Statistics and Computing

, Volume 26, Issue 3, pp 591–607

Sequential Monte Carlo for counting vertex covers in general graphs

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Abstract

In this paper we describe a sequential importance sampling (SIS) procedure for counting the number of vertex covers in general graphs. The optimal SIS proposal distribution is the uniform over a suitably restricted set, but is not implementable. We will consider two proposal distributions as approximations to the optimal. Both proposals are based on randomization techniques. The first randomization is the classic probability model of random graphs, and in fact, the resulting SIS algorithm shows polynomial complexity for random graphs. The second randomization introduces a probabilistic relaxation technique that uses Dynamic Programming. The numerical experiments show that the resulting SIS algorithm enjoys excellent practical performance in comparison with existing methods. In particular the method is compared with cachet—an exact model counter, and the state of the art SampleSearch, which is based on Belief Networks and importance sampling.

Keywords

Vertex cover Counting problem  Sequential importance sampling Dynamic programming Relaxation Random graphs 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Radislav Vaisman
    • 1
  • Zdravko I. Botev
    • 2
  • Ad Ridder
    • 3
  1. 1.School of Mathematics and PhysicsUniversity of QueenslandBrisbaneAustralia
  2. 2.The University of New South WalesSydneyAustralia
  3. 3.Faculty of Economics and Business AdministrationVrije UniversityAmsterdamThe Netherlands

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