A Polynomial-Time Algorithm to Approximate the Mixed Volume within a Simply Exponential Factor

Abstract

Let K=(K ₁,…,K _n ) be an n-tuple of convex compact subsets in the Euclidean space R ⁿ, and let V(⋅) be the Euclidean volume in R ⁿ. The Minkowski polynomial V _K is defined as V _K (λ ₁,…,λ _n )=V(λ ₁ K ₁+⋅⋅⋅+λ _n K _n ) and the mixed volume V(K ₁,…,K _n ) as

$$V(K_{1},\ldots,K_{n})=\frac{\partial^{n}}{\partial\lambda_{1}\cdots\partial \lambda_{n}}V_{\mathbf{K}}(\lambda_{1},\ldots,\lambda_{n}).$$

Our main result is a poly-time algorithm which approximates V(K ₁,…,K _n ) with multiplicative error e ⁿ and with better rates if the affine dimensions of most of the sets K _i are small. Our approach is based on a particular approximation of log (V(K ₁,…,K _n )) by a solution of some convex minimization problem. We prove the mixed volume analogues of the Van der Waerden and Schrijver–Valiant conjectures on the permanent. These results, interesting on their own, allow us to justify the abovementioned approximation by a convex minimization, which is solved using the ellipsoid method and a randomized poly-time time algorithm for the approximation of the volume of a convex set.