On the Complexity of Computing the Volume of a Polyhedron

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 Scholar2 G. Elekes, A geometric inequality and the complexity of computing volume, Discrete Comput. Geom. , 1 (1986), 289–292 87k: 68138 0611. 52010 CrossrefISIGoogle Scholar3 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 Scholar4 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 Scholar5 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 Scholar6 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 Scholar7 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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