Fast Unimodular Counting

Abstract

This paper describes methods for counting the number of nonnegative integer solutions of the system Ax = b when A is a nonnegative 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 2O(√dlogd) (where d is 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.

The Ehrhart Polynomial of the Birkhoff Polytope

“…We feel that providing any general results on this function would go beyond the scope of this article and will hopefully be the subject of a future project. On the other hand, we adjusted our algorithm to compute values of T (a, b) for three (fixed) pairs (a, b), which have been previously computed by Mount [17] and DeLoera and Sturmfels [9]. The first example is which reportedly took 35 minutes/10 days with the DeLoera-Sturmfels algorithm [9], 0.3/2.9 seconds with ours.…”

“…It possesses fascinating combinatorial properties [4,5,6,8,23] and relates to many mathematical areas [10,14]. A long-standing open problem is the determination of the relative volume of B n , which had been known only up to n = 8 [7,17]. In this paper, we propose a new method of calculating this volume and use it to compute vol B 9 and vol B 10 .…”

“…In fact, as far as we are aware of, the volume of B 8 was computed using essentially this method, combined with some nice computational tricks [7,17].…”

Algebraic unimodular counting

“…Our benchmark on unimodular counting is the work of Mount [18,19]. His approach is based on interpolating the chamber polynomials, by evaluating φ A (b) for sufficiently many right hand sides b, coupled with divide-and-conquer decompositions and advanced parallel computation techniques.…”

“…On each chamber, the function (r, c) → #Σ rc is a polynomial of degree nine in the eight variables r 1 , r 2 , r 3 , r 4 , c 1 , c 2 , c 3 , c 4 . Mount [19] computed (interpolation schemes for) all 3694 polynomials. He reported a 3 hour calculation for each chamber, adding up to a total of 6 weeks of distributed computing for preprocessing all chamber polynomials.…”