif
          (10)
          and the optimal matching contains no more than http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq516_HTML.gif edges belonging to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq517_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq518_HTML.gif denote the last edge added to the matching by the greedy algorithm, and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq519_HTML.gif denote the optimal matching. From the above discussion, it is clear that for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq520_HTML.gif , we have
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ11_HTML.gif
          (11)
          Note that by convention http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq521_HTML.gif . Summing over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq522_HTML.gif , we obtain that
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ12_HTML.gif
          (12)

          proving the first part of Theorem 12.

          To prove the second part, we consider a network graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq523_HTML.gif as shown in Figure 5. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq524_HTML.gif denote the link at the center. In this example, we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq525_HTML.gif . One possible matching obtained using the greedy algorithm is shown in Figure 5(a). Note that the weight of this matching is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq526_HTML.gif . However, the weight of an optimal matching is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq527_HTML.gif as shown in Figure 5(b). Thus, the greedy algorithm may return a matching whose weight is off by a factor of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq528_HTML.gif in comparison to the weight of an optimal matching.
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Fig5_HTML.jpg
          Figure 5

          Comparison between a matching returned by the greedy algorithm and an optimal matching. All links in the graph have the same weight, and links included in each matching are marked in red. The greedy algorithm may schedule link http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq529_HTML.gif at the center of the graph while it is possible to schedule http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq530_HTML.gif links at the same time.A matching returned by the greedy algorithmAn optimal matching

          Clearly, Figure 5 shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq531_HTML.gif can be of the order of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq532_HTML.gif in general graphs, and correspondingly, the performance of Algorithm 3 can be arbitrarily small when compared with the optimal performance. However, if the network graphs are governed by some geometric constraints, we can show that Algorithm 3 approximates the optimal scheduler by a constant.

          Theorem 13.

          The weight of the matching returned by Algorithm 3 is within a factor of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq533_HTML.gif of the weight of an optimal matching in case of geometric unit-disk graphs.

          Proof.

          From Theorem 12, it suffices to show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq534_HTML.gif for any geometric unit-disk graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq535_HTML.gif . To this end, we show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq536_HTML.gif for all edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq537_HTML.gif .

          At a time slot, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq538_HTML.gif denote a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq539_HTML.gif -valid matching chosen by Algorithm 3. We consider the set of edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq540_HTML.gif for an edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq541_HTML.gif . For each edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq542_HTML.gif , we draw a disk http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq543_HTML.gif of radius http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq544_HTML.gif centered at the mid-point of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq545_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq546_HTML.gif denote two edges in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq547_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq548_HTML.gif . If there are no such pair of edges, then we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq549_HTML.gif . Otherwise, it is clear from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq550_HTML.gif , two disks http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq551_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq552_HTML.gif do not intersect with each other as shown in Figure 6.

          Now we consider a large disk http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq553_HTML.gif of radius http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq554_HTML.gif centered at the mid-point of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq555_HTML.gif . Since we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq556_HTML.gif for all edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq557_HTML.gif , all disks http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq558_HTML.gif should be contained in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq559_HTML.gif . However, since no two disks intersect, the number of disks http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq560_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq561_HTML.gif is bounded by
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ13_HTML.gif
          (13)
          for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq562_HTML.gif . Hence, we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq563_HTML.gif for all edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq564_HTML.gif , which implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq565_HTML.gif .
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Fig6_HTML.jpg
          Figure 6

          For an edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq566_HTML.gif , we draw a large disk http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq567_HTML.gif of radius http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq568_HTML.gif centered at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq569_HTML.gif . Then for each edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq570_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq571_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq572_HTML.gif -valid matching, we can draw a small disjoint disk http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq573_HTML.gif of radius http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq574_HTML.gif . By counting the number of small disks within http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq575_HTML.gif , we can estimate a bound on the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq576_HTML.gif -hop interference degree http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq577_HTML.gif .

          Note that Algorithm 3 has complexity of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq578_HTML.gif and can be implemented in a distributed manner [35].

          Remark 14.

          The above results imply that PTAS of Algorithm 2 and the greedy algorithm of Algorithm 3 achieves a guaranteed fraction of weights. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq579_HTML.gif denote a class of scheduling policies such that at each time slot, the weight of chosen schedule is no less than http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq580_HTML.gif . Then Algorithms 2 and 3 belong to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq581_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq582_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq583_HTML.gif , respectively. This property needs to be highlighted since distributed rate control algorithms that can deliver the performance of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq584_HTML.gif scheduling policy to end-users have been recently developed [8].

          5. Throughput Guarantees of Scheduling Policies

          Polynomial time algorithms developed in the earlier section can be used to construct scheduling policies that achieve a constant fraction of the capacity region. For example, it can be easily shown that a scheduling policy that belongs to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq585_HTML.gif achieves at least http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq586_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq587_HTML.gif denotes the capacity region of the underlying network graph.

          Although PTAS and the greedy algorithm achieve a guaranteed fraction of the capacity region, they require centralized control and/or a high complexity, which restrict their practical implementation within a limited class of wireless networks. In this section, we focus on throughput performance of scheduling policies. We show that even simpler scheduling policies that can be easily implemented in a distributed fashion have a provable throughput guarantee. Specifically, we show that the maximal scheduling policy of [8, 15] which is an http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq588_HTML.gif scheduling policy can also achieve a guaranteed fraction of the capacity region in geometric unit-disk graphs, when all transmissions are carried out at certain fixed rates (i.e., rate control is not exercised).

          5.1. Distributed Implementation for Geometric Unit-Disk Graphs

          We start with the following definition of the maximal scheduling policy.

          Definition 15.

          A subset http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq589_HTML.gif of edges is a maximal schedule if each edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq590_HTML.gif either has an empty queue, or satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq591_HTML.gif . A scheduling policy is said to be a maximal scheduling policy if it chooses one of the maximal schedules for transmission at each time slot.

          In words, the maximal scheduling policy ensures that if there are any packets waiting to be transmitted over an edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq592_HTML.gif , then either http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq593_HTML.gif or one of edges that interfere with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq594_HTML.gif is scheduled for transmission. Note that an optimal solution to MWKVMP and the greedy algorithm are one of maximal scheduling policies while PTAS of Algorithm 2 is not a maximal scheduling policy.

          Now we consider a network with single-hop fixed-rate sessions. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq595_HTML.gif denote the capacity region of the network, that is, the set of session arrival rates for which the network can be stabilized under some scheduling policy. We have the following theorem.

          Theorem 16.

          In geometric unit-disk graphs under the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq596_HTML.gif -hop interference model, any maximal scheduling policy can stabilize the network system for any set of session arrival rates within http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq597_HTML.gif .

          Proof.

          It has been shown in [15] that any maximal scheduling policy achieves at least http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq598_HTML.gif fraction of the capacity region. In other words, it stabilizes the network system for any set of arrival rates within http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq599_HTML.gif . From Theorem 13, we have that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq600_HTML.gif in geometric unit-disk graphs, and hence, the result follows.

          Note that a simple distributed maximal scheduling policy can be developed by extending the randomized maximal scheduling of [8] to the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq601_HTML.gif -hop interference model. In this case, the complexity of the policy will be http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq602_HTML.gif .

          Remark 17.

          Theorem 16 implies that the maximal scheduling policy can achieve http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq603_HTML.gif in the sense of time average. It can be contrasted with the results of PTAS and the greedy algorithm provided in Section 4, where they guarantee a constant fraction of weights at each time slot. Their average performance can be higher than the guaranteed fraction of weights. For example, it has been recently shown that the greedy algorithm achieves http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq604_HTML.gif [36].

          5.2. Throughput Guarantees in Nongeometric Network Graphs

          The results provided in the previous section are encouraging as they indicate that one can develop simple distributed algorithms whose worst-case throughput is a nonvanishing fraction of the capacity region. Note that the results are obtained by admitting an arbitrarily small fraction of weights at a time slot, on the basis of geometric properties of unit-disk graphs. Although we have shown in Corollary 6 that MWKVMP cannot be approximated within a constant factor in general network graphs, it still remains unclear whether a simple distributed algorithm like the maximal scheduling policy can achieve a constant fraction of the capacity region in general network graphs. In the following, we show that the geometric constraints are indeed crucial to achieve the constant fraction of capacity region. To elaborate, we show that the greedy algorithm (and thus, the maximal scheduling policy as well) can achieve an arbitrarily small fraction of the capacity region in general network graphs.

          We begin with some definitions. For a graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq605_HTML.gif , we consider a subset of edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq606_HTML.gif , and denote the set of all possible matching matchings on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq607_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq608_HTML.gif . Also let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq609_HTML.gif denote the convex hull of set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq610_HTML.gif , that is,
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ14_HTML.gif
          (14)

          Recently, it has been shown in [36, 37] that for a vector http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq611_HTML.gif and all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq612_HTML.gif , we can construct an arrival rate http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq613_HTML.gif such that the queues of all edges in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq614_HTML.gif keep increasing under the greedy scheduling algorithm of Algorithm 3. Note that the optimal scheduler can serve the arrival rate http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq615_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq616_HTML.gif . Therefore, in order to show that the greedy algorithm achieves no greater than http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq617_HTML.gif , it suffices to find a subset http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq618_HTML.gif and two vectors http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq619_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq620_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq621_HTML.gif implies a component-wise inequality, that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq622_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq623_HTML.gif .

          Now we provide a systematic construction of network graphs such that we can find a subset of edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq624_HTML.gif and two vectors http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq625_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq626_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq627_HTML.gif . Once we find those two vectors, we have the upper bound http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq628_HTML.gif of the performance of the greedy algorithm.

          Lemma 18.

          There is a network graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq629_HTML.gif under the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq630_HTML.gif -hop interference model with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq631_HTML.gif such that two vectors http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq632_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq633_HTML.gif satisfy http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq634_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq635_HTML.gif .

          Proof.

          We first describe our systematic construction of a graph, and then find two vectors in a subset of edges of the constructed network graph. Note that these two vectors should be a combination of maximal matchings in the subset of edges and one must be component-wise greater than the other by a factor of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq636_HTML.gif .

          We construct the network graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq637_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq638_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq639_HTML.gif as follows.

          Phase 1 (horizontal edges; see Figure 7(a) for an example of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq640_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq641_HTML.gif ).

          ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq642_HTML.gif ) Start with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq643_HTML.gif (or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq644_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq645_HTML.gif is odd) vertices. Place vertices on a cycle and name them in counter-clockwise order as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq646_HTML.gif . Connect each vertex http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq647_HTML.gif to its immediate neighbor http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq648_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq649_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq650_HTML.gif represents a modular addition by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq651_HTML.gif .

          ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq652_HTML.gif ) Make the circle a centerless wheel by connecting each vertex http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq653_HTML.gif to the opposite vertex http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq654_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq655_HTML.gif . All vertices can be connected because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq656_HTML.gif is an even number. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq657_HTML.gif denote the constructed wheel graph.

          ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq658_HTML.gif ) Connect http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq659_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq660_HTML.gif using http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq661_HTML.gif -hop edges for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq662_HTML.gif . That is, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq663_HTML.gif , add http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq664_HTML.gif vertices between http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq665_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq666_HTML.gif , say http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq667_HTML.gif , and connect them in sequence with edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq668_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq669_HTML.gif . Also, add edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq670_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq671_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq672_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq673_HTML.gif , connect http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq674_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq675_HTML.gif directly.

          ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq676_HTML.gif ) Repeat ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq677_HTML.gif ) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq678_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq679_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq680_HTML.gif .

          ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq681_HTML.gif ) Construct another wheel graph http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq682_HTML.gif by duplicating http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq683_HTML.gif , and name vertices on the wheel of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq684_HTML.gif accordingly with superscript http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq685_HTML.gif .

          Phase 2 (vertical edges; see Figure 7(b) for an example of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq686_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq687_HTML.gif ).

          ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq688_HTML.gif ) Connect http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq689_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq690_HTML.gif using http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq691_HTML.gif -hop edges for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq692_HTML.gif . That is, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq693_HTML.gif , add vertices http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq694_HTML.gif between http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq695_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq696_HTML.gif , and connect them with edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq697_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq698_HTML.gif .

          (2) Repeat ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq699_HTML.gif ) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq700_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq701_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq702_HTML.gif .

          As an example, all horizontal edges and a part of vertical edges of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq703_HTML.gif are shown in Figures 7(a) and 7(b).

          Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq704_HTML.gif be the set of edges inside two wheels, that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq705_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq706_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq707_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq708_HTML.gif . Links in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq709_HTML.gif are presented as solid black lines in Figure 7(a). Note that edges constructed in ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq710_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq711_HTML.gif ) of Phase 1 and in ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq712_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq713_HTML.gif ) of Phase 2 are designed to control interference among edges within and between wheels. If an edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq714_HTML.gif is active in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq715_HTML.gif (or in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq716_HTML.gif ), then edges constructed by ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq717_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq718_HTML.gif ) of Phase 1 allow at most http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq719_HTML.gif other edges to be active in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq720_HTML.gif (or in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq721_HTML.gif ). Hence, we can activate at most http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq722_HTML.gif edges in each wheel (see Figure 7(c)). However, the inter-wheel interference by vertical edges may reduce the number of active edges. In ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq723_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq724_HTML.gif ) of Phase 2, we have constructed http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq725_HTML.gif vertical edges per each vertex of each wheel. Since the vertical edges have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq726_HTML.gif -hop, an active edge in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq727_HTML.gif can interfere with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq728_HTML.gif edges in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq729_HTML.gif and vice versa. Assume that edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq730_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq731_HTML.gif are active in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq732_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq733_HTML.gif , respectively. We can have at most http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq734_HTML.gif more active edges in each wheel, that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq735_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq736_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq737_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq738_HTML.gif . However, if we choose edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq739_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq740_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq741_HTML.gif interferes with all edges of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq742_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq743_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq744_HTML.gif interferes with all edges of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq745_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq746_HTML.gif , then we have only two active edges as a maximal matching in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq747_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq748_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq749_HTML.gif (two red lines in Figure 7(d)). We design the network graph carefully such that a maximal matching can include from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq750_HTML.gif active edges to two active edges.

          Now, we find two convex combinations of maximal matchings in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq751_HTML.gif that one is component-wise greater than the other by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq752_HTML.gif . Consider two sets of maximal matchings; one with maximal matchings of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq753_HTML.gif active edges and the other with maximal matchings of 2 active edges. We first let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq754_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq755_HTML.gif and each maximal matching http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq756_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq757_HTML.gif includes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq758_HTML.gif active edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq759_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq760_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq761_HTML.gif . For the other vector, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq762_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq763_HTML.gif and each maximal matching http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq764_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq765_HTML.gif includes only two edges http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq766_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq767_HTML.gif . Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq768_HTML.gif 's and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq769_HTML.gif 's are valid maximal matchings in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq770_HTML.gif All active edges in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq771_HTML.gif are either activated or interfered, and all active edges satisfy the interference constraints. Figures 7(c) and 7(d) illustrate an instance of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq772_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq773_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq774_HTML.gif , respectively. Active edges are colored in red. To clearly show the interference in Figure 7(d), we color a vertex in black if it is interfered by the active edge in the upper wheel, and in gray if it is interfered by the active edge in the lower wheel.

          Using the scheduling of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq775_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq776_HTML.gif , each edge in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq777_HTML.gif is served exactly once during a unit time for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq778_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq779_HTML.gif or for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq780_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq781_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq782_HTML.gif (or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq783_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq784_HTML.gif is odd), we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq785_HTML.gif for all edge http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq786_HTML.gif and thus, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq787_HTML.gif .
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Fig7_HTML.jpg
          Figure 7

          Example of network graph and matchings under the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq788_HTML.gif -hop interference model, in which the greedy algorithm achieves no greater than http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq789_HTML.gif of the optimal performance ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq790_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq791_HTML.gif ). The subset http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq792_HTML.gif are the edges inside the cycles. (Solid black edges in (a).) An instance of maximal matching for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq793_HTML.gif is shown in (c). Active edges are marked in red. By circulating the active edges in (c), we can obtain similar maximal matchings. Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq794_HTML.gif consists of those maximal matching with an identical weight. Similarly we can construct http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq795_HTML.gif from maximal matchings like (d). Note that both http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq796_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq797_HTML.gif serve all edges in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq798_HTML.gif for the same amount of time, but a maximal matching of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq799_HTML.gif has http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq800_HTML.gif times more active edges than a maximal matching of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq801_HTML.gif . Hence, it can be shown that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq802_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq803_HTML.gif . To make sure that the schedule of (d) is maximal, we color vertices interfered by the active edge in the upper wheel in black, and vertices interfered by the active edge in the lower wheel in gray.Topology; horizontal edgesTopology; a part of vertical edgesA maximal matching of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq804_HTML.gif A maximal matching of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq805_HTML.gif

          Lemma 18 immediately implies the following proposition.

          Proposition 19.

          Under the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq806_HTML.gif -hop interference model, Algorithm 3 can achieve an arbitrarily small fraction of the optimal throughput.

          Proof.

          From Lemma 18 and the techniques of [37], we can find a traffic arrival with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq807_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq808_HTML.gif such that the system is unstable under the greedy scheduling algorithm. However, the optimal scheduling policy can support http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq809_HTML.gif , which follows that the achievable rate of the greedy algorithm is not greater than http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq810_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq811_HTML.gif can be arbitrarily large from our graph construction, the performance ratio can be arbitrarily small.

          Proposition 19 lets us know that it is hard, if possible, to characterize the performance limits of the greedy algorithm (and thus the maximal scheduling policy as well) in arbitrary network graphs under the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq812_HTML.gif -hop interference model.

          6. Concluding Remarks

          We consider the problem of throughput-optimal scheduling in wireless networks subject to interference constraints, which are modeled using a family of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq813_HTML.gif -hop interference models. Under the assumption that each node transmits at a fixed power level (which can be different for different nodes), the optimal scheduling problems are shown to be weighted matching problems with constraints determined by the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq814_HTML.gif -hop interference model. These weighted matching problems are termed Maximum Weighted http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq815_HTML.gif -Valid Matching Problems (MWKVMP s).

          For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq816_HTML.gif , MWKVMP corresponds to the well-studied Maximum Weighted Matching Problem (MWMP) in the literature, which can be solved in polynomial time. We show that MWKVMP is NP-Hard for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq817_HTML.gif and provided upper and lower bounds on its approximability.

          By restricting the problem to geometric unit-disk graphs, under which connectivities are determined by geometric distance between nodes, we show that MWKVMP admits a PTAS within a factor arbitrarily close to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq818_HTML.gif , and the "natural" greedy matching algorithm yields a 49-approximation to the optimal solution for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq819_HTML.gif . We emphasize that both PTAS and the greey scheduling schemes achieve a guaranteed fraction of weights at every time slot. Combining these with the results in [8], it follows that both can be used to develop a joint solution of scheduling and rate control with provable (end-to-end) performance guarantees with multihop traffics.

          However, since PTAS and the greedy algorithm have a polynomial time complexity and require centralized control, their implementations in practice are restricted within a limited class of wireless networks. We complement these results by further focusing on the throughput performance of scheduling policies. Specifically, we show that the maximal scheduling policy that is amenable to distributed implementation achieves http://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq820_HTML.gif fraction of the capacity region under a setting with single-hop traffic and fixed rate transmissions. These results are encouraging as they indicate that one can develop simple distributed algorithms whose worst-case throughput is a nonvanishing fraction of the optimal throughput in the case of a wide class of wireless networks. Finally, we highlight that the geometric constraints are crucial for the maximal scheduling policy to achieve the throughput guarantees. We show that even the greedy scheduling algorithm, in the worst case, can result in an arbitrarily small efficiency without these constraints.

          Declarations

          Acknowledgments

          This work has been supported in part by the NSF Awards CNS-0626703 and CNS-0721236, and the ARO MURI Award W911NF-08-1-0238, USA, and in part by the New Professor Research Program of KUT (2010), Korea.

          Authors’ Affiliations

          (1)
          Department of EECE, Korea University of Technology and Education
          (2)
          D. E. Shaw & Co., L.P.
          (3)
          Departments of ECE & CSE, The Ohio State University
          (4)
          Department of ECE, University of Waterloo

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          © Changhee Joo et al. 2010

          This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.