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
in polynomial time while we can approximate them within
. We begin with the following definitions.

Definition 1.

For a given graph
and number
, *KVMP* is a decision process that determines whether
has a
-valid matching of size
.

Definition 2.

For a given graph
, number
, and weight
, *WKVMP* is a decision process that determines whether
has a
-valid matching of size
and total weight
.

The following theorem shows that *WKVMP*
, which also implies that *KVMP*
.

Theorem 3.

WKVMP
for all
.

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
edges that are at a distance of
or more from each other and have a total weight of
or not. Thus, whether the set of edges constitute a
-valid matching of size
with a total weight of
can be verified in polynomial time. Hence, *WKVMP*
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
.

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
.

We now analyze the approximability of *MKVMP* for
. We have the following result.

Theorem 5.

Let
be a constant such that
. Then, MKVMP (and hence, MWKVMP) for
is not approximable within
for any
, unless
. Further, it is not approximable within
for any
, unless
is equivalent to Zero-error Probabilistic Polynomial time
problems [24].

Before we prove Theorem 5, we introduce some terminology. Consider a graph
. 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
-hop distance. Hastad [25] has shown that *MISP* is not approximable within
for any
unless NP
P, and it is not approximable within
for any
unless NP
ZPP. We are now ready to prove Theorem 5.

Proof.

We show that given a graph
, we can construct a graph
in polynomial time such that a
-valid matching of
has cardinality no smaller than that of the maximum independent set of
. Then we show that both
and
are
, which is equal to
. Finally, we will show that given a
-valid matching in
, one can obtain an independent set of vertices in
with the same cardinality in polynomial time.

Suppose that *MKVMP* admits a polynomial time
-approximation scheme (PTAS). Given a graph
, one can construct the corresponding graph
in polynomial time, and use the PTAS for *MKVMP* to obtain a
-valid matching of size at least
times the cardinality of any maximum independent set of
. Then we can map it back to an independent set of vertices in
with the same cardinality, in polynomial time. This would then result in a
-approximation scheme for *MISP* of
, which, in view of the results in [25], would imply Theorem 5.

We next discuss how to construct the graph
from
in polynomial time. We first consider even
.

We denote the resultant graph as

. Figure

2 illustrates an example of a graph

along with the constructed graph

when

. It is straightforward to see that the graph

can be constructed in polynomial (in

and

) time. Also, we have

Now, suppose that
constitutes an independent set of vertices in
. It is then clear that
constitutes a
-valid matching in
. To see this, observe that since
constitutes an independent set of vertices in
, we have
for all
with
. Hence, by the construction of
, we have
for
. Then it follows that the graph
has a
-valid matching of cardinality not smaller than the cardinality of the maximum independent set of
.

It remains to show that given a
-valid matching
in
, one can, in polynomial time, obtain an independent set of vertices in
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
and
. We check that the resulting set
from Algorithm 1 is indeed an independent set in
. It suffices to show that
for all
. Suppose that there exist two vertices
such that
. It then follows that there must exist edges
such that
, which contradicts our assumption that
is a
-valid matching.

Next, we discuss how to construct the graph

for

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

, we now place a triplet of vertices

(see Figure

3), and connect the pairs of vertices

and

with an edge. In the second step, for each edge

, we connect the vertices

through a sequence of

edges and

vertices. We denote the resulting graph as

. We now have

Similarly, we can check that the graph
has a
-valid matching of cardinality no smaller than the cardinality of the maximum independent set of
. Suppose
constitutes an independent set of vertices in
. We have
for all
with
in
. Then by the construction of
, we have
for all
, and the result follows.

We show that given a
-valid matching in
, we can obtain an independent set of vertices in
with the same cardinality in polynomial time. The construction algorithm is the same as Algorithm 1, except for the following three lines:

- (i)

- (ii)
Line 8: **else if**
is of the form
**then**

- (iii)
Line 11: **if**
**then**

We check that the resulting set
is an independent set in
as follows. Suppose that
is not an independent set. Then there exist two vertices
such that
. Then by the construction of
, there must exist two edges
such that
for
, which contradict our assumption that
is a
-valid matching. The running time of the algorithm is also bounded above by a polynomial in
and
.

For
, we can construct the graph
as in
case, and prove the corresponding results accordingly. We omit the details.

**Algorithm 1:** Constructing an independent set
in
from a
-valid matching
in
, when
is even (
).

**while**
**do**

Pick an edge

**if**
is of the form
**then**

**else if**
is of the form
**then**

**else if**
is of the form
**then**

**else if**
is of the form
**then**

**if**
**then**

**else**

**end if**

**end if**

**end while**

From
and
in the above proof, the following result follows from Theorem 5.

Corollary 6.

MKVMP (and hence, MWKVMP) for
is not approximable within
for any
, unless
. Further, it is not approximable within
for any
, unless
.

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
.

Theorem 7.

MWKVMP can be approximated within a factor of
.

The following Corollary is an immediate consequence of Theorem 7.

Corollary 8.

MKVMP can be approximated within a factor of
.

We define the Vertex Weighted Maximum Independent Set Problem (*VWMISP*), which is the following variation of the Maximum Independent Set Problem (*MISP*). Let
denote the weight of vertex
. *VWMISP* is to find an independent set
of vertices that maximizes
. It is shown in [26] that *VWMISP* is approximable within
. We now proceed to the proof of Theorem 7.

Proof.

Given a network graph
, we construct a graph
from
in polynomial time, and approximately solve *VWMISP* in
using the results of [26]. We can then obtain the corresponding
-valid matching in
from the independent set in
.

We first construct
from
as follows. For each edge
, we generate a vertex
in
with weight
. If two edges
satisfy
, we connect the corresponding vertices
with an edge. The resulting graph
is often called the *conflict graph* of
. Clearly, we have
, and we can construct the conflict graph in polynomial time. From the construction, it is clear that for a
-valid matching in
, there exists an independent set of vertices in
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
with weight at least
times the weight of an optimal independent set. From the independent set in
, we can reconstruct a
-valid matching in
with the same weight due to
, in polynomial time.