On the Complexity of Computing the Volume of a Polyhedron

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Previous article Next article On the Complexity of Computing the Volume of a PolyhedronM. E. Dyer and A. M. FriezeM. E. Dyer and A. M. Friezehttps://doi.org/10.1137/0217060PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractWe show that computing the volume of a polyhedron given either as a list of facets or as a list of vertices is as hard as computing the permanent of a matrix.[1] I. Bárány and , Z. Füredi, Computing the volume is difficult, Proc. 18th Annual ACM Symposium on Theory of Computing, 1986, 442–447 Google Scholar[2] G. Elekes, A geometric inequality and the complexity of computing volume, Discrete Comput. Geom., 1 (1986), 289–292 87k:68138 0611.52010 CrossrefISIGoogle Scholar[3] Michael R. Garey and , David S. Johnson, Computers and intractability, W. H. Freeman and Co., San Francisco, Calif., 1979x+338, A Guide to the Theory of NP-Completeness 80g:68056 0411.68039 Google Scholar[4] László Lovász, An algorithmic theory of numbers, graphs and convexity, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 50, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1986iv+91 87m:68066 0606.68039 LinkGoogle Scholar[5] Alexander Schrijver, Theory of linear and integer programming, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons Ltd., Chichester, 1986xii+471 88m:90090 0665.90063 Google Scholar[6] Leslie G. Valiant, The complexity of enumeration and reliability problems, SIAM J. Comput., 8 (1979), 410–421 10.1137/0208032 80f:68055 0419.68082 LinkISIGoogle Scholar[7] S. Gill Williamson, Combinatorics for computer science, Computers and Math Series, Computer Science Press, Rockville, MD, 1985xliii+479 88b:05002 Google ScholarKeywordsvolumepolyhedracomplexity# P-complete Previous article Next article FiguresRelatedReferencesCited ByDetails Generalized adaptive partition-based method for two-stage stochastic linear programs : Geometric oracle and analysisOperations Research Letters, Vol. 50, No. 5 | 1 Sep 2022 Cross Ref Semi-discrete optimal transport: hardness, regularization and numerical solutionMathematical Programming, Vol. 91 | 25 July 2022 Cross Ref Practical volume approximation of high-dimensional convex bodies, applied to modeling portfolio dependencies and financial crisesComputational Geometry | 1 Jul 2022 Cross Ref Exploiting Sparsity for Semi-Algebraic Set Volume ComputationFoundations of Computational Mathematics, Vol. 22, No. 1 | 26 March 2021 Cross Ref An algorithm for computing phase space structures in chemical reaction dynamics using 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No.98CB36280) Cross Ref Volume 17, Issue 5| 1988SIAM Journal on Computing849-1061 History Submitted:01 July 1987Accepted:15 November 1987Published online:31 July 2006 InformationCopyright © 1988 © Society for Industrial and Applied MathematicsKeywordsvolumepolyhedracomplexity# P-completeMSC codes68A20PDF Download Article & Publication DataArticle DOI:10.1137/0217060Article page range:pp. 967-974ISSN (print):0097-5397ISSN (online):1095-7111Publisher:Society for Industrial and Applied Mathematics