# Counting Integral Points in Polytopes via Numerical Analysis of Contour Integration

In this paper, we address the problem of counting integer points in a rational polytope described by P(y) = { x ∈R^m Ax = y, x ≥ 0}, where A is an n × m integer matrix and y is an n-dimensional integer vector. We study the Z-transformation approach initiated by Brion-Vergne, Beck, and Lasserre-Zeron from the numerical analysis point of view, and obtain a new algorithm on this problem: If A is nonnegative, then the number of integer points in P(y) can be computed in O(poly (n,m, y_∞) (y_∞ + 1)^n) time and O(poly (n,m, y_∞)) space.This improves, in terms of space complexity, a naive DP algorithm with O((y_∞ + 1)^n)-size DP table. Our result is based on the standard error analysis to the numerical contour integration for the inverse Z-transform, and establish a new type of an inclusion-exclusion formula for integer points in P(y). We apply our result to hypergraph b-matching, and obtain a O(poly( n,m,b_∞) (b_∞ +1)^(1-1/k)n) time algorithm for counting b-matchings in a k-partite hypergraph with n vertices and m hyperedges. This result is viewed as a b-matching generalization of the classical result by Ryser for k=2 and its multipartite extension by Björklund-Husfeldt.

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