Exact algorithms for OWA-optimization in multiobjective spanning tree problems

TL;DR

This paper introduces exact algorithms for solving multiobjective spanning tree problems optimized with Ordered Weighted Average criteria, including formulations, bounds, and computational comparisons, addressing the problem's NP-hardness.

Contribution

It proposes mixed integer programming formulations, optimality conditions, and bounds for OWA-based spanning tree problems, enhancing solution efficiency and scope.

Findings

The problem is weakly NP-hard.

Decreasing OWA weights allow for specific formulations.

Bounds improve pruning in exact algorithms.

Abstract

This paper deals with the multiobjective version of the optimal spanning tree problem. More precisely, we are interested in determining the optimal spanning tree according to an Ordered Weighted Average (OWA) of its objective values. We first show that the problem is weakly NP-hard. In the case where the weights of the OWA are strictly decreasing, we then propose a mixed integer programming formulation, and provide dedicated optimality conditions yielding an important reduction of the size of the program. Next, we present two bounds that can be used to prune subspaces of solutions either in a shaving phase or in a branch and bound procedure. The validity of these bounds does not depend on specific properties of the weights (apart from non-negativity). All these exact resolution algorithms are compared on the basis of numerical experiments, according to their respective validity scopes.