2003/04/28 by Ravi Kannan
Mathematics · #math.PR #msc:68W20 #msc:60G50
published as Proceedings of the ICM, Beijing 2002, vol. 3, 673--684
arxiv created 2003/04/28 · arxiv updated 2009/11/30
A wide class of ``counting'' problems have been studied in Computer Science. Three typical examples are the estimation of - (i) the permanent of an n× n 0-1 matrix, (ii) the partition function of certain n- particle Statistical Mechanics systems and (iii) the volume of an n- dimensional convex set. These problems can be reduced to sampling from the steady state distribution of implicitly defined Markov Chains with exponential (in n) number of states. The focus of this talk is the proof that such Markov Chains converge to the steady state fast (in time polynomial in n). A combinatorial quantity called conductance is used for this purpose. There are other techniques as well which we briefly outline. We then illustrate on the three examples and briefly mention other examples.