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Certifying Solvers for Clique and Maximum Common (Connected) Subgraph Problems

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Principles and Practice of Constraint Programming (CP 2020)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 12333))

Abstract

An algorithm is said to be certifying if it outputs, together with a solution to the problem it solves, a proof that this solution is correct. We explain how state of the art maximum clique, maximum weighted clique, maximal clique enumeration and maximum common (connected) induced subgraph algorithms can be turned into certifying solvers by using pseudo-Boolean models and cutting planes proofs, and demonstrate that this approach can also handle reductions between problems. The generality of our results suggests that this method is ready for widespread adoption in solvers for combinatorial graph problems.

The first and fourth authors were funded by the Swedish Research Council (VR) grant 2016-00782. The fourth author was also supported by the Independent Research Fund Denmark (DFF) grant 9040-00389B. The third, fifth and sixth authors were supported by the Engineering and Physical Sciences Research Council [grant numbers EP/P026842/1 and EP/M508056/1]. Some code development used resources provided by the Swedish National Infrastructure for Computing (SNIC).

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Notes

  1. 1.

    We are not aware of any obstacles for providing formal verification for this translation step. However, since this translation is so simple, in this paper we focus on the more challenging task of formally verifying the correctness of solvers’ reasoning.

  2. 2.

    In a PB setting, unit propagation is equivalent to achieving integer bounds consistency [9] on all constraints. This is identical to SAT unit propagation on clausal constraints, but is stronger in general.

  3. 3.

    In all of what follows, these variables are equivalent under the exchange of f and g, and so we may halve the number of variables needed by exchanging f and g if \(f > g\). We do this in practice, but omit this in the description for clarity.

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Authors and Affiliations

  1. Lund University, Lund, Sweden

    Stephan Gocht & Jakob Nordström

  2. University of Copenhagen, Copenhagen, Denmark

    Stephan Gocht & Jakob Nordström

  3. University of Glasgow, Glasgow, Scotland

    Ross McBride, Ciaran McCreesh, Patrick Prosser & James Trimble

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  1. Stephan Gocht
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    Helmut Simonis

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Gocht, S., McBride, R., McCreesh, C., Nordström, J., Prosser, P., Trimble, J. (2020). Certifying Solvers for Clique and Maximum Common (Connected) Subgraph Problems. In: Simonis, H. (eds) Principles and Practice of Constraint Programming. CP 2020. Lecture Notes in Computer Science(), vol 12333. Springer, Cham. https://doi.org/10.1007/978-3-030-58475-7_20

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