Two metaheuristics for solving the connected multidimensional maximum bisection problem

Abstract

In this paper, a connected multidimensional maximum bisection problem is considered. This problem is a generalization of a standard NP-hard maximum bisection problem, where each graph edge has a vector of weights and induced subgraphs must be connected. We propose two metaheuristic approaches, a genetic algorithm (GA) and an electromagnetism-like metaheuristic (EM). The GA uses modified integer encoding of individuals, which enhances the search process and enables usage of standard genetic operators. The EM, besides standard attraction–repulsion mechanism, is extended with a scaling procedure, which additionally moves EM points closer to local optima. A specially constructed penalty function, used for both approaches, is performed as a practical technique for temporarily including infeasible solutions into the search process. Both GA and EM use the same local search procedure based on 1-swap improvements. Computational results were obtained on instances from literature with up to 500 vertices and 60,000 edges. EM reaches all known optimal solutions on small-size instances, while GA reaches all known optimal solutions except for one case. Both proposed methods give results on medium-size and large-scale instances, which are out of reach for exact methods.