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Counting Weighted Spanning Trees to Solve Constrained Minimum Spanning Tree Problems

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Integration of AI and OR Techniques in Constraint Programming (CPAIOR 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10335))

Abstract

Building on previous work about counting the number of spanning trees of an unweighted graph, we consider the case of edge-weighted graphs. We present a generalization of the former result to compute in pseudo-polynomial time the exact number of spanning trees of any given weight, and in particular the number of minimum spanning trees. We derive two ways to compute solution densities, one of them exhibiting a polynomial time complexity. These solution densities of individual edges of the graph can be used to sample weighted spanning trees uniformly at random and, in the context of constraint programming, to achieve domain consistency on the binary edge variables and, more importantly, to guide search through counting-based branching heuristics. We exemplify our contribution using constrained minimum spanning tree problems.

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Notes

  1. 1.

    We generalize slightly their definition in order to include multigraphs, which may occur when we contract edges in the context of cp search.

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Author information

Authors and Affiliations

  1. École Polytechnique de Montréal, Montreal, Canada

    Antoine Delaite & Gilles Pesant

  2. CIRRELT, Université de Montréal, Montreal, Canada

    Gilles Pesant

Authors
  1. Antoine Delaite
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  2. Gilles Pesant
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Corresponding author

Correspondence to Gilles Pesant .

Editor information

Editors and Affiliations

  1. University of Padua, Padua, Italy

    Domenico Salvagnin

  2. University of Bologna, Bologna, Italy

    Michele Lombardi

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Delaite, A., Pesant, G. (2017). Counting Weighted Spanning Trees to Solve Constrained Minimum Spanning Tree Problems. In: Salvagnin, D., Lombardi, M. (eds) Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2017. Lecture Notes in Computer Science(), vol 10335. Springer, Cham. https://doi.org/10.1007/978-3-319-59776-8_14

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  • DOI: https://doi.org/10.1007/978-3-319-59776-8_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-59775-1

  • Online ISBN: 978-3-319-59776-8

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