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On the Parameterized Complexity of the Acyclic Matching Problem

A matching is a set of edges in a graph with no common endpoint. A matching M is called acyclic if the induced subgraph on the endpoints of the edges in M is acyclic. Given a graph G and an integer k, Acyclic Matching Problem seeks for an acyclic matching of size k in G. The problem is known to be NP-complete. In this paper, we investigate the complexity of the problem in different aspects. First, we prove that the problem remains NP-complete for the class of planar bipartite graphs of maximum degree three and arbitrarily large girth. Also, the problem remains NP-complete for the class of planar line graphs with maximum degree four. Moreover, we study the parameterized complexity of the problem. In particular, we prove that the problem is W[1]-hard on bipartite graphs with respect to the parameter k. On the other hand, the problem is fixed parameter tractable with respect to k, for line graphs, C_4-free graphs and every proper minor-closed class of graphs (including bounded tree-width and planar graphs).

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1 Introduction

Throughout the paper, all graphs are simple, undirected and loopless. Given a graph , a subset of pairwise nonadjacent edges is called a matching of . Let denote the subgraph of induced on the all endpoints of the matching . A matching of is called an acyclic matching if is acyclic. Also, a matching of is called an induced matching if contains no edges except the edges in . It is clear that every induced matching is an acyclic matching. In the Acyclic (Induced) Matching problem, the input is a graph and a positive integer and the task is to find an acyclic (induced) matching of size in .

The Induced Matching problem was first introduced by Stockmeyer and Vazirani [30] as a variant of the maximum matching problem and proved to be NP-complete in general graphs. This problem is known to be NP-complete for planar graphs of maximum degree [19], bipartite graphs of maximum degree three, -free bipartite graphs [31], regular graphs for , line-graphs, chair-free graphs, and Hamiltonian graphs [20]. The problem is known to be polynomial time solvable for many classes of graphs such as trees [32], chordal graphs [5] and line-graphs of Hamiltonian graphs [20] (for more information see e.g. [7, 16, 17, 6, 22]). On the parameterized complexity of the problem, Moser and Sidkar [23] showed that the problem is W[1]-hard with respect to the parameter on bipartite graphs, and on the other hand, it becomes fixed-parameter tractable on planar graphs, graphs with girth at least 6, line graphs and graphs with bounded treewidth. Also, Song [29] showed that the Induced Matching problem is W[1]-hard with respect to the parameter for Hamiltonian bipartite graphs and cannot be solved in time , where is the number of vertices of the input graph, unless the 3-SAT problem can be solved in subexponential time.

The Acyclic Matching problem first introduced by Goddard et al. [15] and is proved to be NP-complete in general graphs. In this paper, we focus on the complexity of the Acyclic Matching problem and for the first time investigate the parameterized complexity of the problem with respect to some known parameters such as , and treewidth and on some subclasses of graphs such as bipartite graphs, line graphs and proper minor-closed graphs. First, we survey most known results about the problem.

In [26], Panda and Pradhan proved that the Acyclic Matching problem is NP-complete for planar bipartite graphs and Perfect elimination bipartite graphs. Here, we will improve this result and prove that the problem remains NP-complete for planar perfect elimination bipartite graphs with maximum degree three and girth at least , for every fixed integer (see Theorem 1). In [25], Panda and Chaudhary proved that the Acyclic Matching problem is NP-complete for comb-convex bipartite graphs and dually chordal graphs. They also proved that the problem is hard to approximate within a ratio of for any , unless P = NP and this problem is APX-complete for -regular graphs for .

The Acyclic Matching problem is known to be polynomial time solvable for the class of bipartite permutation graphs, chain graphs [26], free graphs, free graphs [13], chordal graphs [4], split graphs and proper interval graphs [25]. In [13], Furst and Rautenbach proved that for a given graph , deciding that the size of the maximum acyclic matching of is equal to the size of the maximum matching of is NP-hard even for bipartite graphs with a perfect matching and the maximum degree . This result contrasts with a result by Kobler and Rotics [20] that deciding if the size of the maximum induced matching of is equal to the size of the maximum matching is polynomially solvable for general graphs. Also, Furst and Rautenbach et al. [3] proved that every graph with vertices, the maximum degree , and no isolated vertex, has an acyclic matching of size at least , and they explained how to find such an acyclic matching in polynomial time. Moreover, they provided a -approximation algorithm for the Acyclic Matching problem, based on greedy and local search strategies.

Parameter Result
- NP-hard for planar perfect elimination bipartite graphs with maximum degree three and girth at least (Theorem 1)
- NP-hard for planar line graphs with maximum degree (Theorem 10)
W[1]-hard on bipartite graphs (Theorem 3)
FPT for line graphs (Theorem 12)
FPT for every proper minor-closed class of graphs (Theorem 15)
FPT (Theorem 13)
FPT (Theorem 14)
Table 1: Overview of new NP-hardness and parameterized results for the Acyclic Matching problem. Herein, and denote the number of cycles with length four and the treewidth of the input graph, respectively.

In this paper, we prove some new hardness results about the Acyclic Matching problem and investigate its parameterized complexity. Our main results are summarized in Table 1. The organization of forthcoming sections is as follows. In Section 2, we give the necessary notations and definitions which are needed later. In Section 3, we provide our hardness results for the Acyclic Matching Problem including the fact that the problem remains NP-complete for the class of planar perfect elimination bipartite graphs with maximum degree three and girth of arbitrary large as well as the class of planar line graphs with maximum degree four. We also prove that the problem is W[1]-hard on bipartite graphs with respect to the parameter . In Section 4, we prove that the problem is fixed parameter tractable with respect to in some subclasses of graphs including line graphs, -free graphs (graphs with no -cycle as a subgraph) and every proper minor-closed class of graphs (including bounded tree-width and planar graphs).

2 Notations and Conventions

Given a positive integer , the notation stands for the set . The set of neighbors of a vertex in the graph is denoted by and the closed neighborhood of in is defined as . The degree of a vertex in is defined as . For a subset of vertices , and are respectively defined as and . Also, we drop the subscript whenever there is no ambiguity. Two nonadjacent vertices and in is called twin if . An edge is called a pendant edge if one of its endpoints has degree equal to one. For two disjoint subsets of vertices , we say that is complete (resp. incomplete) to , if every vertex in is adjacent (resp. nonadjacent) to every vertex in . Let and be two disjoint subsets of vertices in the graph such that and let be a bijection. We say that is anti-matched to by in if every vertex is adjacent to every vertex in except . An anti-matching is a bipartite graph with the bipartition such that is anti-matched to in by a bijection .

A matching in a graph is a set of edges in which are mutually nonadjacent (i.e. have no common endpoint). The set of all endpoints of the edges in the matching is denoted by and the subgraph of induced on is denoted by . Also, for a subset of vertices , the induced subgraph of on is denoted by . An induced (resp. acyclic) matching is a matching such that has exactly edges (resp. is acyclic). The size of the maximum matching, the maximum induced matching and the maximum acyclic matching are denoted by , and , respectively. The Acyclic (Inudced) Matching problem is defined as follows.

Acyclic Matching (AM)
INSTANCE: A graph and integer . QUERY: Is there an acyclic matching in such that ?
Induced Matching (IM)
INSTANCE: A graph and integer . QUERY: Is there an induced matching in such that ?

The line graph of is a graph denoted by whose vertices are corresponding to the edges of and the vertices and in are adjacent if their corresponding edges in are adjacent. A graph is called a line graph if there is a graph such that .

Let be two integers with . An -grid is the graph with and . A grid graph is defined as an induced subgraph of an -grid for some integers . An elementary -wall is a graph with the vertex set

and the edge set

A subdivision of a graph is a graph which is obtained from by replacing each edge of by a path of arbitrary length. An -wall is a subdivision of an elementary -wall. A graph is called chordless if every cycle in is an induced subgraph of .

3 Hardness Results

Let be a bipartite graph. An edge is said to be a bisimplicial edge if is a complete bipartite subgraph of . Let be an ordering of pairwise nonadjacent edges of . Let and let . The ordering is called a perfect edge elimination ordering of G if is a bisimplicial edge in for and has no edge. A bipartite graph for which there exists a perfect edge elimination ordering is called a perfect elimination bipartite graph. Panda and Pradhan [26] showed that the Acyclic Matching problem is NP-hard for perfect elimination bipartite graphs. In this section, we strengthen their result and prove that the problem remains NP-hard for planar perfect elimination bipartite graphs with maximum degree three and girth at least for any arbitrary positive integer . Our reduction is from the following problem which is known to be NP-hard for -regular planar graphs [9] (in fact, it is the dual of the Feedback Vertex Set Problem).

Induced Acyclic Subgraph (IAS)
INSTANCE: A graph and integer . QUERY: Is there a subset of vertices such that and is acyclic?

Theorem 1.

For every integer , the Acyclic Matching problem is NP-hard for planar perfect elimination bipartite graphs with maximum degree three and girth at least .

Proof.

We prove the theorem by a polynomial reduction from Induced Acyclic Subgraph problem that is known to be NP-hard for 4-regular planar graphs [9]. Let be an instance of Induced Acylcic Subgraph problem where is a 4-regular planar graph on vertices and edges. Now, fix an ordering on the edge set . Also, fix an integer . We are going to construct a graph as follows.

Figure 1: The vertex gadgets and the edge gadgets

For every vertex , consider a vertex gadget and for every edge consider an edge gadget as in Figure 1, where . Then, the vertices of is the disjoint union of vertices of all gadgets , and , . If is an edge of such that is the th edge in the set of edges incident with and th edge in the set of edges incident with , then connect in to in and connect in to in . The edge set of is the set of all these edges along with the edges inside all gadgets , and , .

Now we claim that has an induced acyclic subgraph on vertices if and only if has an acyclic matching of size . In order to prove the claim, let be an induced acyclic subgraph of on vertices. Define,

It is clear that is a matching of size . Suppose that there exists a cycle in . So is crossing through some gadgets for some vertices . Now, we have , for all , since otherwise and it is not possible that passes through . Therefore, form a cycle in which is a contradiction. Thus, is an acyclic matching of size . Conversely, let be an acyclic matching of size in .

Claim 2.

There is an acyclic matching of size at least in such that for every , .

Proof of Claim 2.

We construct from . Let . First, note that one can assume that every edge is a pendant edge, since one can replace every edge in with its adjacent pendant edge and the matching remains acyclic.

Now, consider a gadget , where and , for some . Then, add to and if both and are covered by the matching , then remove the edge in which covers . The obtained matching remains acyclic. This proves the claim. ∎

Now, consider the matching . It has edges from each edge gadget, so it has at least edges from vertex gadgets. The size of the maximum matching of a vertex gadget is , therefore there are at least vertex gadgets that contains all edges and , . Let be the induced subgraph of on the vertex set . We claim that is acyclic. If is a cycle in , then can be extended to a cycle in that crosses through the gadgets , , , , , , , because contains all these gadgets. This contradiction implies that is acyclic. This completes the reduction.

Also, note that the maximum degree of is three. Since is planar, obviously is also planar. Moreover, all cycle lengths in are even and at least which is greater than . So, is bipartite with girth at least . Finally, is a perfect elimination bipartite graph. To see this, note that is a perfect matching of . Consider an ordering of as such that all edges in lie after all remaining edges in . For each , define . One can see that for each , the edge is a bisimplicial edge in because is a star. Also, has no edge because is a perfect matching. Thus, is a perfect edge elimination ordering of . ∎

It is proved in [18] that the Induced Acyclic Subgraph problem is W[1]-hard with respect to the parameter . By a simple reduction from this problem (i.e. adding pendant edges to each vertex of the input graph) one can prove that the Acyclic Matching problem is also W[1]-hard with respect to the parameter . Here, we improve this result and prove that the problem remains W[1]-hard on bipartite graphs.

Theorem 3.

The Acyclic Matching problem for bipartite graphs AMB is W[1]-hard with respect to the parameter .

In order to prove the above theorem, first we need a couple of easy observations.

Proposition 4.

Let be an anti-matching on at least four vertices. Then .

Proof.

It is easy to see that . Now, let be an acyclic matching of . If , then is a supergraph of an anti-matching on vertices which contains a cycle . This contradiction implies that . ∎

A blow-up of a graph is a graph which is obtained from by a finite sequence of the following operation: take a vertex and replace with a nonempty stable set of new vertices and connect every vertex in to every neighbor of in .

Proposition 5.

Let be a graph and be a blow-up of . Then, .

Proof.

Let be obtained from by blow-up of a vertex . It is clear that is an induced subgraph of and thus . Now, let be the maximum acyclic matching of and let . If contains more than one vertex in , then contains a cycle (since all vertices in are twin). Thus, and so is an induced subgraph of . This shows that . ∎

Proof of Theorem 3.

We give a three-stage parameterized reduction. First, consider the following four problems.

Acyclic Matching on Bipartite Graphs (AMB)
INSTANCE: A bipartite graph and a positive integer . QUERY: Is there an acyclic matching of size in ?

Multicolor Acyclic Matching on Bipartite Graphs (MAMB)
INSTANCE: A bipartite graph and a -partition of . QUERY: Is there a multicolor acyclic matching for ? (A matching in is called multicolor if for every , .)

Irredundant Set (IS)
INSTANCE: A graph and integer . QUERY: Is there an irredundant set in of size ? (An irredundant set in is a set such that for every , .)

Multicolor Irredundant Set (MIS)
INSTANCE: A graph and a -partition of . QUERY: Is there a multicolor irredundant set for ? (A multicolor irredundant set for is a set such that for every , and for , there is a vertex such that .)

We provide a sequence of parameterized reductions as follows . In all problems the parameter is . Downey et al. [12] proved that the Irredundant Set problem is W[1]-hard.

Parameterize reduction from IS to MIS.

Let be an instance of IS. Construct a graph with the vertex set such is adjacent to for every and and if is adjacent to in , then is adjacent to for every . Also, for every , let and . Then with the partition of is an instance of MIS. If is a yes-instance with an irredundant set , then we claim that is a multicolor irredundant set of . To see this, note that for every , there is a vertex . Thus, . On the other hand, suppose that is a yes-instance of MIS. Then, there is a multicolor irredundant set in . First, note that is a set of size in , otherwise if for , then which is a contradiction with the fact that is an irredundant set. Now, for every , there is a vertex . Therefore, . Hence, is an irredundant set of size for and is a yes-instance for IS.

Parameterized reduction from MIS to MAMB.

Let be an instance of MIS with the partition of . Construct the bipartite graph where is the disjoint union of the set and . For every vertices , if , then is adjacent to for every , . Also, for every , is adjacent to all vertices in . Finally, is adjacent to and for every , is adjacent to and . Now, consider the -partition of such that for , and for , . First, suppose that is a multicolor irredundant set for , where for each , and there is a vertex . Now, consider the multicolor matching of size in . Note that induces a tree on the set . So, if there is a cycle in , then there must be an edge between and for some which is impossible since . Hence, is a multicolor acyclic matching. On the other hand, suppose that is arbitrary multicolor acyclic matching in . Since for each , is a set of size two, . Also, for every , contains an edge with both endpoints in . So, , for some . We claim that is a multicolor irredundant set in . To see this, note that for each , . If , for some , then and are adjacent in and so contains a cycle which is a contradiction. Hence, and is a multicolor irredundant set for .

Parameterized reduction from MAMB to AMB.

This is the main part of the proof. Let the bipartite graph with the bipartition and the -partition of be an instance of MAMB. For each , let be the set of all edges of with both endpoints in . We are going to construct a bipartite graph and choose an integer , where is a constant that will be determined in the proof, such that has a multicolor acyclic matching of size if and only if contains an acyclic matching of size . First, note that for each , intersects both parts and , otherwise is a No instance. In the construction of , we use the following three gadgets (To illustrate the construction, see Figure 2 as an example).

Figure 2: An example of the construction of from .
  • is an induced matching where is the bipartition of .

  • is the bipartite graph with the bipartition where . The vertices and are respectively complete to and and is anti-matched to .

  • is the bipartite graph with the bipartition , where and is a bijection. The vertices and are respectively complete to and , and are respectively adjacent to and , and is anti-matched to by .

  • is the bipartite graph comprised with the union of two disjoint copies of , say with bijection and with bijection and the following additional edges. Let be a bijection. Then, is anti-matched to by and is anti-matched to by the bijection . Also, and are respectively complete to and .

  • is the bipartite graph on the vertex set with edges , , , , , and .

It should be noted that in the following construction, for instance if is a copy of , by abuse of notation, we use for the (sets of) vertices in corresponding to . The graph is comprised with some vertex-disjoint copies of the above gadgets, where there are a number of edges between these gadgets, as defined in the following.

First, for every part in , set a disjoint copy of , where and every vertex in and its counterpart in are corresponding to an edge in . For each edge , let us call its corresponding vertices in and by and respectively.

Also, for every pair , , , let and be the disjoint copies of and respectively, where and every vertex and its three counterparts , and are corresponding to a pair for some and . To simplify notations, for each pair , let us call its corresponding vertex in by . Similar notations are used for its corresponding vertices in and .

Let be the disjoint union of all , , all and , , with the following additional edges: For each , , the connections between and are as follows; and are respectively complete to and . Also, every vertex in corresponding to the edge , is adjacent to all vertices in for every and . Moreover, every vertex in corresponding to the edge , is adjacent to all vertices in for every and . Now, for each , , the connections between and are as follows; for every pair , if the endpoint of in is adjacent to the endpoint of in , then the vertex (resp. ) is adjacent to the vertices and (resp. and ) in , otherwise, the vertex (resp. ) is adjacent to the vertex (resp. ) in . There is no more edges in .

Now, let be obtained from as follows. Let be the gadget defined as above such that (we will determine the constant shortly in Claim 6). For each , add two disjoint copies of , say and , to and connect every vertex in (resp. ) to (resp. ). Also, for each , , add five disjoint copies of , say , , , and , to and connect every vertex in , , , and respectively to every vertex in , , , and .

Now, suppose that contains a multicolor acyclic matching , where , for each . Now, we construct an acyclic matching of size in . Define the following subsets of .