Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra

TL;DR

This paper presents an efficient, local algorithm to compute the highest coefficients of weighted Ehrhart quasi-polynomials for rational polytopes, extending Barvinok's method to handle weights and providing practical computational results.

Contribution

It introduces a novel local algorithm that computes weighted Ehrhart coefficients in closed form, generalizing Barvinok's unweighted approach to weighted cases.

Findings

Algorithm efficiently computes highest coefficients of weighted Ehrhart quasi-polynomials.

Method handles polynomial weights and provides coefficients as step polynomials.

Computational experiments show competitiveness with existing software.

Abstract

This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algorithm for computing the highest degree coefficients of the weighted Ehrhart quasi-polynomial for a rational simple polytope in varying dimension, when the weights of the lattice points are given by a polynomial function h. Our technique is based on a refinement of an algorithm of A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), pp. 1449--1466] in the unweighted case (i.e., h = 1). In contrast to Barvinok's method, our method is local, obtains an approximation on the level of generating functions, handles the general weighted case, and provides the coefficients in closed form as step polynomials of the dilation. To demonstrate the practicality of our approach we report on computational experiments which show even our simple implementation can compete with state of the art software.