A Benders Decomposition Approach for the k-Defensive Domination Problem

TL;DR

This paper introduces a Benders decomposition approach with novel heuristics to efficiently solve the computationally challenging k-defensive domination problem in network security and disaster management.

Contribution

It develops a Benders decomposition framework with new cut strategies and heuristics, including the first feasible solution method, improving solution efficiency for complex instances.

Findings

The proposed method solves instances unsolvable by classical formulations.

Heuristics achieve up to 98% improvement over trivial bounds.

The algorithm effectively handles diverse network structures.

Abstract

The k-defensive domination problem is a powerful modeling tool for strategic decision-making in network security and disaster/emergency management, where multiple nodes may be simultaneously under attack. Despite its practical relevance, the problem has been poorly studied, largely due to its high computational difficulty. This study investigates the application of Benders decomposition to the k-defensive domination problem, aiming to improve computational efficiency over standard integer programming formulations. Several cut generation strategies, including a combinatorial approach and the simultaneous addition of multiple cuts, are proposed. Theoretical results on the strength of feasibility cuts are presented. In addition, two novel enhancement strategies are proposed: a clique-cover-based heuristic, the first feasible solution method in the literature for this problem, achieving up to 98% improvement over the trivial upper bound, and an initial cut generation heuristic that in some cases resolves the problem without further branching. Experiments on Erdos-Renyi, chordal, and Barabasi-Albert instances show that the algorithm variant involving all of the proposed components is able to solve instances that remain out of reach for classical formulations and standard Benders decomposition.