An FPTAS for the Volume of Some $$\mathcal{V}$$-polytopes—It is Hard to Compute the Volume of the Intersection of Two Cross-Polytopes

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 \(\mathcal{V}\)-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). 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., \(L_1\)-balls) is #P-hard, unlike the cases of \(L_{\infty }\)-balls or \(L_2\)-balls.