An FPTAS for the Volume of Some v-polytopes—It is Hard to Compute the Volume of the Intersection of Two Cross-Polytopes

Ei Ando, Shuji Kijima

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

Abstract

Given an n-dimensional convex body by a membership oracle in general, it is known that any polynomial-time deterministic algorithm cannot approximate its volume within ratio. There is a substantial progress on randomized approximation such as Markov chain Monte Carlo for a high-dimensional volume, and for many #P-hard problems, while only a few #P-hard problems are known to yield deterministic approximation. Motivated by the problem of deterministically approximating the volume of a -polytope, that is a polytope with a small number of vertices and (possibly) exponentially many facets, this paper investigates the problem of computing the volume of a “knapsack dual polytope,” which is known to be #P-hard due to Khachiyan (1989). We reduce an approximate volume of a knapsack dual polytope to that of the intersection of two cross-polytopes, and give FPTASs for those volume computations. Interestingly, computing the volume of the intersection of two cross-polytopes (i.e.,-balls) is #P-hard, unlike the cases of-balls or -balls.

Original languageEnglish
Title of host publicationComputing and Combinatorics - 23rd International Conference, COCOON 2017, Proceedings
EditorsYixin Cao, Jianer Chen
PublisherSpringer Verlag
Pages13-24
Number of pages12
ISBN (Print)9783319623887
DOIs
Publication statusPublished - Jan 1 2017
Event23rd International Conference on Computing and Combinatorics, COCOON 2017 - Hong Kong, China
Duration: Aug 3 2017Aug 5 2017

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume10392 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other23rd International Conference on Computing and Combinatorics, COCOON 2017
CountryChina
CityHong Kong
Period8/3/178/5/17

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All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Ando, E., & Kijima, S. (2017). An FPTAS for the Volume of Some v-polytopes—It is Hard to Compute the Volume of the Intersection of Two Cross-Polytopes. In Y. Cao, & J. Chen (Eds.), Computing and Combinatorics - 23rd International Conference, COCOON 2017, Proceedings (pp. 13-24). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10392 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-62389-4_2