An Efficient Matheuristic for the Minimum-Weight Dominating Set Problem

TL;DR

This paper presents a hybrid matheuristic combining tabu search and integer programming to efficiently solve the minimum weight dominating set problem, with extensive experiments demonstrating its effectiveness.

Contribution

It introduces a novel hybrid algorithm with adaptive penalties, perturbations, and neighborhood exploration for the minimum weight dominating set problem.

Findings

The proposed method outperforms existing approaches on various instances.

Component analysis shows the effectiveness of adaptive penalties and neighborhood classes.

The algorithm achieves high-quality solutions efficiently across multiple problem instances.

Abstract

A minimum dominating set in a graph is a minimum set of vertices such that every vertex of the graph either belongs to it, or is adjacent to one vertex of this set. This mathematical object is of high relevance in a number of applications related to social networks analysis, design of wireless networks, coding theory, and data mining, among many others. When vertex weights are given, minimizing the total weight of the dominating set gives rise to a problem variant known as the minimum weight dominating set problem. To solve this problem, we introduce a hybrid matheuristic combining a tabu search with an integer programming solver. The latter is used to solve subproblems in which only a fraction of the decision variables, selected relatively to the search history, are left free while the others are fixed. Moreover, we introduce an adaptive penalty to promote the exploration of intermediate infeasible solutions during the search, enhance the algorithm with perturbations and node elimination procedures, and exploit richer neighborhood classes. Extensive experimental analyses on a variety of instance classes demonstrate the good performance of the algorithm, and the contribution of each component in the success of the search is analyzed.