An FPTAS for the Volume Computationof 0-1 Knapsack Polytopes Based on Approximate Convolution Integral

Abstract

Computing high dimensional volumes is a hard problem, even for approximation. It is known that no polynomial-time deterministic algorithm can approximate with ratio $$1.999^n$$ the volumes of convex bodies in the $$n$$ dimension as given by membership oracles. Several randomized approximation techniques for #P-hard problems has been developed in the three decades, while some deterministic approximation algorithms are recently developed only for a few #P-hard problems. For instance, Stefankovic, Vempala and Vigoda (2012) gave an FPTAS for counting 0-1 knapsack solutions (i.e., integer points in a 0-1 knapsack polytope) based on an ingenious dynamic programming. Motivated by a new technique for designing FPTAS for #P-hard problems, this paper is concerned with the volume computation of $$0$$ - $$1$$ knapsack polytopes: it is given by $$\{{\varvec{x}} \in \mathbb {R}^n \mid {\varvec{a}}^{\top } {\varvec{x}} \le b,\ 0 \le x_i \le 1\ (i=1,\ldots ,n)\}$$ with a positive integer vector $${\varvec{a}}$$ and a positive integer $$b$$ as an input, the volume computation of which is known to be #P-hard. Li and Shi (2014) gave an FPTAS for the problem by modifying the dynamic programming for counting solutions. This paper presents a new technique based on approximate convolution integral for a deterministic approximation of volume computations, and provides an FPTAS for the volume computation of 0-1 knapsack polytopes.