Stochastic Enumeration with Importance Sampling
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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.
KeywordsRandomized 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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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.
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