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Clause Learning and New Bounds for Graph Coloring

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Principles and Practice of Constraint Programming (CP 2018)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11008))

Abstract

Graph coloring is a major component of numerous allocation and scheduling problems.

We introduce a hybrid CP/SAT approach to graph coloring based on exploring Zykov’s tree: for two non-neighbors, either they take a different color and there might as well be an edge between them, or they take the same color and we might as well merge them. Branching on whether two neighbors get the same color yields a symmetry-free tree with complete graphs as leaves, which correspond to colorings of the original graph.

We introduce a new lower bound for this problem based on Mycielskian graphs; a method to produce a clausal explanation of this bound for use in a CDCL algorithm; and a branching heuristic emulating Brelaz on the Zykov tree.

The combination of these techniques in both a branch-and-bound and in a bottom-up search outperforms Dsatur and other SAT-based approaches on standard benchmarks both for finding upper bounds and for proving lower bounds.

G. Katsirelos—Partially supported by the french “Agence nationale de la Recherche”, project DEMOGRAPH, reference ANR-16-C40-0028.

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Notes

  1. 1.

    We abuse the neighborhood notation and write \(N({S})\) for \(\bigcup _{u \in S}N({u})\).

  2. 2.

    We assume the graph is connected, otherwise \(u \) may not always exist.

  3. 3.

    Sources available at: https://bitbucket.org/gkatsi/minicsp.

  4. 4.

    Sources available at: https://bitbucket.org/gkatsi/gc-cdcl/src/master/.

  5. 5.

    cdcl exceeded the memory limit on 4 instances, and CPLEX on 16 instances.

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Author information

Authors and Affiliations

  1. LAAS-CNRS, Université de Toulouse, CNRS, Toulouse, France

    Emmanuel Hebrard

  2. MIAT, UR-875, INRA, Toulouse, France

    George Katsirelos

Authors
  1. Emmanuel Hebrard
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  2. George Katsirelos
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Corresponding authors

Correspondence to Emmanuel Hebrard or George Katsirelos .

Editor information

Editors and Affiliations

  1. Carnegie Mellon University, Pittsburgh, PA, USA

    John Hooker

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Hebrard, E., Katsirelos, G. (2018). Clause Learning and New Bounds for Graph Coloring. In: Hooker, J. (eds) Principles and Practice of Constraint Programming. CP 2018. Lecture Notes in Computer Science(), vol 11008. Springer, Cham. https://doi.org/10.1007/978-3-319-98334-9_12

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  • DOI: https://doi.org/10.1007/978-3-319-98334-9_12

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  • Online ISBN: 978-3-319-98334-9

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