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A unifying model for locally constrained spanning tree problems

Author

Listed:
  • Luiz Viana

    (Universidade Federal do Ceará)

  • Manoel Campêlo

    (Universidade Federal do Ceará)

  • Ignasi Sau

    (Université de Montpellier, CNRS)

  • Ana Silva

    (Universidade Federal do Ceará)

Abstract

Given a graph G and a digraph D whose vertices are the edges of G, we investigate the problem of finding a spanning tree of G that satisfies the constraints imposed by D. The restrictions to add an edge in the tree depend on its neighborhood in D. Here, we generalize previously investigated problems by also considering as input functions $$\ell $$ ℓ and u on E(G) that give a lower and an upper bound, respectively, on the number of constraints that must be satisfied by each edge. The produced feasibility problem is denoted by G-DCST, while the optimization problem is denoted by G-DCMST. We show that G-DCST is $$\texttt {NP}$$ NP -complete even if functions $$\ell $$ ℓ and u are taken under tight assumptions, as well as G and D. On the positive side, we prove two polynomial results, one for G-DCST and another for G-DCMST, and also give a simple exponential-time algorithm along with a proof that it is asymptotically optimal under the ETH. Finally, we prove that other previously studied constrained spanning tree (CST) problems can be modeled within our framework, namely, the Conflict CST, the Forcing CST, the At Least One/All Dependency CST, the Maximum Degree CST, the Minimum Degree CST, and the Fixed-Leaves Minimum Degree CST.

Suggested Citation

  • Luiz Viana & Manoel Campêlo & Ignasi Sau & Ana Silva, 2021. "A unifying model for locally constrained spanning tree problems," Journal of Combinatorial Optimization, Springer, vol. 42(1), pages 125-150, July.
  • Handle: RePEc:spr:jcomop:v:42:y:2021:i:1:d:10.1007_s10878-021-00740-2
    DOI: 10.1007/s10878-021-00740-2
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    References listed on IDEAS

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    1. Ulrich Pferschy & Joachim Schauer, 2013. "The maximum flow problem with disjunctive constraints," Journal of Combinatorial Optimization, Springer, vol. 26(1), pages 109-119, July.
    2. Frank Gurski & Carolin Rehs, 2019. "The Knapsack Problem with Conflict Graphs and Forcing Graphs of Bounded Clique-Width," Operations Research Proceedings, in: Bernard Fortz & Martine Labbé (ed.), Operations Research Proceedings 2018, pages 259-265, Springer.
    3. Luis Bicalho & Alexandre Cunha & Abilio Lucena, 2016. "Branch-and-cut-and-price algorithms for the Degree Constrained Minimum Spanning Tree Problem," Computational Optimization and Applications, Springer, vol. 63(3), pages 755-792, April.
    4. Frank Gurski & Carolin Rehs, 2019. "Solutions for the knapsack problem with conflict and forcing graphs of bounded clique-width," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(3), pages 411-432, June.
    5. Camerini, P. M. & Galbiati, G. & Maffioli, F., 1980. "Complexity of spanning tree problems: Part I," European Journal of Operational Research, Elsevier, vol. 5(5), pages 346-352, November.
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