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An Incremental SAT-Based Approach to the Graph Colouring Problem

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Principles and Practice of Constraint Programming (CP 2019)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11802))

Abstract

We propose and evaluate a new CNF encoding based on Zykov’s tree for computing the chromatic number of a graph. Zykov algorithms are branch-and-bound procedures, that branch on pairings of vertices that express whether or not two non-adjacent vertices have the same colour. Thus, vertices with the same colour are contracted whereas edges are added between vertices when they have different colours. Such pairings make possible the use of a well-known recurrence relation, that states that the chromatic number of a graph cannot be lower than the chromatic number of its subgraphs. Our encoding associates with any graph and integer k a CNF formula that is satisfiable if and only if the chromatic number of the graph is at least k. We first show that any colouring satisfying a complete pairing always required a fixed number of colours. Then, we establish a CNF encoding that counts the number of colours required by a pairing. However, due to a large number of clauses required to encode transitivity constraints on pairings, a direct encoding does not scale well in practice. To avoid this pitfall, we designed a CEGAR-based (Counter-Example Guided Abstraction Refinement) approach that only encodes a part of the problem and then adds the missing constraints in an incremental way until a valid solution with k colours is found or the unsatisfiability of the problem is proven, meaning that the chromatic number of the graph is greater than k. We show that our encoding scheme performs in many cases significantly better than the state-of-the-art approaches to the graph colouring problem.

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Notes

  1. 1.

    The historical name is misleading: it either merges vertices or adds non-existing edges.

  2. 2.

    The source are accessible at: https://github.com/Mystelven/picasso.

  3. 3.

    https://mat.tepper.cmu.edu/COLOR03/.

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Acknowledgements

Part of this work was supported by the French Ministry for Higher Education and Research, the Haut-de-France Regional Council through the “Contrat de Plan État Région (CPER) DATA” and by the IDEX UCA\(^{\textsc {jedi}}\).

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Authors and Affiliations

  1. CRIL, Artois University and CNRS, 62300, Lens, France

    Gael Glorian, Jean-Marie Lagniez & Nicolas Szczepanski

  2. I3S, Côte d’Azur University and CNRS, Nice, France

    Valentin Montmirail

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  1. Gael Glorian
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  2. Jean-Marie Lagniez
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  4. Nicolas Szczepanski
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Correspondence to Gael Glorian .

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  1. INRA, Castanet Tolosan, France

    Thomas Schiex

  2. INRA, Castanet Tolosan, France

    Simon de Givry

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Glorian, G., Lagniez, JM., Montmirail, V., Szczepanski, N. (2019). An Incremental SAT-Based Approach to the Graph Colouring Problem. In: Schiex, T., de Givry, S. (eds) Principles and Practice of Constraint Programming. CP 2019. Lecture Notes in Computer Science(), vol 11802. Springer, Cham. https://doi.org/10.1007/978-3-030-30048-7_13

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