[Paper Review] Computing parametric rational generating functions with a primal Barvinok algorithm

Matthias Köppe, Sven Verdoolaege|arXiv (Cornell University)|May 24, 2007

Advanced Combinatorial Mathematics11 references4 citations

TL;DR

This paper presents a primal Barvinok algorithm for computing parametric rational generating functions of integer points in polyhedra, avoiding dual-space inclusion-exclusion by using half-open polyhedra to express linear identities in indicator functions. The key contribution is a practical, efficient method for parametric counting that operates directly in the primal space, enabling fast evaluation of piecewise quasipolynomial counting functions.

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.

Motivation & Objective

• To develop an efficient algorithm for computing parametric rational generating functions of integer points in polyhedra, particularly for families of polytopes parameterized by a right-hand side vector.
• To overcome the computational complexity of inclusion–exclusion formulas traditionally used in dual-space Barvinok methods by expressing linear identities in indicator functions using only half-open full-dimensional polyhedra.
• To enable practical computation of parametric counting functions, such as vector partition functions, by operating in the primal space without relying on complex dual decompositions.
• To provide a constructive method for representing piecewise quasipolynomial counting functions as rational generating functions that can be efficiently evaluated for any parameter value.

Proposed method

• The method uses half-open variants of full-dimensional polyhedra to represent linear identities in the space of indicator functions, eliminating the need for inclusion–exclusion over intersecting faces.
• It applies Barvinok’s signed decomposition to triangulate vertex cones into half-open simplicial cones, each with a sign to account for overcounting.
• For each half-open cone, the rational generating function is computed using a closed-form formula based on the cone’s generators and sign.
• The generating function of the full parametric polytope is assembled as a signed sum over these half-open cones, preserving correctness through the indicator function identities.
• The parametric generating function is then specialized by setting all variables to 1 to extract the counting function value.
• The algorithm operates entirely in the primal space, avoiding the combinatorial overhead of dual-space triangulations and inclusion–exclusion.

Experimental results

Research questions

• RQ1 Can linear identities in the space of indicator functions of polyhedra be fully expressed using only half-open full-dimensional polyhedra, without resorting to inclusion–exclusion over face intersections?
• RQ2 Is it possible to construct a primal Barvinok algorithm that avoids dual-space computations and remains computationally efficient for parametric counting problems?
• RQ3 How can the rational generating function of a parametric polytope be computed efficiently in the primal space while maintaining correctness and piecewise quasipolynomial structure?
• RQ4 What is the role of half-open decompositions in simplifying the representation of generating functions for integer points in polyhedra?
• RQ5 Can this primal approach be effectively applied to compute vector partition functions and other parametric counting functions with low computational overhead?

Key findings

• The paper proves that all linear identities in the space of indicator functions of polyhedra can be expressed purely using half-open full-dimensional polyhedra, eliminating the need for inclusion–exclusion formulas.
• A primal Barvinok algorithm is constructed that operates directly in the primal space, avoiding the combinatorial complexity of dual-space methods.
• The method enables efficient computation of parametric rational generating functions by decomposing vertex cones into half-open simplicial cones with signed coefficients.
• The resulting algorithm computes the counting function as a specialization of the generating function at z = 1, yielding a piecewise quasipolynomial function.
• The approach is practically efficient and avoids the computational bottlenecks associated with inclusion–exclusion in dual-space Barvinok algorithms.
• The method provides a constructive and algorithmically implementable framework for parametric integer point counting in fixed dimension.