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Triangle-based consistencies for cost function networks

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Abstract

Cost Function Networks (aka Weighted CSP) allow to model a variety of problems, such as optimization of deterministic and stochastic graphical models including Markov random Fields and Bayesian Networks. Solving cost function networks is thus an important problem for deterministic and probabilistic reasoning. This paper focuses on local consistencies which define essential tools to simplify Cost Function Networks, and provide lower bounds on their optimal solution cost. To strengthen arc consistency bounds, we follow the idea of triangle-based domain consistencies for hard constraint networks (path inverse consistency, restricted or max-restricted path consistencies), describe their systematic extension to cost function networks, study their relative strengths, define enforcing algorithms, and experiment with them on a large set of benchmark problems. On some of these problems, our improved lower bounds seem necessary to solve them.

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Notes

  1. We use the terminology of Cost Function Networks by similarity to Constraint Networks. The Weighted Constraint Satisfaction Problem (WCSP) is the problem of solving a CFN. For outsiders, guessing what a Cost Function Network could be, is also much easier.

  2. There exists tiny variations on the definition of AC for CFNs. This paper uses the definition in [20] which simplifies the definition in [10] by not considering the propagation of inconsistent tuples.

  3. http://mulcyber.toulouse.inra.fr/projects/toulbar2/ version 0.9.6 branch maxrpc.

  4. We only excluded from the set all 35 Minizinc instances as well as two subcategories (UAI/DBN, 108 instances and CSP/warehouse, 55 instances) that contain no triangle of binary cost functions. Over the original 3,018 original instances, 2,820 remain.

  5. All the instances are available at http://genoweb.toulouse.inra.fr/degivry/evalgm.

  6. https://mulcyber.toulouse.inra.fr/scm/viewvc.php/trunk/?root=costfunctionlib

  7. http://www.cril.univ-artois.fr/CPAI08,../~lecoutre/benchmarks.html/

  8. http://www.cs.huji.ac.il/project/PASCAL/realBoard.php

  9. http://maxsat.ia.udl.cat:81/13/benchmarks/

  10. http://hci.iwr.uni-heidelberg.de/opengm2

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Acknowledgements

This work has been partly funded by the “Agence nationale de la Recherche”, reference ANR-10-BLA-0214

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Authors and Affiliations

  1. MIAT, UR 875, Université de Toulouse, INRA, Castanet-Tolosan, France

    Hiep Nguyen, Simon de Givry & Thomas Schiex

  2. University of Montpellier, Montpellier, France

    Christian Bessiere

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  1. Hiep Nguyen
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  2. Christian Bessiere
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  3. Simon de Givry
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  4. Thomas Schiex
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Correspondence to Thomas Schiex.

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Nguyen, H., Bessiere, C., Givry, S.d. et al. Triangle-based consistencies for cost function networks. Constraints 22, 230–264 (2017). https://doi.org/10.1007/s10601-016-9250-1

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