An Improved Algorithm for Parameterized Edge Dominating Set Problem

Abstract

An edge dominating set of a graph G = (V, E) is a subset M ⊆ E of edges such that each edge in E \ M is incident to at least one edge in M. In this paper, we consider the parameterized edge dominating set problem which asks us to test whether a given graph has an edge dominating set with size bounded from above by an integer k or not, and we design an O * (2.2351 k)-time and polynomialspace algorithm. This is an improvement over the previous best time bound of O * (2.3147 k). We also show that a related problem: the parameterized weighted edge dominating set problem can be solved in O * (2.2351 k) time and polynomial space.

An Exact Algorithm for Lowest Edge Dominating Set

“…In this paper, we have studied Lowest Edge Dominating Set, which asks us to test whether a given graph G has an edge dominating set whose size is equal to μ(G)/2 , a lower bound on the size of an edge dominating set of G. We proved that the problem remains NP-complete and showed that it admits an O * (2.0801 μ(G)/2 )-time and polynomial-space algorithm, whose time bound is better than the currently best bound O * (2.2351 μ(G)/2 ) to Parameterized Edge Dominating Set [7]. We see that the bottleneck of the time bound is attained by the branching on a vertex in a component in G[U 2 ] mentioned in Lemma 12 with r = 3.…”

“…There arises a further question: for another parameter Δ ≥ 0, the problem of testing whether a given graph G has an edge dominating set of size at most μ(G)/2 + Δ or not can be solved in O * (2.2351 μ(G)/2 · 2.2351 Δ ) time by setting k = μ(G)/2 + Δ in the O * (2.2351 k )-time algorithm to Parameterized Edge Dominating Set [7]. Does the problem admit an algorithm with a better time bound, say O * (2.0801 μ(G)/2 · 2.2351 Δ )?…”

(In)approximability of maximum minimal FVS

“…Many other classical optimization problems have recently been studied in the MaxMin or MinMax framework, such as Max Min Separator [25], Max Min Cut [21], Min Max Knapsack (also known as the Lazy Bureaucrat Problem) [3,23,24], and Max Min Edge Cover [32,26]. Some problems in this area also arise naturally in other forms and have been extensively studied, such as Min Max Matching (also known as Edge Dominating Set [29]) and Grundy Coloring, which can be seen as a Max Min version of Coloring [2,6].…”

On Kernelization for Edge Dominating Set under Structural Parameters

“…The parameterized complexity of edge dominating set has been studied in a number of papers [11,12,32,33,34,35,10,19]. Structural parameters were studied, e.g., by Escoffier et al [10] who obtained an O * (1.821 ) time algorithm where is the vertex cover size of the input graph, and by Kobler and Rotics [23] who gave a polynomial-time algorithm for graphs of bounded clique-width.…”

“…In the edge dominating set problem (EDS) we are given a graph G = (V, E) and an integer k, and need to determine whether there is a set F ⊆ E of at most k edges that are incident with all (other) edges of G. It is known that this is equivalent to the existence of a maximal matching of size at most k. The edge dominating set problem is NP-hard but admits a simple 2-approximation by taking any maximal matching of G. It can be solved in time O * (2.2351 k ) 1 [19], making it fixed-parameter tractable for parameter k. Additionally, for EDS any given instance (G, k) can be efficiently reduced to an equivalent one (G , k ) with only O(k 2 ) vertices and O(k 3 ) edges [34] (this is called a kernelization).…”