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

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, 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, and (3) calculate the volume of the object in polynomial time. It suffices to be able to answer all three questions approximately. We show that this holds for a large class of objects. It implies that Klee's measure problem can be approximated efficiently even though it is #P-hard and hence cannot be solved exactly in polynomial time in the number of dimensions unless P = NP . Our algorithm also allows to efficiently approximate 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 − ε -approximation for certain boxes unless NP = BPP , but give a simple additive polynomial-time ε-approximation.