A branch and cut algorithm for minimum spanning trees under conflict constraints

TL;DR

This paper presents a new branch and cut algorithm for solving the NP-hard minimum spanning tree problem with conflict constraints, significantly improving dual bounds and providing new certificates of optimality.

Contribution

The paper introduces a novel branch and cut approach using an IP formulation with valid inequalities, enhancing solution quality for conflict-constrained minimum spanning trees.

Findings

Dual bounds are significantly stronger than previous methods.

The approach provides new feasibility and optimality certificates.

Computational results show improved solution quality.

Abstract

We study approaches for the exact solution of the \NP--hard minimum spanning tree problem under conflict constraints. Given a graph $G(V,E)$ and a set $C \subset E \times E$ of conflicting edge pairs, the problem consists of finding a conflict-free minimum spanning tree, i.e. feasible solutions are allowed to include at most one of the edges from each pair in $C$. The problem was introduced recently in the literature, with several results on its complexity and approximability. Some formulations and both exact and heuristic algorithms were also discussed, but computational results indicate considerably large duality gaps and a lack of optimality certificates for benchmark instances. In this paper, we build on the representation of conflict constraints using an auxiliary conflict graph $\hat{G}(E,C)$, where stable sets correspond to conflict-free subsets of $E$. We introduce a general preprocessing method and a branch and cut algorithm using an IP formulation with exponentially sized classes of valid inequalities for both the spanning tree and the stable set polytopes. Encouraging computational results indicate that the dual bounds of our approach are significantly stronger than those previously available, already in the initial LP relaxation, and we are able to provide new feasibility and optimality certificates.