Minimum Maximal Matching Is NP-Hard in Regular Bipartite Graphs

Demange, Marc; Ekim, Tınaz

doi:10.1007/978-3-540-79228-4_32

Cited by 27 publications

(23 citation statements)

References 8 publications

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“…Yannakakis and Gavril [18] then showed that EDS remains NP-hard when restricted to planar or bipartite graphs of maximum degree 3, and gave a polynomial-time algorithm for MMM in trees. Additional hard and polynomially solvable classes of graphs were then given by Horton et al [13], and much more recently by Demange and Ekim [6]. A fundamental inapproximability result was given by Chlebìk and Chlebìkovà [5], who showed that it is NP-Hard to approximate EDS (and hence also MMM) within any factor better than 7/6.…”

Section: Related Workmentioning

confidence: 96%

Improved approximation bounds for edge dominating set in dense graphs

Cardinal¹,

Langerman²,

Levy³

2009

Theoretical Computer Science

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a b s t r a c tWe analyze the simple greedy algorithm that iteratively removes the endpoints of a maximum-degree edge in a graph, where the degree of an edge is the sum of the degrees of its endpoints. This algorithm provides a 2-approximation to the minimum edge dominating set and minimum maximal matching problems. We refine its analysis and give an expression of the approximation ratio that is strictly less than 2 in the cases where the input graph has n vertices and at least n 2 edges, for > 1/2. This ratio is shown to be asymptotically tight for > 1/2.

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Section: Related Workmentioning

confidence: 96%

Improved approximation bounds for edge dominating set in dense graphs

Cardinal¹,

Langerman²,

Levy³

2009

Theoretical Computer Science

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“…Then, one can solve MWMM on G c by considering the graph G w where the weights on vertices satisfy the above condition and by solving MVWMM on G w as described in Remark 1. Note that in this situation the t-variable and optimality cuts (13) are no longer needed because any perfect matching in subgraph G[ŷ] has the same total weight (equal to the sum of the weights of saturated vertices), which is already minimized by the objective function (8). On the other hand, if no set of vertex weights satisfies the above condition, then one cannot transform MWMM on G c to an equivalent MVWMM instance.…”

Section: Using Mvwmm In the Solution Procedures For Mwmmmentioning

confidence: 99%

“…Although the problem of finding a maximum matching on a given graph is polynomially solvable by Edmonds's augmenting path algorithm [11], MMM is NP-hard on general graphs [15] and on several restricted graph classes. Examples include bipartite or planar graphs with maximum degree 3 [33], planar bipartite graphs, planar cubic graphs [17], and k-regular bipartite graphs for any fixed k ≥ 3 [8]. In contrast, MMM is polynomially solvable in certain restricted graph classes.…”

Section: Introduction and Literature Surveymentioning

confidence: 99%

Decomposition algorithms for solving the minimum weight maximal matching problem

2013

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We investigate the problem of finding a maximal matching that has minimum total weight on a given edgeweighted graph. Although the minimum weight maximal matching problem is NP-hard in general, polynomial time exact or approximation algorithms on several restricted graph classes are given in the literature. In this article, we propose an exact algorithm for solving several variants of the problem on general graphs. In particular, we develop integer programming (IP) formulations for the problem and devise a decomposition algorithm, which is based on a combination of IP techniques and combinatorial matching algorithms. Our computational tests on a large suite of randomly generated graphs show that our decomposition approach significantly improves the solvability of the problem compared to the underlying IP formulation.

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“…The MMM problem is in general NP-hard [4] and therefore the MWMM problem is in general NP-hard as well, whereas the problems of finding a minimum/maximum weighted matching or minimum/maximum weighted perfect matching are all solvable in polynomial time, see, e.g., [5]. In particular, the MMM problem has been shown to be NP-hard in bipartite graphs with maximum degree 3 [4], planar cubic graphs [6], and -regular bipartite graphs for any ≥ 3 [7]. On the positive side, it has been shown that the MMM problem can be solved in polynomial time in some classes of graphs including trees [8], series-parallel graphs [9], and block graphs [10].…”

Section: Introductionmentioning

confidence: 99%

“…Yannakakis and Gavril provide such an application in a telephone switching network in [4]. For other applications of the MWMM, the reader is referred to [7].…”

Section: Introductionmentioning

confidence: 99%