Approximating edge dominating set in dense graphs

Schmied, Richard; Viehmann, Claus

doi:10.1016/j.tcs.2011.10.001

Cited by 17 publications

(9 citation statements)

References 20 publications

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“…Note that under UGC, since min vertex cover cannot be approximated to within 2 − ε for any ε > 0 [18], we get that for any ε > 0, edge dominating set is not (3/2 − ε)-approximable in polynomial time, which is the same lower bound recently achieved in [22].…”

Section: An Improved Polynomial-time Lower Boundsupporting

confidence: 68%

“…Carr et al [7] proved a (2 + 1 10 )-approximation algorithm for weighted edge dominating set (the generalization of edge dominating set where weights are assigned to the edges of the input graph and the objective becomes to determine a minimum total-weight edge dominating set), the ratio of which was later improved to 2 by Fujito and Nagamochi [16]. Improved results have also been obtained in sparse graphs [6] and in dense graphs [22]. However, providing an approximation algorithm with ratio (strictly) smaller than 2, or proving that such algorithm does not exist (under some likely complexity hypothesis) still remains as an open problem.…”

Section: Introductionmentioning

confidence: 99%

“…Chlebik and Chlebikova [9] proved that it is NPhard to approximate it within any factor better than 7 6 . Assuming the unique game conjecture (UGC), [22] showed some inapproximability results on dense instances, a corollary of which is that for every ε > 0 edge dominating set is inapproximable within ratio 3/2 − ε (under UGC).…”

Section: Introductionmentioning

confidence: 99%

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New Results on Polynomial Inapproximabilityand Fixed Parameter Approximability of Edge Dominating Set

et al. 2014

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Abstract. An edge dominating set in a graph G = (V, E) is a subset S of edges such that each edge in E − S is adjacent to at least one edge in S. The edge dominating set problem, to find an edge dominating set of minimum size, is a basic and important NP-hard problem that has been extensively studied in approximation algorithms and parameterized complexity. In this paper, we present improved hardness results and parameterized approximation algorithms for edge dominating set. More precisely, we first show that it is NP-hard to approximate edge dominating set in polynomial time within a factor better than 1.18. Next, we give a parameterized approximation schema (with respect to the standard parameter) for the problem and, finally, we develop an O * (1.821 τ )-time exact algorithm where τ is the size of a minimum vertex cover of G.

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Section: An Improved Polynomial-time Lower Boundsupporting

confidence: 68%

Section: Introductionmentioning

confidence: 99%

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New Results on Polynomial Inapproximabilityand Fixed Parameter Approximability of Edge Dominating Set

et al. 2014

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“…Gotthilf, Lewenstein and Rainschmidt came up with a 2´c logn n -approximation for the general case [8]. Schmied and Viehmann have a better-than-two constant ratio for dense graphs [18].…”

Section: Related Workmentioning

confidence: 99%

Tight Approximation Ratio for Minimum Maximal Matching

Dudycz

Lewandowski

Marcinkowski

2019

Integer Programming and Combinatorial Optimization

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We study a combinatorial problem called Minimum Maximal Matching, where we are asked to find in a general graph the smallest matching that can not be extended. We show that this problem is hard to approximate with a constant smaller than 2, assuming the Unique Games Conjecture.As a corollary we show, that Minimum Maximal Matching in bipartite graphs is hard to approximate with constant smaller than 4 3 , with the same assumption. With a stronger variant of the Unique Games Conjecture -that is Small Set Expansion Hypothesis -we are able to improve the hardness result up to the factor of 3 2 .

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“…Examples include trees [24], block graphs [18], series-parallel graphs [26], bipartite permutation graphs, and co-triangulated graphs [29]. Various approximation algorithms for MMM and MWMM have been proposed in the literature (see for instance [4,9,13,14,16,23,28]). Another line of research on MMM and MWMM considers the development of exponential time exact combinatorial algorithms [12,32].…”

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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Approximating edge dominating set in dense graphs

Cited by 17 publications

References 20 publications

New Results on Polynomial Inapproximabilityand Fixed Parameter Approximability of Edge Dominating Set

New Results on Polynomial Inapproximabilityand Fixed Parameter Approximability of Edge Dominating Set

Tight Approximation Ratio for Minimum Maximal Matching

Decomposition algorithms for solving the minimum weight maximal matching problem

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