TY - JOUR

T1 - An FPTAS for the volume of some V-polytopes — It is hard to compute the volume of the intersection of two cross-polytopes

AU - Ando, Ei

AU - Kijima, Shuji

N1 - Funding Information:
A preliminary version appeared in [2] . This work was/is partly supported by Grant-in-Aid for Scientific Research on Innovative Areas MEXT Japan “Exploring the Limits of Computation (ELC)” ( JP24106008 , JP24106005 ), and JST PRESTO Grant Number JPMJPR16E4 , Japan.
Funding Information:
A preliminary version appeared in [2]. This work was/is partly supported by Grant-in-Aid for Scientific Research on Innovative Areas MEXT Japan ?Exploring the Limits of Computation (ELC)? (JP24106008, JP24106005), and JST PRESTO Grant Number JPMJPR16E4, Japan.
Publisher Copyright:
© 2020 Elsevier B.V.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/9/12

Y1 - 2020/9/12

N2 - 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/logn)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.

AB - 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/logn)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.

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U2 - 10.1016/j.tcs.2020.05.029

DO - 10.1016/j.tcs.2020.05.029

M3 - Article

AN - SCOPUS:85085505230

SN - 0304-3975

VL - 833

SP - 87

EP - 106

JO - Theoretical Computer Science

JF - Theoretical Computer Science

ER -