Induced Matching Problem Research Articles

Disconnected matchings

In 2005, Goddard et al. (2005) [12] asked the computational complexity of determining the maximum cardinality of a matching whose vertex set induces a disconnected graph. In this paper, we answer this question. In fact, we consider the generalized problem of finding c-disconnected matchings; such matchings are ones whose vertex sets induce subgraphs with at least c connected components. We show that, for every fixed c≥2, this problem is NP-complete even if we restrict the input to bounded diameter bipartite graphs, while can be solved in polynomial time if c=1. For the case when c is part of the input, we show that the problem is NP-complete for chordal graphs, while being solvable in polynomial time for interval graphs. Finally, we explore the parameterized complexity of the problem. We present an FPT algorithm under the treewidth parameterization, and an XP algorithm for graphs with a polynomial number of minimal separators when parameterized by c. We complement these results by showing that, unless NP⊆coNP/poly, the related Induced Matching problem does not admit a polynomial kernel when parameterized by vertex cover and size of the matching nor when parameterized by vertex deletion distance to clique and size of the matching. As for Connected Matching, we show how to obtain a maximum connected matching in linear time given an arbitrary maximum matching in the input.

On the induced matching problem in hamiltonian bipartite graphs

Abstract In this paper, we study the parameterized complexity of the induced matching problem in hamiltonian bipartite graphs and the inapproximability of the maximum induced matching problem in hamiltonian bipartite graphs. We show that, given a hamiltonian bipartite graph, the induced matching problem is W[1]-hard and cannot be solved in time n o ⁢ ( k ) {n^{o(\sqrt{k})}} , where n is the number of vertices in the graph, unless the 3SAT problem can be solved in subexponential time. In addition, we show that unless NP = P {\operatorname{NP}=\operatorname{P}} , a maximum induced matching in a hamiltonian bipartite graph cannot be approximated within a ratio of n 1 / 4 - ϵ {n^{1/4-\epsilon}} , where n is the number of vertices in the graph.

Maximum weight induced matching in some subclasses of bipartite graphs

A subset $$M\subseteq E$$ of edges of a graph $$G=(V,E)$$ is called a matching in G if no two edges in M share a common vertex. A matching M in G is called an induced matching if G[M], the subgraph of G induced by M, is the same as G[S], the subgraph of G induced by $$S=\{v \in V |$$ v is incident on an edge of $$M\}$$ . The Maximum Induced Matching problem is to find an induced matching of maximum cardinality. Given a graph G and a positive integer k, the Induced Matching Decision problem is to decide whether G has an induced matching of cardinality at least k. The Maximum Weight Induced Matching problem in a weighted graph $$G=(V,E)$$ in which the weight of each edge is a positive real number, is to find an induced matching such that the sum of the weights of its edges is maximum. It is known that the Induced Matching Decision problem and hence the Maximum Weight Induced Matching problem is known to be NP-complete for general graphs and bipartite graphs. In this paper, we strengthened this result by showing that the Induced Matching Decision problem is NP-complete for star-convex bipartite graphs, comb-convex bipartite graphs, and perfect elimination bipartite graphs, the subclasses of the class of bipartite graphs. On the positive side, we propose polynomial time algorithms for the Maximum Weight Induced Matching problem for circular-convex bipartite graphs and triad-convex bipartite graphs by making polynomial time reductions from the Maximum Weight Induced Matching problem in these graph classes to the Maximum Weight Induced Matching problem in convex bipartite graphs.

Linear programming based approximation for unweighted induced matchings—Breaking the [formula omitted] barrier

A matching in a graph is induced if no two of its edges are joined by an edge, and finding a large induced matching is a very hard problem. Lin et al. (2018) provide an approximation algorithm with ratio Δ for the weighted version of the induced matching problem on graphs of maximum degree Δ. Their approach is based on an integer linear programming formulation whose integrality gap is at least Δ−1, that is, their approach offers only little room for improvement in the weighted case. For the unweighted case though, we conjecture that the integrality gap is at most 58Δ+O(1), and that also the approximation ratio can be improved at least to this value. We provide primal–dual approximation algorithms with ratios (1−ϵ)Δ+12 for general Δ with ϵ≈0.02005, and 73 for Δ=3. Furthermore, we prove a best-possible bound on the fractional induced matching number in terms of the order and the maximum degree.

New kernels for several problems on planar graphs

In this paper, we study the kernelization of the Induced Matching problem on planar graphs, the Parameterized Planar 4-Cycle Transversal problem and the Parameterized Planar Edge-Disjoint 4-Cycle Packing problem. For the Induced Matching problem on planar graphs, based on the Gallai-Edmonds decomposition structure, a kernel of size 26k is presented, which improves the previous best result 28k. For the Parameterized Planar 4-Cycle Transversal problem, by partitioning the vertices in a given instance into several parts and analyzing the size of each part independently, a kernel with at most 51k−22 vertices is obtained, which improves the previous best result 74k. Based on the kernelization process of the Parameterized Planar 4-Cycle Transversal problem, a kernel of size 51k−22 can also be obtained for the Parameterized Planar Edge-Disjoint 4-Cycle Packing problem, which improves the previous best result 96k.

Integer Programming Formulations and Benders Decomposition for the Maximum Induced Matching Problem

We investigate the maximum induced matching problem (MIM), which is the problem of finding an induced matching having the largest cardinality on an undirected graph. The problem is known to be NP-hard for general graphs. We first propose a vertex-based integer programming formulation for MIM, which is more compact compared to an edge-based formulation found in the literature. We also introduce the maximum weight induced matching problem (MWIM), which generalizes MIM so that vertices and edges have weights. We adapt the edge-based formulation to MWIM, and propose a quadratic programming formulation of MWIM based on our vertex-based formulation. We then linearize our quadratic programming formulation, and devise a Benders decomposition algorithm that exploits a special structure of the linearized formulation. We also propose valid inequalities and formulation tightening procedures to improve the efficiency of our approach. Our computational tests on a large suite of randomly generated graphs show that our vertex-based formulation and decomposition approach significantly improve the solvability of MIM and MWIM, especially on dense graphs. The online appendix and data are available at https://doi.org/10.1287/ijoc.2017.0764 .

Exact algorithms for Maximum Induced Matching

This paper studies exact algorithms for the Maximum Induced Matching problem, in which an n-vertex graph is given and we are asked to find a set of maximum number of edges in the graph such that no pair of edges in the set have a common endpoint or are adjacent by another edge. This problem has applications in many different areas. We give several structural properties of the problem and show that the problem can be solved in O⁎(1.4231n) time and polynomial space or O⁎(1.3752n) time and exponential space.

On Minimum Maximal Distance-k Matchings

We study the computational complexity of several problems connected with the problems of finding a maximal distance-k matching of minimum cardinality or minimum weight. We introduce the class of k-equimatchable graphs which is an edge analogue of k-equipackable graphs. We prove that the recognition of k-equimatchable graphs is co-NP-complete for any fixed k≥2. We also prove that the problem of finding a minimum weight maximal distance-2l matching in chordal graphs is hard to approximate within a factor of εln|V(G)| for a fixed ε unless P=NP. Finally, we show NP-hardness of the minimum maximal induced matching problem in several restricted graph classes.

Moderately exponential time algorithms for the maximum induced matching problem

An induced matching\(M\subseteq E\) in a graph \(G=(V, E)\) is a matching such that no two edges in \(M\) are joined by any third edge of the graph. The Maximum Induced Matching problem is to find an induced matching of maximum cardinality. It is NP-hard. Branch-and-reduce algorithms proposed in the previous results for the Maximum Induced Matching problem use a standard branching rule: for a vertex \(v\), it branches into \(deg(v)+1\) subproblems that either \(v\) is not an endvertex of any edge in \(M\) or \(v\) and one of its neighbor are endvertices of an edge in \(M\). In this paper, we give a simple branch-and-reduce algorithm consisting of a boundary condition, a reduction rule, and a branching rule. Especially, the branching rule only branches the original problem into two subproblems. When the algorithm meets the input instance satisfying the boundary condition, we reduce the Maximum Induced Matching problem to the Maximum Independent Set problem. By using the measure-and-conquer approach to analyze the running time of the algorithm, we show that this algorithm runs in time \(O^{*}(1.4658^n)\) which is faster than previously known algorithms. By adding two branching rules in the simple exact algorithm, we obtain an \(O^{*}(1.4321^n)\)-time algorithm for the Maximum Induced Matching problem. Moreover, we give a moderately exponential time \(\rho \)-approximation algorithm, \(0 < \rho < 1\), for the Maximum Induced Matching problem. For \(\rho =0.5\), the algorithm runs in time \(O^{*}(1.3348^n)\).

Dominating Sets and Induced Matchings in Orthogonal Ray Graphs

SUMMARY An orthogonal ray graph is an intersection graph of horizontal and vertical rays (closed half-lines) in the plane. Such a graph is 3-directional if every vertical ray has the same direction, and 2-directional if every vertical ray has the same direction and every horizontal ray has the same direction. We derive some structural properties of orthogonal ray graphs, and based on these properties, we introduce polynomial-time algorithms that solve the dominating set problem, the induced matching problem, and the strong edge coloring problem for these graphs. We show that for 2-directional orthogonal ray graphs, the dominating set problem can be solved in O(n 2 log 5 n) time, the weighted dominating set problem can be solved in O(n4 logn) time, and the number of dominating sets of a fixed size can be computed in O(n 6 logn) time, where n is the number of vertices in the graph. We also show that for 2-directional orthogonal ray graphs, the weighted induced matching problem and the strong edge coloring problem can be solved in O(n 2 + mlogn) time, where m is the number of edges in the graph. Moreover, we show that for 3-directional orthogonal ray graphs, the induced matching problem can be solved in O(m 2 ) time, the weighted induced matching problem can be solved in O(m 4 ) time, and the strong edge coloring problem can be solved in O(m 3 ) time. We finally show that the weighted induced matching problem can be solved in O(m 6 ) time for orthogonal ray graphs.

Induced Matchings in Subcubic Planar Graphs

Induced Matchings in Subcubic Planar Graphs

Breaking the [formula omitted]-barrier for Irredundance: Two lines of attack

The lower and the upper irredundance numbers of a graph G, denoted ir ( G ) and IR ( G ) , respectively, are conceptually linked to the domination and independence numbers and have numerous relations to other graph parameters. It has been an open question whether determining these numbers for a graph G on n vertices admits exact algorithms running in time faster than the trivial Θ ( 2 n ⋅ poly ( n ) ) enumeration, also called the 2 n -barrier. The main contributions of this article are exact exponential-time algorithms breaking the 2 n -barrier for irredundance. We establish algorithms with running times of O ⁎ ( 1.99914 n ) for computing ir ( G ) and O ⁎ ( 1.9369 n ) for computing IR ( G ) . Both algorithms use polynomial space. The first algorithm uses a parameterized approach to obtain (faster) exact algorithms. The second one is based, in addition, on a reduction to the Maximum Induced Matching problem providing a branch-and-reduce algorithm to solve it.

On the induced matching problem

We study extremal questions on induced matchings in certain natural graph classes. We argue that these questions should be asked for twinless graphs, that is graphs not containing two vertices with the same neighborhood. We show that planar twinless graphs always contain an induced matching of size at least n / 40 while there are planar twinless graphs that do not contain an induced matching of size ( n + 10 ) / 27 . We derive similar results for outerplanar graphs and graphs of bounded genus. These extremal results can be applied to the area of parameterized computation. For example, we show that the induced matching problem on planar graphs has a kernel of size at most 40 k that is computable in linear time; this significantly improves the results of Moser and Sikdar (2007). We also show that we can decide in time O ( 91 k + n ) whether a planar graph contains an induced matching of size at least k.

The parameterized complexity of the induced matching problem

Given a graph G and an integer k ≥ 0 , the NP-complete Induced Matching problem asks whether there exists an edge subset M of size at least k such that M is a matching and no two edges of M are joined by an edge of G . The complexity of this problem on general graphs, as well as on many restricted graph classes has been studied intensively. However, other than the fact that the problem is W [1]-hard on general graphs, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we provide first-time fixed-parameter tractability results for planar graphs, bounded-degree graphs, graphs with girth at least six, bipartite graphs, line graphs, and graphs of bounded treewidth. In particular, we give a linear-size problem kernel for planar graphs.

Approximability results for the maximum and minimum maximal induced matching problems

An induced matching M of a graph G is a set of pairwise non-adjacent edges such that their end-vertices induce a 1-regular subgraph. An induced matching M is maximal if no other induced matching contains M . The Minimum Maximal Induced Matching problem asks for a minimum maximal induced matching in a given graph. The problem is known to be NP -complete. Here we show that, if P ≠ NP , for any ε > 0 , this problem cannot be approximated within a factor of n 1 − ε in polynomial time, where n denotes the number of vertices in the input graph. The result holds even if the graph in question is restricted to being bipartite. For the related problem of finding an induced matching of maximum size ( Maximum Induced Matching), it is shown that, if P ≠ NP , for any ε > 0 , the problem cannot be approximated within a factor of n 1 / 2 − ε in polynomial time. Moreover, we show that Maximum Induced Matching is NP -complete for planar line graphs of planar bipartite graphs.

Maximum Induced Matchings for Chordal Graphs in Linear Time

The Maximum Induced Matching (MIM) Problem asks for a largest set of pairwise vertex-disjoint edges in a graph which are pairwise of distance at least two. It is well-known that the MIM problem is NP-complete even on particular bipartite graphs and on line graphs. On the other hand, it is solvable in polynomial time for various classes of graphs (such as chordal, weakly chordal, interval, circular-arc graphs and others) since the MIM problem on graph G corresponds to the Maximum Independent Set problem on the square G *=L(G)2 of the line graph L(G) of G, and in some cases, G * is in the same graph class; for example, for chordal graphs G, G * is chordal. The construction of G *, however, requires ${\mathcal{O}}(m^{2})$time, where m is the number of edges in G. Is has been an open problem whether there is a linear-time algorithm for the MIM problem on chordal graphs. We give such an algorithm which is based on perfect elimination order and LexBFS.

Generalizing the induced matching by edge capacity constraints

Given an edge-weighted graph, the induced matching problem is an edge packing problem, which asks to find a maximum weight edge set such that every edge in the graph is adjacent to at most one edge in the set. In this paper, we generalize this problem by introducing edge capacities and propose an approximation algorithm to the problem. The analysis of this algorithm is based on a relationship between two polytopes for an LP relaxation of the above problem and the capacitated b -matching problem.

A Linear Algorithm for Computing of a Minimum Weight Maximal Induced Matching in an Edge-Weighted Tree

An induced matching M of a graph G is a set of pairwise non-adjacent edges such that their end-vertices induce a 1-regular subgraph. An induced matching M is maximal if no other induced matching contains M. The Minimum Induced Matching problem asks for a minimum maximal induced matching in a given graph. The problem is known to be NP-complete. Here we show that, if G is a tree then this problem can be solved in linear time.

Induced matchings in intersection graphs

An induced matching in a graph G is a set of edges, no two of which meet a common node or are joined by an edge of G; that is, an induced matching is a matching which forms an induced subgraph. Induced matchings in graph G correspond precisely to independent sets of nodes in the square of the line-graph of G, which we denote by [ L( G)] 2. Often, if G has a nice representation as an intersection graph, we can obtain a nice representation of [ L( G)] 2 as an intersection graph. Then, if the independent set problem is polytime-solvable in [ L( G)] 2, the induced matching problem is polytime-solvable in G. In particular, we show that if G is a polygon-circle graph, then so is [ L( G)] 2, and the same holds for asteroidal triple-free and interval-filament graphs. It follows that the induced matching problem is polytime-solvable in these classes. Gavril's interval-filament graphs include cocomparability and polygon-circle graphs, and the latter include circle graphs, circular-arc graphs, chordal graphs, and outerplanar graphs.

Finding a maximum induced matching in weakly chordal graphs

An induced matching in a graph G is a set of edges, no two of which meet a common vertex or are joined by an edge of G; that is, an induced matching is a matching which forms an induced subgraph. It is known that finding an induced matching of maximum cardinality in a graph is NP-hard. We show that a maximum induced matching in a weakly chordal graph can be found in polynomial time. This generalizes previously known results for the induced matching problem. This also demonstrates that the maximum induced matching problem in chordal bipartite graphs can be solved in polynomial time while the problem is known to be NP-hard for bipartite graphs in general.