Maximum entropy Gaussian approximation for the number of integer points and volumes of polytopes

Abstract

We describe a maximum entropy approach for computing volumes and counting integer points in polyhedra. To estimate the number of points from a particular set X in R^n in a polyhedron P in R^n, by solving a certain entropy maximization problem, we construct a probability distribution on the set X such that a) the probability mass function is constant on the intersection of P and X and b) the expectation of the distribution lies in P. This allows us to apply Central Limit Theorem type arguments to deduce computationally efficient approximations for the number of integer points, volumes, and the number of 0-1 vectors in the polytope. As an application, we obtain asymptotic formulas for volumes of multi-index transportation polytopes and for the number of multi-way contingency tables.