Exact Optimization for Minimum Dominating Sets

TL;DR

This paper presents ParDS, an exact algorithm for the NP-hard Minimum Dominating Set problem, featuring advanced bounds and reduction rules, outperforming previous methods in speed and instance solvability.

Contribution

Introduction of ParDS, a novel exact algorithm with improved bounds and reduction techniques for solving the MDS problem more efficiently.

Findings

Achieves fastest solving times in 70% of benchmark categories

Up to 3,411 times speed-up on individual instances

Successfully solves 16 out of 43 previously unsolvable instances within 5 hours

Abstract

The Minimum Dominating Set (MDS) problem is a well-established combinatorial optimization problem with numerous real-world applications. Its NP-hard nature makes it increasingly difficult to obtain exact solutions as the graph size grows. This paper introduces ParDS, an exact algorithm developed to address the MDS problem within the branch-and-bound framework. ParDS features two key innovations: an advanced linear programming technique that yields tighter lower bounds and a set of novel reduction rules that dynamically simplify instances throughout the solving process. Compared to the leading exact algorithms presented at IJCAI 2023 and 2024, ParDS demonstrates theoretically superior lower-bound quality. Experimental results on standard benchmark datasets highlight several significant advantages of ParDS: it achieves fastest solving times in 70% of graph categories, especially on large, sparse graphs, delivers a speed-up of up to 3,411 times on the fastest individual instance, and successfully solves 16 out of 43 instances that other algorithms were unable to resolve within the 5-hour time limit. These findings establish ParDS as a state-of-the-art solution for exactly solving the MDS problem.