Computing parametric rational generating functions with a primal Barvinok algorithm

TL;DR

This paper introduces a primal-space Barvinok algorithm that efficiently computes parametric rational generating functions by avoiding complex inclusion-exclusion formulas, simplifying boundary effect calculations.

Contribution

The authors prove that boundary effects can be handled without inclusion-exclusion formulas, enabling a practical primal-space Barvinok algorithm for generating functions.

Findings

Primal-space approach simplifies computations.

Boundary effects are managed without inclusion-exclusion.

Algorithm demonstrates practical efficiency.

Abstract

Computations with Barvinok's short rational generating functions are traditionally being performed in the dual space, to avoid the combinatorial complexity of inclusion--exclusion formulas for the intersecting proper faces of cones. We prove that, on the level of indicator functions of polyhedra, there is no need for using inclusion--exclusion formulas to account for boundary effects: All linear identities in the space of indicator functions can be purely expressed using half-open variants of the full-dimensional polyhedra in the identity. This gives rise to a practically efficient, parametric Barvinok algorithm in the primal space.