Stochastic Enumeration with Importance Sampling

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Abstract

Many hard problems in the computational sciences are equivalent to counting the leaves of a decision tree, or, more generally, by summing a cost function over the nodes. These problems include calculating the permanent of a matrix, finding the volume of a convex polyhedron, and counting the number of linear extensions of a partially ordered set. Many approximation algorithms exist to estimate such sums. One of the most recent is Stochastic Enumeration (SE), introduced in 2013 by Rubinstein. In 2015, Vaisman and Kroese provided a rigorous analysis of the variance of SE, and showed that SE can be extended to a fully polynomial randomized approximation scheme for certain cost functions on random trees. We present an algorithm that incorporates an importance function into SE, and provide theoretical analysis of its efficacy. We also present the results of numerical experiments to measure the variance of an application of the algorithm to the problem of counting linear extensions of a poset, and show that introducing importance sampling results in a significant reduction of variance as compared to the original version of SE.

Keywords

Randomized algorithms Monte Carlo sampling Importance sampling Sequential importance sampling Linear extensions Decision tree Counting 

Mathematics Subject Classification (2010)

05C05 65C05 05C85 05C81 60J80 

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Notes

Acknowledgments

The author would like to thank Isabel Beichl and Francis Sullivan for the idea for this project. The author would also like to thank the Applied and Computational Mathematics Division of the Information Technology Laboratory at the National Institute of Standards and Technology for hosting the author as a guest researcher during the preparation of this article.

References

  1. Beichl I, Sullivan F (1999) Approximating the permanent via importance sampling with applications to the dimer covering problem. J Comput Phys 149:128–147MathSciNetCrossRefMATHGoogle Scholar
  2. Beichl I, Jensen A, Sullivan F (2017) A sequential importance sampling algorithm for estimating linear extensions, preprintGoogle Scholar
  3. Blitzstein J, Diaconis P (2011) A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Math 6(4):489–522MathSciNetCrossRefMATHGoogle Scholar
  4. Chen PC (1992) Heuristic sampling: a method for predicting the performance of tree searching programs. SIAM J Comput 21(2):295–315CrossRefMATHGoogle Scholar
  5. Cloteaux B, Valentin LA (2011) Counting the leaves of trees. Congr Numerantium 207:129–139MathSciNetMATHGoogle Scholar
  6. Harris D, Sullivan F, Beichl I (2014) Fast sequential importance sampling to estimate the graph reliability polynomial. Algorithmica 68(4):916–939MathSciNetCrossRefMATHGoogle Scholar
  7. Kahn AB (1962) Topological sorting of large networks. Commun ACM 5 (11):558–562.  https://doi.org/10.1145/368996.369025 CrossRefMATHGoogle Scholar
  8. Karp RM, Luby M (1983) Monte-carlo algorithms for enumeration and reliability problems. In: Proceedings of the 24th annual symposium on foundations of computer science, SFCS ’83. IEEE Computer Society, pp 56–64Google Scholar
  9. Knuth DE (1975) Estimating the efficiency of backtrack programs. Math Comput 29(129):121–136MathSciNetCrossRefMATHGoogle Scholar
  10. Rubinstein R (2013) Stochastic enumeration method for counting NP-hard problems. Methodol Comput Appl Probab 15(2):249–291MathSciNetCrossRefMATHGoogle Scholar
  11. Rubinstein R, Ridder A, Vaisman R (2014) Fast sequential Monte Carlo methods for counting and optimization. Wiley, New YorkGoogle Scholar
  12. Vaisman R, Kroese DP (2017) Stochastic enumeration method for counting trees. Methodol Comput Appl Probab 19(1):31–73.  https://doi.org/10.1007/s11009-015-9457-4 MathSciNetCrossRefMATHGoogle Scholar
  13. Valiant LG (1979) The complexity of enumeration and reliability problems. SIAM J Comput 8(3):410–421MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.George Mason UniversityFairfaxUSA

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