Closed-Form Decomposition for Simplicial Cones and PDBarv Algorithm for Lattice Point Counting

TL;DR

This paper introduces a closed-form expression for lattice point generating functions and a hybrid PDBarv algorithm that combines primal and dual methods, significantly improving computational efficiency in lattice point counting within rational polytopes.

Contribution

It presents a new closed-form formula for generating functions and a hybrid algorithm that accelerates lattice point counting by combining primal and dual decompositions.

Findings

Over 20% average performance improvement in dimension 5

Effective primal space decomposition via closed-form expression

Enhanced efficiency in higher-dimensional lattice point counting

Abstract

Counting lattice points within a rational polytope is a foundational problem with applications across mathematics and computer science. A key approach is Barvinok's algorithm, which decomposes the lattice point generating function of cones to that of unimodular cones. However, standard implementations face difficulties: the original primal method struggles with points on cone boundaries, while the alternative dual method can be slow for certain cone types. This paper introduces two main contributions. First, We derive a closed-form expression for these generating functions using arbitrary lattice point decompositions, enabling more effective primal space decomposition. Second, by decomposing both the cone and its dual cone starting from the side with a smaller index, we develop a novel algorithm called \textup{PDBarv}. This hybrid approach integrates the primal and dual Barvinok algorithms with a novel acceleration strategy, achieving an average computational performance improvement of over 20\% in dimension 5 and even better in higher dimensions.