A dual-mode local search algorithm for solving the minimum dominating set problem

Given a graph G = (V, E), the minimum dominating set (MinDS) problem is to identify a smallest set of vertices D such that every vertex in V \ D is adjacent to at least one vertex in D . The MinDS problem is a classic NP-hard problem and has been extensively studied because of its many disparate applications in network analysis. To solve this problem in practice, many heuristic approaches have been proposed to obtain a high-quality solution within a given time limit. However, existing heuristic algorithms are limited by various tie-breaking cases when selecting vertices, which slows down the effectiveness of the algorithms. In this paper, we design an efficient local search algorithm for MinDS, named DmDS - a dual-mode local search framework that probabilistically chooses between two distinct vertex-swapping schemes. We further address limitations of other algorithms by introducing vertex selection criterion based on the frequency of vertices added to solutions to address tie-breaking cases, and by improving the quality of the initial solution via a greedy strategy with perturbation. We evaluate DmDS against the state-of-the-art algorithms on real-world datasets, consisting of 382 instances (or families) with up to tens of millions of vertices. Experimental results show that DmDS obtains the best performance in accuracy for almost all instances and finds significantly better solutions than state-of-the-art MinDS algorithms on a broad range of large real-world graphs; specifically, DmDS computes the smallest solution on 352 (out of 382) instances, and on 119 instances DmDS finds smaller solutions than all other algorithms in our comparison.