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Randomized Algorithms with Splitting: Why the Classic Randomized Algorithms Do Not Work and How to Make them Work

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  • Reuven Rubinstein

    (Israel Institute of Technology)

Abstract

We show that the original classic randomized algorithms for approximate counting in NP-hard problems, like for counting the number of satisfiability assignments in a SAT problem, counting the number of feasible colorings in a graph and calculating the permanent, typically fail. They either do not converge at all or are heavily biased (converge to a local extremum). Exceptions are convex counting problems, like estimating the volume of a convex polytope. We also show how their performance could be dramatically improved by combining them with the classic splitting method, which is based on simulating simultaneously multiple Markov chains. We present several algorithms of the combined version, which we simple call the splitting algorithms. We show that the most advance splitting version coincides with the cloning algorithm suggested earlier by the author. As compared to the randomized algorithms, the proposed splitting algorithms require very little warm-up time while running the MCMC from iteration to iteration, since the underlying Markov chains are already in steady-state from the beginning. What required is only fine tuning, i.e. keeping the Markov chains in steady-state while moving from iteration to iteration. We present extensive simulation studies with both the splitting and randomized algorithms for different NP-hard counting problems.

Suggested Citation

  • Reuven Rubinstein, 2010. "Randomized Algorithms with Splitting: Why the Classic Randomized Algorithms Do Not Work and How to Make them Work," Methodology and Computing in Applied Probability, Springer, vol. 12(1), pages 1-50, March.
  • Handle: RePEc:spr:metcap:v:12:y:2010:i:1:d:10.1007_s11009-009-9126-6
    DOI: 10.1007/s11009-009-9126-6
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    References listed on IDEAS

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    1. Stephen Baumert & Archis Ghate & Seksan Kiatsupaibul & Yanfang Shen & Robert L. Smith & Zelda B. Zabinsky, 2009. "Discrete Hit-and-Run for Sampling Points from Arbitrary Distributions Over Subsets of Integer Hyperrectangles," Operations Research, INFORMS, vol. 57(3), pages 727-739, June.
    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. 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. Hao Ma & Henk A. P. Blom, 2022. "Random Assignment Versus Fixed Assignment in Multilevel Importance Splitting for Estimating Stochastic Reach Probabilities," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2313-2338, December.
    2. Paul Dupuis & Bahar Kaynar & Ad Ridder & Reuven Rubinstein & Radislav Vaisman, 2011. "Counting with Combined Splitting and Capture-Recapture Methods," Tinbergen Institute Discussion Papers 11-062/4, Tinbergen Institute.
    3. Kleijnen, Jack P.C. & Ridder, A.A.N. & Rubinstein, R.Y., 2010. "Variance Reduction Techniques in Monte Carlo Methods," Other publications TiSEM 87680d1a-53c1-4107-ada4-7, Tilburg University, School of Economics and Management.

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