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

  • Reuven Rubinstein
Article

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.

Keywords

Combinatorial optimization Counting Cross-entropy Gibbs sampler Importance sampling Rare-event Randomized algorithms Splitting 

AMS 2000 Subject Classifications

65C05 60C05 68W20 90C59 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Faculty of Industrial Engineering and TechnionIsrael Institute of TechnologyHaifaIsrael

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