Approximating the volume of unions and intersections of high-dimensional geometric objects

TL;DR

This paper presents a fast approximation scheme for computing the volume of unions of high-dimensional geometric objects using weak oracles, and establishes complexity results for intersections.

Contribution

It introduces a polynomial-time FPRAS for volume approximation of unions with weak oracles and proves #P-hardness and hardness results for intersections.

Findings

FPRAS for volume of unions with weak oracles

#P-hardness of intersection volume computation for boxes

Existence of additive polynomial-time approximation for intersections

Abstract

We consider the computation of the volume of the union of high-dimensional geometric objects. While showing that this problem is #P-hard already for very simple bodies (i.e., axis-parallel boxes), we give a fast FPRAS for all objects where one can: (1) test whether a given point lies inside the object, (2) sample a point uniformly, (3) calculate the volume of the object in polynomial time. All three oracles can be weak, that is, just approximate. This implies that Klee's measure problem and the hypervolume indicator can be approximated efficiently even though they are #P-hard and hence cannot be solved exactly in time polynomial in the number of dimensions unless P=NP. Our algorithm also allows to approximate efficiently the volume of the union of convex bodies given by weak membership oracles. For the analogous problem of the intersection of high-dimensional geometric objects we prove #P-hardness for boxes and show that there is no multiplicative polynomial-time $2^{d^{1-\epsilon}}$-approximation for certain boxes unless NP=BPP, but give a simple additive polynomial-time $\epsilon$-approximation.