Maximum Induced Matching Problem Research Articles

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.

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.

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 \(O^{*}(1.3348^n)\).

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.

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.