Local Search for Minimum Weight Dominating Set with Two-Level Configuration Checking and Frequency Based Scoring Function (Extended Abstract)

Abstract

The Minimum Weight Dominating Set (MWDS) problem is an important generalization of the Minimum Dominating Set (MDS) problem with extensive applications. This paper proposes a new local search algorithm for the MWDS problem, which is based on two new ideas. The first idea is a heuristic called two-level configuration checking (CC 2 ), which is a new variant of a recent powerful configuration checking strategy (CC) for effectively avoiding the recent search paths. The second idea is a novel scoring function based on the frequency of being uncovered of vertices. Our algorithm is called CC 2 FS, according to the names of the two ideas. The experimental results show that, CC 2 FS performs much better than some state-of-the-art algorithms in terms of solution quality on a broad range of MWDS benchmarks. IntroductionGiven an undirected graph G, a dominating set D is a subset of vertices such that every vertex not in D is adjacent to at least one member of D. The Minimum Dominating Set (MDS) problem consists in identifying the smallest dominating set in a graph. The Minimum Weight Dominating Set (MWDS) problem is a generalized version of MD-S. In the MWDS problem, each vertex is associated with a positive value as its weight, and the task is to find a dominating set that minimizes the total weight of the vertices in it. The MWDS problem has played a prominent role in various real-world domains [Shen and Li, 2010;Golovach et al., 2013], such as social networks, communication networks, and industrial applications.Most practical algorithms for solving the MWDS problem are heuristic algorithms [Jovanovic et al., 2010;Potluri and Singh, 2013;Nitash and Singh, 2014; Chaurasia and Singh, 2015;Bouamama and Blum, 2016]. However, the efficiency of existing heuristic algorithm are still not satisfactory, especially for hard and large-scaled instances (as will be shown in our experiments). The reason may be that the heuristic * This paper is an extended abstract of an article in the Journal of Artificial Intelligence Research [Wang et al., 2017]

A polynomial-time approximation to a minimum dominating set in a graph

“…Recently, Wang et al [40] and [41] have proposed the heuristics for the weighted version MWDS of our domination problem. Repeatedly, a frequency based scoring function is used to choose randomly the next vertex that is added to the current dominating set.…”

“…Repeatedly, a frequency based scoring function is used to choose randomly the next vertex that is added to the current dominating set. The algorithm from [40] initially, reduces the size of the MWDS problem and then at the second stage constructs the first approximation to the MWDS problem by making random choices. At the third stage, the output of the second stage is reduced by eliminating some redundant vertices applying a kind of purification procedure.…”

“…As a simple illustrative example, in Figure 7 we depict a graph for which the dominating set generated by Greedy Algorithm is {1, 2, 3}. Our purification procedure eliminates redundant vertex 1, which will be left in the dominating set constructed by the algorithm in [40].…”