A Dual-mode Local Search Algorithm for Solving the Minimum Dominating Set Problem

TL;DR

This paper introduces DmDS, a dual-mode local search algorithm for the NP-hard minimum dominating set problem, which improves solution quality and efficiency on large real-world graphs by addressing tie-breaking and initial solution quality.

Contribution

The paper proposes a novel dual-mode local search framework with probabilistic vertex-swapping and new selection criteria, enhancing solution quality and computational efficiency for MinDS.

Findings

DmDS outperforms existing algorithms in accuracy on large datasets.

It finds significantly better solutions on real-world graphs.

The approach scales well to instances with tens of millions of vertices.

Abstract

Given a graph, the minimum dominating set (MinDS) problem is to identify a smallest set $D$ of vertices such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The MinDS problem is a classic $\mathcal{NP}$-hard problem and has been extensively studied because of its many disparate applications in network analysis. To solve this problem efficiently, many heuristic approaches have been proposed to obtain a good solution within an acceptable time limit. However, existing MinDS heuristic algorithms are always 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 the MinDS problem, 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 a new strategy to improve the quality of the initial solution via a greedy-based strategy integrated with perturbation. We evaluate DmDS against the state-of-the-art algorithms on seven datasets, consisting of 346 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 much better solutions than state-of-the-art MinDS algorithms on a broad range of large real-world graphs.