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Stochastic Enumeration Method for Counting Trees

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  • Radislav Vaisman

    (The University of Queensland)

  • Dirk P. Kroese

    (The University of Queensland)

Abstract

The problem of estimating the size of a backtrack tree is an important but hard problem in the computational sciences. An efficient solution of this problem can have a major impact on the hierarchy of complexity classes. The first randomized procedure, which repeatedly generates random paths through the tree, was introduced by Knuth. Unfortunately, as was noted by Knuth and a few other researchers, the estimator can introduce a large variance and become ineffective in the sense that it underestimates the cost of the tree. Recently, a new sequential algorithm called Stochastic Enumeration (SE) method was proposed by Rubinstein et al. The authors showed numerically that this simple algorithm can be very efficient for handling different counting problems, such as counting the number of satisfiability assignments and enumerating the number of perfect matchings in bipartite graphs. In this paper we introduce a rigorous analysis of SE and show that it results in significant variance reduction as compared to Knuth’s estimator. Moreover, we establish that for almost all random trees the SE algorithm is a fully polynomial time randomized approximation scheme (FPRAS) for the estimation of the overall tree size.

Suggested Citation

  • Radislav Vaisman & Dirk P. Kroese, 2017. "Stochastic Enumeration Method for Counting Trees," Methodology and Computing in Applied Probability, Springer, vol. 19(1), pages 31-73, March.
    • Handle: RePEc:spr:metcap:v:19:y:2017:i:1:d:10.1007_s11009-015-9457-4
    DOI: 10.1007/s11009-015-9457-4
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    References listed on IDEAS

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    1. Yuguo Chen & Persi Diaconis & Susan P. Holmes & Jun S. Liu, 2005. "Sequential Monte Carlo Methods for Statistical Analysis of Tables," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 109-120, March.
    2. Reuven Rubinstein, 2009. "The Gibbs Cloner for Combinatorial Optimization, Counting and Sampling," Methodology and Computing in Applied Probability, Springer, vol. 11(4), pages 491-549, December.
    3. Yuguo Chen & Junyi Xie & Jun S. Liu, 2005. "Stopping‐time resampling for sequential Monte Carlo methods," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 199-217, April.
    4. Zdravko I. Botev & Dirk P. Kroese, 2008. "An Efficient Algorithm for Rare-event Probability Estimation, Combinatorial Optimization, and Counting," Methodology and Computing in Applied Probability, Springer, vol. 10(4), pages 471-505, December.
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    Cited by:

    1. Alathea Jensen, 2018. "Stochastic Enumeration with Importance Sampling," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1259-1284, December.
    2. Radislav Vaisman & Ilya B. Gertsbakh, 2023. "Optimal balanced chain decomposition of partially ordered sets with applications to operating cost minimization in aircraft routing problems," Public Transport, Springer, vol. 15(1), pages 199-225, March.

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