🤖 AI Summary
This paper addresses the NP-hard Minimum Dominating Set (MDS) problem by proposing ParDS, an exact algorithm based on the branch-and-bound framework. Methodologically, ParDS introduces two key innovations: (1) a high-accuracy linear programming relaxation to derive tighter theoretical lower bounds, and (2) dynamic graph reduction rules that simplify instances in real time during search, substantially shrinking the state space. Both components offer provable theoretical advantages over existing approaches in terms of bound quality and reduction efficiency. Experimental evaluation demonstrates that ParDS achieves the fastest runtime on 70% of benchmark graph classes, with up to 3,411× speedup on individual instances. Moreover, it successfully solves 16 previously unsolved instances—each exceeding the prior timeout threshold—within five hours. These results significantly extend the practical scalability of exact MDS computation on large-scale sparse graphs.
📝 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.