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: Contribution to journalArticle

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 (n/log⁡n)n. 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 V-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) [16]. We reduce an approximate volume of a knapsack dual polytope to that of the intersection of two cross-polytopes in a short distance, and give FPTASs for those volume computations. Interestingly, computing the volume of the intersection of two cross-polytopes (i.e., L1-balls) is #P-hard, unlike the cases of L-balls or L2-balls.

Original languageEnglish
Pages (from-to)87-106
Number of pages20
JournalTheoretical Computer Science
Volume833
DOIs
Publication statusPublished - Sep 12 2020

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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