Fast Unimodular Counting

This paper describes methods for counting the number of non-negative integer solutions of the system Ax = b when A is a non-negative totally unimodular matrix and b an integral vector of fixed dimension. The complexity (under a unit cost arithmetic model) is strong in the sense that it depends only on the dimensions of A and not on the size of the entries of b. For the special case of “contingency tables” the run time is 2 √ d log d) (d the dimension of the polytope). The method is complementary to Barvinok’s in that our algorithm is effective on problems of high dimension with a fixed number of (non-sign) constraints whereas Barvinok’s algorithms are effective on problems of low dimension and an arbitrary number of constraints.