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From Cliques to Colorings and Back Again

Authors Marijn J. H. Heule, Anthony Karahalios , Willem-Jan van Hoeve

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Author Details

Marijn J. H. Heule
  • Carnegie Mellon University, Pittsburgh, PA, USA
Anthony Karahalios
  • Carnegie Mellon University, Pittsburgh, PA, USA
Willem-Jan van Hoeve
  • Carnegie Mellon University, Pittsburgh, PA, USA

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Marijn J. H. Heule, Anthony Karahalios, and Willem-Jan van Hoeve. From Cliques to Colorings and Back Again. In 28th International Conference on Principles and Practice of Constraint Programming (CP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 235, pp. 26:1-26:10, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


We present an exact algorithm for graph coloring and maximum clique problems based on SAT technology. It relies on four sub-algorithms that alternatingly compute cliques of larger size and colorings with fewer colors. We show how these techniques can mutually help each other: larger cliques facilitate finding smaller colorings, which in turn can boost finding larger cliques. We evaluate our approach on the DIMACS graph coloring suite. For finding maximum cliques, we show that our algorithm can improve the state-of-the-art MaxSAT-based solver IncMaxCLQ, and for the graph coloring problem, we close two open instances, decrease two upper bounds, and increase one lower bound.

Subject Classification

ACM Subject Classification
  • Theory of computation → Constraint and logic programming
  • Mathematics of computing → Graph coloring
  • Graph coloring
  • maximum clique
  • Boolean satisfiability


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