2019/02/28 by Isabel Beichl, Alathea Jensen · 1 citation
Computer Science · Decision Sciences · Engineering · Mathematics · #Importance sampling #Markov Chains and Monte Carlo Methods #Markov chain #Markov chain Monte Carlo #Monte Carlo method #Polynomial #Probabilistic logic #Probability and Risk Models #Rejection sampling #Reliability and Maintenance Optimization #Sampling (signal processing) #Time complexity #cs.DS #math.PR #msc:05C05 #msc:05C81 #msc:05C85 #msc:60J80 #msc:65C05
paper · pdf · doi:10.1145/3385650
published in ACM Journal of Experimental Algorithmics 25, 1-14 (Association for Computing Machinery)
openalex created_date 2019/03/11 · arxiv created 2020/02/28 · openalex publication_date 2020/05/03 · arxiv updated 2020/09/07 · openalex updated_date 2026/08/05
In recent decades, a number of profound theorems concerning approximation of hard counting problems have appeared. These include estimation of the permanent, estimating the volume of a convex polyhedron, and counting (approximately) the number of linear extensions of a partially ordered set. All of these results have been achieved using probabilistic sampling methods, specifically Monte Carlo Markov Chain (MCMC) techniques. In each case, a rapidly mixing Markov chain is defined that is guaranteed to produce, with high probability, an accurate result after only a polynomial number of operations. Although of polynomial complexity, none of these results lead to a practical computational technique, nor do they claim to. The polynomials are of high degree and a non-trivial amount of computing is required to get even a single sample. Our aim in this article is to present practical Monte Carlo methods for one of these problems, counting linear extensions. Like related work on estimating the coefficients of the reliability polynomial, our technique is based on improving the so-called Knuth counting algorithm by incorporating an importance function into the node selection technique giving a sequential importance sampling (SIS) method. We define and report performance on two importance functions.