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

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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.

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Acknowledgments

The authors wish to thank the anonymous reviewers for carefully reading the manuscript and for the suggestions that helped improving our presentation. Phillippe Samer is supported by a grant from CAPES (Coordenadoria de Aperfeiçoamento de Pessoal de Nível Superior, Brazil). Sebastián Urrutia is partially supported by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil) grant 303442/2010-7.

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  1. Universidade Federal de Minas Gerais (UFMG), Belo Horizonte, MG, Brazil

    Phillippe Samer & Sebastián Urrutia

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  1. Phillippe Samer
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  2. Sebastián Urrutia
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Correspondence to Phillippe Samer.

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Samer, P., Urrutia, S. A branch and cut algorithm for minimum spanning trees under conflict constraints. Optim Lett 9, 41–55 (2015). https://doi.org/10.1007/s11590-014-0750-x

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