Statistics and Computing

, Volume 26, Issue 3, pp 591–607

Sequential Monte Carlo for counting vertex covers in general graphs



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.


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


  1. Asmussen, S., Glynn, P.W.: Stochastic Simulation: Algorithms and Analysis. Applications of Mathematics. Springer, New York (2007)MATHGoogle Scholar
  2. Blitzstein, J., Diaconis, P.: A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Math. 6, 487–520 (2010)MathSciNetMATHGoogle Scholar
  3. Bollobás, B.: Random Graphs. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  4. Botev, Z., Kroese, D.: Efficient Monte Carlo simulation via the generalized splitting method. Stat. Comput. 22, 1–16 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. Chen, Y., Diaconis, P., Holmes, S.P., Liu, J.S.: Sequential Monte Carlo methods for statistical analysis of tables. J. Am. Stat. Assoc. 100, 109–120 (2005)MathSciNetCrossRefMATHGoogle Scholar
  6. Cryan, M., Dyer, M.: A polynomial-time algorithm to approximately count contingency tables when the number of rows is constant. J. Comput. Syst. Sci. 67, 291–310 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. Dechter, R., Gogate, V.: A new algorithm for sampling CSP solutions uniformly at random. In: Principles and Practice of Constraint Programming (May 2006)Google Scholar
  8. Dyer, M.: Approximate counting by dynamic programming. In: Proceedings of the 35th ACM Symposium on Theory of Computing, pp. 693–699 (2003)Google Scholar
  9. Dyer, M., Frieze, A., Jerrum, M.: On counting independent sets in sparse graphs. In: 40th Annual Symposium on Foundations of Computer Science, pp. 210–217 (1999)Google Scholar
  10. Gogate, V., Dechter, R.: Approximate counting by sampling the backtrack-free search space. In: Proceedings of the 22nd National Conference on Artificial Intelligence, vol. 1. AAAI’07, pp. 198–203. AAAI Press, Menlo Park (2007)Google Scholar
  11. Gomes, C.P., Hoffmann, J., Sabharwal, A., Selman, B.: From sampling to model counting. In: IJCAI, pp. 2293–2299 (2007)Google Scholar
  12. Harary, F., Hayes, J.P., Wu, H.-J.: A survey of the theory of hypercube graphs. Comput. Math. Appl. 15(4), 277–289 (1988)Google Scholar
  13. Jerrum, M., Sinclair, A.: The Markov chain Monte Carlo method: an approach to approximate counting and integration. In: Hochbaum, D. (ed.) Approximation Algorithms for NP-Hard Problems, pp. 482–520. PWS Publishing, Pacific Grove (1996)Google Scholar
  14. Jerrum, M., Valiant, L.G., Vazirani, V.V.: Random generation of combinatorial structures from a uniform distribution. Theor. Comput. Sci. 43, 169–188 (1986)MathSciNetCrossRefMATHGoogle Scholar
  15. Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. J. ACM 51(4), 671–697 (2004)Google Scholar
  16. Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972)CrossRefGoogle Scholar
  17. Karp, R.M., Luby, M.: Monte-Carlo algorithms for enumeration and reliability problems. In: Proceedings of the 24th Annual Symposium on Foundations of Computer Science, SFCS ’83, pp. 56–64. IEEE Computer Society, Washington, DC (1983)Google Scholar
  18. Karp, R.M., Luby, M., Madras, N.: Monte-Carlo approximation algorithms for enumeration problems. J. Algorithms 10, 429–448 (1989)MathSciNetCrossRefMATHGoogle Scholar
  19. Liu, J., Lu, P.: FPTAS for counting monotone CNF. CoRR. (2013)
  20. Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient SAT solver. In: Annual ACM IEEE Design Automation Conference, pp. 530–535. ACM, New York (2001)Google Scholar
  21. Rasmussen, L.E.: Approximately counting cliques. Random Struct. Algorithms 11(4), 395–411 (1997)MathSciNetCrossRefMATHGoogle Scholar
  22. Rubinstein, R.: The Gibbs cloner for combinatorial optimization, counting and sampling. Methodol. Comput. Appl. Probab. 11, 491–549 (2009)MathSciNetCrossRefMATHGoogle Scholar
  23. Rubinstein, R.: Stochastic enumeration method for counting NP-hard problems. Methodol. Comput. Appl. Probab. 15(2), 249–291 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. Rubinstein, R., Kroese, D.: Simulation and the Monte Carlo Method, 2nd edn. Wiley, New York (2007)CrossRefMATHGoogle Scholar
  25. Rubinstein, R., Dolgin, A., Vaisman, R.: The splitting method for decision making. Commun. Stat. Simul. Comput. 41(6), 905–921 (2012)MathSciNetCrossRefMATHGoogle Scholar
  26. Sang, T., Bacchus, F., Beame, P., Kautz, H., Pitassi, T.: Combining component caching and clause learning for effective model counting. In: Seventh International Conference on Theory and Applications of Satisfiability Testing (2004)Google Scholar
  27. Vadhan, S.P.: The complexity of counting in sparse, regular, and planar graphs. SIAM J. Comput. 31, 398–427 (1997)Google Scholar
  28. Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8(3), 410–421 (1979)MathSciNetCrossRefMATHGoogle Scholar

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

Personalised recommendations