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Representing Fitness Landscapes by Valued Constraints to Understand the Complexity of Local Search

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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))

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Abstract

Local search is widely used to solve combinatorial optimisation problems and to model biological evolution, but the performance of local search algorithms on different kinds of fitness landscapes is poorly understood. Here we introduce a natural approach to modelling fitness landscapes using valued constraints. This allows us to investigate minimal representations (normal forms) and to consider the effects of the structure of the constraint graph on the tractability of local search. First, we show that for fitness landscapes representable by binary Boolean valued constraints there is a minimal necessary constraint graph that can be easily computed. Second, we consider landscapes as equivalent if they allow the same (improving) local search moves; we show that a minimal normal form still exists, but is NP-hard to compute. Next we consider the complexity of local search on fitness landscapes modelled by valued constraints with restricted forms of constraint graph. In the binary Boolean case, we prove that a tree-structured constraint graph gives a tight quadratic bound on the number of improving moves made by any local search; hence, any landscape that can be represented by such a model will be tractable for local search. We build two families of examples to show that both the conditions in our tractability result are essential. With domain size three, even just a path of binary constraints can model a landscape with an exponentially long sequence of improving moves. With a treewidth two constraint graph, even with a maximum degree of three, binary Boolean constraints can model a landscape with an exponentially long sequence of improving moves.

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Acknowledgments

David A. Cohen was supported by Leverhulme Trust Grant RPG-2018-161.

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Authors and Affiliations

  1. Department of Computer Science, University of Oxford, Oxford, UK

    Artem Kaznatcheev & Peter G. Jeavons

  2. Department of Translational Hematology and Oncology Research, Cleveland Clinic, Cleveland, OH, USA

    Artem Kaznatcheev

  3. Department of Computer Science, Royal Holloway, University of London, Egham, UK

    David A. Cohen

Authors
  1. Artem Kaznatcheev
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  2. David A. Cohen
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  3. Peter G. Jeavons
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Corresponding author

Correspondence to Peter G. Jeavons .

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Editors and Affiliations

  1. INRA, Castanet Tolosan, France

    Thomas Schiex

  2. INRA, Castanet Tolosan, France

    Simon de Givry

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Kaznatcheev, A., Cohen, D.A., Jeavons, P.G. (2019). Representing Fitness Landscapes by Valued Constraints to Understand the Complexity of Local Search. 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_18

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  • DOI: https://doi.org/10.1007/978-3-030-30048-7_18

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-30047-0

  • Online ISBN: 978-3-030-30048-7

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