Approximating Edge Dominating Set in Dense Graphs

Schmied, Richard; Viehmann, Claus

doi:10.1007/978-3-642-20877-5_5

Cited by 4 publications

(6 citation statements)

References 16 publications

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“…By this result and Theorem 1, we get the following corollary. 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].…”

supporting

confidence: 70%

“…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%

See 2 more Smart Citations

New Results on Polynomial Inapproximability and Fixed Parameter Approximability of edge dominating set

Escoffier

Monnot

Paschos

et al. 2012

Parameterized and Exact Computation

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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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supporting

confidence: 70%

Section: Introductionmentioning

confidence: 99%

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New Results on Polynomial Inapproximability and Fixed Parameter Approximability of edge dominating set

Escoffier

Monnot

Paschos

et al. 2012

Parameterized and Exact Computation

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show abstract

“…which implies (2). Equality in (2) implies equality in (11), which, by Theorem 5, implies that every component of G ′ is either K ∆−1,∆−1 or K ∆ , where ∆ has to be even in the latter case. Since G is ∆-regular, and there are exactly 2∆ − 2 edges between {u, v} and…”

Section: Proofsmentioning

confidence: 86%

Bounding and approximating minimum maximal matchings in regular graphs

Baste¹,

Fürst²,

Henning³

et al. 2019

Preprint

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The edge domination number γ e (G) of a graph G is the minimum size of a maximal matching in G. It is well known that this parameter is computationally very hard, and several approximation algorithms and heuristics have been studied. In the present paper, we provide best possible upper bounds on γ e (G) for regular and non-regular graphs G in terms of their order and maximum degree. Furthermore, we discuss algorithmic consequences of our results and their constructive proofs.

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“…32 and 33] Table 2: Complexity status for various values of p and q: on tournaments Related Work: On undirected graphs Edge Dominating Set, also known as Maximum Minimal Matching, is NP-complete even on bipartite, planar, bounded degree graphs as well as other special cases [37,25]. It can be approximated within a factor of 2 [20] (or better in some special cases [9,32,2]), but not a factor better than 7/6 [10] unless P=NP. The problem has been the subject of intense study in the parameterized and exact algorithms community [35], producing a series of improved FPT algorithms [18,4,19,33]; the current best is given in [26].…”

Section: Range Of P Qmentioning

confidence: 99%

New Results on Directed Edge Dominating Set

Belmonte¹,

Hanaka²,

Katsikarelis³

et al. 2019

Preprint

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We study a family of generalizations of Edge Dominating Set on directed graphs called Directed (p, q)-Edge Dominating Set. In this problem an arc (u, v) is said to dominate itself, as well as all arcs which are at distance at most q from v, or at distance at most p to u.First, we give significantly improved FPT algorithms for the two most important cases of the problem, (0, 1)-dEDS and (1, 1)-dEDS (that correspond to versions of Dominating Set on line graphs), as well as polynomial kernels. We also improve the best-known approximation for these cases from logarithmic to constant. In addition, we show that (p, q)-dEDS is FPT parameterized by p + q + tw, but W-hard parameterized by tw (even if the size of the optimal is added as a second parameter), where tw is the treewidth of the underlying graph of the input.We then go on to focus on the complexity of the problem on tournaments. Here, we provide a complete classification for every possible fixed value of p, q, which shows that the problem exhibits a surprising behavior, including cases which are in P; cases which are solvable in quasi-polynomial time but not in P; and a single case (p = q = 1) which is NP-hard (under randomized reductions) and cannot be solved in sub-exponential time, under standard assumptions.

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Approximating Edge Dominating Set in Dense Graphs

Cited by 4 publications

References 16 publications

New Results on Polynomial Inapproximability and Fixed Parameter Approximability of edge dominating set

New Results on Polynomial Inapproximability and Fixed Parameter Approximability of edge dominating set

Bounding and approximating minimum maximal matchings in regular graphs

New Results on Directed Edge Dominating Set

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