Computing mixed discriminants, mixed volumes, and permanents

Abstract

We construct a probabilistic polynomial time algorithm that computes the mixed discriminant of given n positive definite n × n matrices within a 2O(n) factor. As a corollary, we show that the permanent of an n × n nonnegative matrix and the mixed volume of n ellipsoids in ℝn can be computed within a 2O(n) factor by probabilistic polynomial time algorithms. Since every convex body can be approximated by an ellipsoid, the last algorithm can be used for approximating in polynomial time the mixed volume of n convex bodies in ℝn within a factor nO(n).