ISSN 0333-3590 A Branch-and-Reduce Algorithm for Finding a Minimum Independent Dominating Set in Graphs

An independent dominating set D of a graph G = (V, E) is a subset of vertices such that every vertex in V \ D has at least one neighbour in D and D is an independent set, i.e. no two vertices in D are adjacent. Finding a minimum independent dominating set in a graph is an NP-hard problem. Whereas it is hard to cope with this problem using parameterized and approximation algorithms, there is a simple exact O(1.4423)time algorithm solving the problem by enumerating all maximal independent sets. In this paper we improve the latter result, providing the first non trivial algorithm computing a minimum independent dominating set of a graph in time O(1.3575). Furthermore, we give a lower bound of Ω(1.3247) on the worst-case running time of this algorithm, showing that the running time analysis is almost tight. Finally we show that for the class of c-dense graphs (graphs respecting |E| ≥ c|V | for a constant c, 0 < c < 1/2) an O(1.3575 √ 1−2c)-time algorithm solves the problem.