Improved approximation bounds for edge dominating set in dense graphs

Cardinal, Jean; Langerman, Stefan; Levy, Eythan

doi:10.1016/j.tcs.2008.12.036

Cited by 21 publications

(10 citation statements)

References 20 publications

(17 reference statements)

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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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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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“…There also exist algorithms that approach γ(G) within a factor strictly smaller than 2, see e.g. [6,7,12,19], yet most of this work seems to have focused on the general case or on very dense graphs. In this paper, we instead focus on sparse graphs.…”

Section: Introductionmentioning

confidence: 99%

Minimum maximal matchings in cubic graphs

van Batenburg

2020

Preprint

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We prove that every connected cubic graph with n vertices has a maximal matching of size at most 5 12 n + 1 2 . This confirms the cubic case of a conjecture of Baste, Fürst, Henning, Mohr and Rautenbach (2019) on regular graphs. More generally, we prove that every graph with n vertices and m edges and maximum degree at most 3 has a maximal matching of size at most 4n−m 6 + 1 2 . These bounds are attained by the graph K3,3, but asymptotically there may still be some room for improvement. Moreover, the claimed maximal matchings can be found efficiently. As a corollary, we have a 25 18 + O 1 n -approximation algorithm for minimum maximal matching in connected cubic graphs.

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“…There are also some results in the literature that investigates the MMM problem from an approximation point of view, see e.g., [11,12]. The MWMM problem is used in applications where it is desirable to obtain a maximum weighted matching in a network (with positive edge weights) when the user has no control on the edges that are going to be selected in the matching.…”

Section: Introductionmentioning

confidence: 99%