New parameterized algorithms for edge dominating set

An edge dominating set of a graph G=(V,E) is a subset M of edges in the graph such that each edge in E-M is incident with at least one edge in M. In an instance of the parameterized edge dominating set problem we are given a graph G=(V,E) and an integer k and we are asked to decide whether G has an edge dominating set of size at most k. In this paper we show that the parameterized edge dominating set problem can be solved in O^*(2.3147^k) time and polynomial space. We show that this problem can be reduced to a quadratic kernel with O(k^3) edges.

💡 Research Summary

The paper studies the parameterized Edge Dominating Set (EDS) problem: given a graph G = (V, E) and an integer k, decide whether there exists an edge dominating set of size at most k. An edge dominating set M has the property that every edge not in M shares at least one endpoint with an edge of M. The authors present two deterministic algorithms that improve the best known running times for the problem, and they also provide a compact quadratic kernel.

Background and reduction to vertex covers
A fundamental observation is that the set of vertices incident to an edge dominating set is a vertex cover of G, and conversely, for any vertex cover C one can construct a minimum (C, I)-edge dominating set (where I is an independent set disjoint from C) in polynomial time by taking a maximum matching in G