Open Access

On the Complexity of Scheduling in Wireless Networks

  • Changhee Joo1Email author,
  • Gaurav Sharma2,
  • Ness B. Shroff3 and
  • Ravi R. Mazumdar4
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 https://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 https://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 https://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 https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ1_HTML.gif
(1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq6_HTML.gif denotes the set of wireless links; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq7_HTML.gif is the vector of link rates https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq9_HTML.gif ; https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq11_HTML.gif ; https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq15_HTML.gif hops, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq16_HTML.gif is a positive integer. By varying https://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 https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq23_HTML.gif -valid matchings for an integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq24_HTML.gif .

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq26_HTML.gif . Letting https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq28_HTML.gif as follows: for two edges https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq29_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq30_HTML.gif , let
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq31_HTML.gif a " https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq32_HTML.gif -valid matching" if for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq33_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq34_HTML.gif , we have https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq37_HTML.gif denote the set of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq38_HTML.gif -Valid matchings of the graph https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq39_HTML.gif . We consider the following scheduling problems:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ3_HTML.gif
(3)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq40_HTML.gif denotes the weight of edge https://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 https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ4_HTML.gif
(4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq44_HTML.gif denotes the cardinality of the set https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq47_HTML.gif , by satisfying the https://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 https://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 https://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 " https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq53_HTML.gif -hop interference model with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq54_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq55_HTML.gif , respectively.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Fig1_HTML.jpg
Figure 1

The https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq59_HTML.gif of the capacity region, where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq61_HTML.gif of the optimal throughput, where https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq64_HTML.gif -hop or https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq66_HTML.gif -hop interference model with https://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 https://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 https://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 https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq75_HTML.gif and number https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq77_HTML.gif has a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq78_HTML.gif -valid matching of size https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq79_HTML.gif .

Definition 2.

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

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

Theorem 3.

WKVMP https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq91_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq92_HTML.gif for all https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq97_HTML.gif -valid matching of size https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq98_HTML.gif with a total weight of https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq104_HTML.gif be a constant such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq105_HTML.gif . Then, MKVMP (and hence, MWKVMP) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq106_HTML.gif is not approximable within https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq107_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq108_HTML.gif , unless https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq109_HTML.gif . Further, it is not approximable within https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq110_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq111_HTML.gif , unless https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq116_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq117_HTML.gif unless NP https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq119_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq120_HTML.gif unless NP https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq122_HTML.gif , we can construct a graph https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq123_HTML.gif in polynomial time such that a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq124_HTML.gif -valid matching of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq126_HTML.gif . Then we show that both https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq127_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq128_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq129_HTML.gif , which is equal to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq131_HTML.gif -valid matching in https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq134_HTML.gif -approximation scheme (PTAS). Given a graph https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq135_HTML.gif , one can construct the corresponding graph https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq137_HTML.gif -valid matching of size at least https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq141_HTML.gif -approximation scheme for MISP of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq143_HTML.gif from https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq145_HTML.gif .

  1. (1)

    For each vertex https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq146_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq147_HTML.gif , we place a pair of vertices https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq149_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq150_HTML.gif , we connect the vertices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq151_HTML.gif through a sequence of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq152_HTML.gif edges and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq154_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq155_HTML.gif being the vertex adjacent to vertex https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq156_HTML.gif .

     
We denote the resultant graph as https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq158_HTML.gif along with the constructed graph https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq159_HTML.gif when https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq161_HTML.gif can be constructed in polynomial (in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq162_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq163_HTML.gif ) time. Also, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ5_HTML.gif
(5)

Now, suppose that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq165_HTML.gif . It is then clear that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq166_HTML.gif constitutes a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq167_HTML.gif -valid matching in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq168_HTML.gif . To see this, observe that since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq170_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq171_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq172_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq173_HTML.gif . Hence, by the construction of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq174_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq175_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq176_HTML.gif . Then it follows that the graph https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq177_HTML.gif has a https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq180_HTML.gif -valid matching https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq181_HTML.gif in https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq184_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq185_HTML.gif . We check that the resulting set https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq187_HTML.gif . It suffices to show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq188_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq189_HTML.gif . Suppose that there exist two vertices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq190_HTML.gif such that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq192_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq193_HTML.gif , which contradicts our assumption that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq194_HTML.gif is a https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq196_HTML.gif for https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq200_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq202_HTML.gif , we connect the vertices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq203_HTML.gif through a sequence of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq204_HTML.gif edges and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq206_HTML.gif . We now have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq207_HTML.gif has a https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq209_HTML.gif . Suppose https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq211_HTML.gif . We have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq212_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq213_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq214_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq215_HTML.gif . Then by the construction of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq216_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq217_HTML.gif for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq219_HTML.gif -valid matching in https://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 https://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   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq222_HTML.gif is of the form https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq223_HTML.gif or https://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   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq225_HTML.gif is of the form https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq226_HTML.gif   then

     
  3. (iii)

    Line 11: if   https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq228_HTML.gif is an independent set in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq229_HTML.gif as follows. Suppose that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq231_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq232_HTML.gif . Then by the construction of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq233_HTML.gif , there must exist two edges https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq234_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq235_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq236_HTML.gif , which contradict our assumption that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq237_HTML.gif is a https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq239_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq240_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq241_HTML.gif , we can construct the graph https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq242_HTML.gif as in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq244_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq245_HTML.gif from a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq246_HTML.gif -valid matching https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq247_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq248_HTML.gif , when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq249_HTML.gif is even ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq250_HTML.gif ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A graph https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq287_HTML.gif along with the graph https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq289_HTML.gif .

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

A graph https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq290_HTML.gif along with the graph https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq292_HTML.gif .

From https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq293_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq295_HTML.gif is not approximable within https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq296_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq297_HTML.gif , unless https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq298_HTML.gif . Further, it is not approximable within https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq299_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq300_HTML.gif , unless https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq305_HTML.gif denote the weight of vertex https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq307_HTML.gif of vertices that maximizes https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq310_HTML.gif , we construct a graph https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq311_HTML.gif from https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq314_HTML.gif -valid matching in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq315_HTML.gif from the independent set in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq316_HTML.gif .

We first construct https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq317_HTML.gif from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq318_HTML.gif as follows. For each edge https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq319_HTML.gif , we generate a vertex https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq320_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq321_HTML.gif with weight https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq322_HTML.gif . If two edges https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq323_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq324_HTML.gif , we connect the corresponding vertices https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq325_HTML.gif with an edge. The resulting graph https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq327_HTML.gif . Clearly, we have https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq329_HTML.gif -valid matching in https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq332_HTML.gif with weight at least https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq334_HTML.gif , we can reconstruct a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq335_HTML.gif -valid matching in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq336_HTML.gif with the same weight due to https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq340_HTML.gif -valid matching in geometric graphs. Let https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq343_HTML.gif -valid matching accordingly as (2), for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq344_HTML.gif , we let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ7_HTML.gif
(7)

A set of edges https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq345_HTML.gif is said to be a " https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq346_HTML.gif -valid matching" if for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq347_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq348_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq349_HTML.gif . We also denote the set of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq350_HTML.gif -valid matchings of the graph https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq351_HTML.gif by https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq353_HTML.gif with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq355_HTML.gif -valid matching with weight at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq356_HTML.gif times the weight of an optimal https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq357_HTML.gif -valid matching in polynomial time, where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq359_HTML.gif , we calculate the smallest possible https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq360_HTML.gif that satisfies https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq362_HTML.gif and height https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq363_HTML.gif , and denote each grid by https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq365_HTML.gif , we partition the set of edges https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq366_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq367_HTML.gif disjoint sets https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq369_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq370_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq371_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq372_HTML.gif be the smallest subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq373_HTML.gif such that all edges in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq374_HTML.gif are of the form https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq375_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq376_HTML.gif . Also, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq377_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq378_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq379_HTML.gif . For each subgraph https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq380_HTML.gif , we find a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq381_HTML.gif -valid matching of size at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq382_HTML.gif times the size of the optimal https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq383_HTML.gif -valid matching in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq385_HTML.gif , the union of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq386_HTML.gif -valid matchings for subgraphs https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq387_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq388_HTML.gif -valid matching for the graph https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq390_HTML.gif -valid matching with weight at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq391_HTML.gif times the weight of an optimal https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq392_HTML.gif -valid matching in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq394_HTML.gif , and achieves https://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].
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Fig4_HTML.jpg
Figure 4

Graph partition at iteration https://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 https://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 https://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 https://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+ https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq400_HTML.gif )-approximation scheme for MWKVMP https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq401_HTML.gif

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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq406_HTML.gif and height https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq407_HTML.gif . Each grid is denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq408_HTML.gif .

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

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq412_HTML.gif   Partition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq413_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq414_HTML.gif disjoint sets https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq416_HTML.gif such

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

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

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

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

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq429_HTML.gif    Partition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq430_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq431_HTML.gif disjoint sets https://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

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

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

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

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

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

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

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

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

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

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

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

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

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq460_HTML.gif https://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 https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq467_HTML.gif -valid matching algorithm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq468_HTML.gif

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

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

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

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

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

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

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

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

Definition 9.

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

Definition 10.

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

Definition 11.

The https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq492_HTML.gif -hop interference degree of the graph https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq493_HTML.gif , denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq494_HTML.gif , is defined as
https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq498_HTML.gif for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq500_HTML.gif edges belonging to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq502_HTML.gif . Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq504_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq505_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq507_HTML.gif that are not in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq509_HTML.gif edges belonging to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq511_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq512_HTML.gif , let https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq515_HTML.gif to the matching that satisfies
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq516_HTML.gif edges belonging to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq517_HTML.gif . Let https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq520_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ11_HTML.gif
(11)
Note that by convention https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq521_HTML.gif . Summing over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq522_HTML.gif , we obtain that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq523_HTML.gif as shown in Figure 5. Let https://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 https://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 https://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 https://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 https://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.
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq531_HTML.gif can be of the order of https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq534_HTML.gif for any geometric unit-disk graph https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq535_HTML.gif . To this end, we show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq536_HTML.gif for all edges https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq537_HTML.gif .

At a time slot, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq538_HTML.gif denote a https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq540_HTML.gif for an edge https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq541_HTML.gif . For each edge https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq542_HTML.gif , we draw a disk https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq543_HTML.gif of radius https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq544_HTML.gif centered at the mid-point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq545_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq546_HTML.gif denote two edges in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq547_HTML.gif with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq549_HTML.gif . Otherwise, it is clear from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq550_HTML.gif , two disks https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq551_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq553_HTML.gif of radius https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq554_HTML.gif centered at the mid-point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq555_HTML.gif . Since we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq556_HTML.gif for all edges https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq557_HTML.gif , all disks https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq558_HTML.gif should be contained in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq560_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq561_HTML.gif is bounded by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Equ13_HTML.gif
(13)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq562_HTML.gif . Hence, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq563_HTML.gif for all edge https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq564_HTML.gif , which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq565_HTML.gif .
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_Fig6_HTML.jpg
Figure 6

For an edge https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq566_HTML.gif , we draw a large disk https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq567_HTML.gif of radius https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq568_HTML.gif centered at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq569_HTML.gif . Then for each edge https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq570_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq571_HTML.gif is a https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq573_HTML.gif of radius https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq576_HTML.gif -hop interference degree https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq581_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq582_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq585_HTML.gif achieves at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq586_HTML.gif , where https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq592_HTML.gif , then either https://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 https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq599_HTML.gif . From Theorem 13, we have that https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq605_HTML.gif , we consider a subset of edges https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq607_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq608_HTML.gif . Also let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq609_HTML.gif denote the convex hull of set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq610_HTML.gif , that is,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq611_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq612_HTML.gif , we can construct an arrival rate https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq615_HTML.gif if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq617_HTML.gif , it suffices to find a subset https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq618_HTML.gif and two vectors https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq619_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq620_HTML.gif , where https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq622_HTML.gif for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq624_HTML.gif and two vectors https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq625_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq626_HTML.gif with https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq629_HTML.gif under the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq630_HTML.gif -hop interference model with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq631_HTML.gif such that two vectors https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq632_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq633_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq634_HTML.gif for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq636_HTML.gif .

We construct the network graph https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq637_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq638_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq640_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq641_HTML.gif ).

( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq642_HTML.gif ) Start with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq643_HTML.gif (or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq644_HTML.gif if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq646_HTML.gif . Connect each vertex https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq647_HTML.gif to its immediate neighbor https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq648_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq649_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq650_HTML.gif represents a modular addition by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq651_HTML.gif .

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

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

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

( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq681_HTML.gif ) Construct another wheel graph https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq682_HTML.gif by duplicating https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq684_HTML.gif accordingly with superscript https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq686_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq687_HTML.gif ).

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

(2) Repeat ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq699_HTML.gif ) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq700_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq701_HTML.gif for https://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 https://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 https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq705_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq706_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq707_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq708_HTML.gif . Links in https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq710_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq711_HTML.gif ) of Phase 1 and in ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq712_HTML.gif ) and ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq714_HTML.gif is active in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq715_HTML.gif (or in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq716_HTML.gif ), then edges constructed by ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq717_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq718_HTML.gif ) of Phase 1 allow at most https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq719_HTML.gif other edges to be active in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq720_HTML.gif (or in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq721_HTML.gif ). Hence, we can activate at most https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq723_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq724_HTML.gif ) of Phase 2, we have constructed https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq726_HTML.gif -hop, an active edge in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq727_HTML.gif can interfere with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq728_HTML.gif edges in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq729_HTML.gif and vice versa. Assume that edges https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq730_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq731_HTML.gif are active in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq732_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq733_HTML.gif , respectively. We can have at most https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq735_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq736_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq737_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq738_HTML.gif . However, if we choose edges https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq739_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq740_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq741_HTML.gif interferes with all edges of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq742_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq743_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq744_HTML.gif interferes with all edges of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq745_HTML.gif in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq747_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq748_HTML.gif and https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq754_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq755_HTML.gif and each maximal matching https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq756_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq757_HTML.gif includes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq758_HTML.gif active edges https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq759_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq760_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq761_HTML.gif . For the other vector, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq762_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq763_HTML.gif and each maximal matching https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq764_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq765_HTML.gif includes only two edges https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq766_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq767_HTML.gif . Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq768_HTML.gif 's and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq769_HTML.gif 's are valid maximal matchings in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq770_HTML.gif All active edges in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq772_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq773_HTML.gif in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq775_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq776_HTML.gif , each edge in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq778_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq779_HTML.gif or for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq780_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq781_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq782_HTML.gif (or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq783_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq784_HTML.gif is odd), we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq785_HTML.gif for all edge https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq786_HTML.gif and thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq787_HTML.gif .
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq789_HTML.gif of the optimal performance ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq790_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq791_HTML.gif ). The subset https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq796_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq797_HTML.gif serve all edges in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq799_HTML.gif has https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq801_HTML.gif . Hence, it can be shown that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq802_HTML.gif for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq804_HTML.gif A maximal matching of https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq807_HTML.gif for all https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq810_HTML.gif . Since https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F418934/MediaObjects/13638_2010_Article_1902_IEq815_HTML.gif -Valid Matching Problems (MWKVMP s).

For https://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 https://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 https://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 https://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 https://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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