Maximum Entropy Gaussian Approximation for the Number of Integer Points and Volumes of Polytopes

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 ⊂ Rn in a polyhedron P ⊂ Rn we construct a probability distribution on the set X by solving a certain entropy maximization problem such that a) the probability mass function is constant on the set P ∩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 in a number of cases. Examples include polytopes of doubly stochastic matrices and polystochastic tensors, polytopes defined by totally unimodular matrices of constraints, and polytopes associated to some covering problems.