An FPTAS for the Volume of a ${\cal V}$-polytope ---It is Hard to Compute The Volume of The Intersection of Two Cross-polytopes

TL;DR

This paper develops a Fully Polynomial-Time Approximation Scheme (FPTAS) for estimating the volume of certain high-dimensional polytopes, specifically the intersection of two cross-polytopes, which is a #P-hard problem.

Contribution

It introduces an FPTAS for the volume of the intersection of two cross-polytopes, linking it to the volume of a knapsack dual polytope, and highlights the complexity of these volume computations.

Findings

FPTAS for the intersection of two cross-polytopes

Volume computation is #P-hard for these polytopes

Reduction from knapsack dual polytope volume estimation

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 some deterministic approximation algorithms are recently developed only for a few #P-hard problems. Motivated by a deterministic approximation of the volume of a ${\cal V}$-polytope, that is a polytope with few vertices and (possibly) exponentially many facets, this paper investigates 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, the volume of the intersection of two cross-polytopes (i.e., $L_1$-balls) is #P-hard, unlike the cases of $L_{\infty}$-balls or $L_2$-balls.