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We describe a maximum entropy approach for computing volumes and countinginteger points in polyhedra. To estimate the number of points from a particularset X in R^n in a polyhedron P in R^n, by solving a certain entropymaximization problem, we construct a probability distribution on the set X suchthat a) the probability mass function is constant on the intersection of P andX and b) the expectation of the distribution lies in P. This allows us to applyCentral Limit Theorem type arguments to deduce computationally efficientapproximations for the number of integer points, volumes, and the number of 0-1vectors in the polytope. As an application, we obtain asymptotic formulas forvolumes of multi-index transportation polytopes and for the number of multi-waycontingency tables.

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