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

Abstract

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.

Tight Approximation Ratio for Minimum Maximal Matching

“…As such, the best known NP-hardness of mmm remains 1.18 by Escoffier, Monnot, Paschos, and Xiao [6] and it is an open problem, whether it can be improved using 2-2 Games Conjecture.…”

“…The result was later improved to 1.18 by Escoffier, Monnot, Paschos, and Xiao [6]. 3 2 -hardness results depending on ugc were also obtained [6,18].…”

“…In 2006 Chlebík and Chlebíková proved, that it is NP-hard to approximate the problem within factor better than 7 6 [4]. The result was later improved to 1.18 by Escoffier, Monnot, Paschos, and Xiao [6]. 3 2 -hardness results depending on ugc were also obtained [6,18].…”

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

“…For these problems, the method gives an (1 + ε)-approximation algorithm that runs in time O * (δ (1−Ω(ε))k ), where δ > 0 denotes a constant for which a O * (δ k )-time algorithm is known for the exact version of the corresponding problem. The approach, in some form or another, is also applicable both to other parameterized problems [278,279] and to non-parameterized problems (e.g., [272]); since the latter is out-of-scope for the survey, we will not discuss the specifics here.…”

Extension of Some Edge Graph Problems: Standard and Parameterized Complexity

“…In the particular case U = ∅, Min Ext 1-DCPS is exactly the well known problem Minimum Maximal Matching where the goal is to find the smallest maximal matching. In [14,15], it is shown that Minimum Maximal Matching is hard to approximate with a factor better than 2 and 1.18, assuming Unique Games Conjecture (UGC) and P = NP, respectively. We complement this bound by showing the following.…”