On the randomized complexity of volume and diameter

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28 Citations (Scopus)

Abstract

The authors give an O(n 7/log sup 2/n) randomised algorithm to approximate the volume of a convex body, and an O(n 6 log n) algorithm to sample a point from the uniform distribution over a convex body. For convex polytopes the algorithm runs in O(n 7/log sup 4/n) steps. Several tools are developed that may be interesting on their own. They extend results of Sinclair-Jerrum (1988) and the authors (1990) on the mixing rate of Markov chains from finite to arbitrary Markov chains. They describe an algorithm to integrate a function with respect to the stationary distribution of a general Markov chain. They also analyze the mixing rate of various random walks on convex bodies, in particular the random walk with steps from the uniform distribution over a unit ball. In several previous positive and negative results, the problem of computing the diameter of a convex body behaved similarly as the volume problem. In contrast to this, they show that there is no polynomial randomized algorithm to compute the diameter within a factor of n 1/4 .

Original languageEnglish
Title of host publicationProceedings - 33rd Annual Symposium on Foundations of Computer Science, FOCS 1992
PublisherIEEE Computer Society
Pages482-491
Number of pages10
ISBN (Electronic)0818629002
DOIs
Publication statusPublished - Jan 1 1992
Event33rd Annual Symposium on Foundations of Computer Science, FOCS 1992 - Pittsburgh, United States
Duration: Oct 24 1992Oct 27 1992

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume1992-October
ISSN (Print)0272-5428

Conference

Conference33rd Annual Symposium on Foundations of Computer Science, FOCS 1992
CountryUnited States
CityPittsburgh
Period10/24/9210/27/92

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ASJC Scopus subject areas

  • Computer Science(all)

Cite this

Lovász, L., & Simonovits, M. (1992). On the randomized complexity of volume and diameter. In Proceedings - 33rd Annual Symposium on Foundations of Computer Science, FOCS 1992 (pp. 482-491). [267803] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS; Vol. 1992-October). IEEE Computer Society. https://doi.org/10.1109/SFCS.1992.267803