A New and Faster Representation for Counting Integer Points in Parametric Polyhedra
D. Gribanov, D. Malyshev, P. Pardalos, N. Zolotykh
TL;DR
This work introduces piece-wise periodic step-polynomials as a new, faster representation for counting integer points in parametric polyhedra $P_y$ defined by $A x \le b + B y$. By constructing a chamber decomposition of the parametric space and expressing $\mathcal{E}_P(y)$ as a sum of periodic polynomial components within each chamber, the authors achieve polynomial-time or fixed-parameter tractable algorithms in key regimes (e.g., fixed co-dimension or fixed parametric dimension), and connect this representation to Ehrhart theory. The methodology leverages Brion’s theorem, polyhedral valuations, and dual-type analyses to derive efficient preprocessing and query-time complexities, with explicit bounds involving structural parameters like $\Delta$, $\nu$, $\mu$, and chamber counts. The results have practical implications for compiler optimization and related counting problems, offering faster evaluation of $|P_y\cap \mathbb{Z}^{n_x}|$ in realistic parameter ranges. Overall, the paper unifies and extends known Ehrhart-type representations under a broader, computation-friendly framework.
Abstract
In this paper, we consider the counting function $E_P(y) = |P_{y} \cap Z^{n_x}|$ for a parametric polyhedron $P_{y} = \{x \in R^{n_x} \colon A x \leq b + B y\}$, where $y \in R^{n_y}$. We give a new representation of $E_P(y)$, called a \emph{piece-wise step-polynomial with periodic coefficients}, which is a generalization of piece-wise step-polynomials and integer/rational Ehrhart's quasi-polynomials. It gives the fastest way to calculate $E_P(y)$ in certain scenarios. The most important cases are the following: 1) We show that, for the parametric polyhedron $P_y$ defined by a standard-form system $A x = y,\, x \geq 0$ with a fixed number of equalities, the function $E_P(y)$ can be represented by a polynomial-time computable function. In turn, such a representation of $E_P(y)$ can be constructed by an $poly\bigl(n, \|A\|_{\infty}\bigr)$-time algorithm; 2) Assuming again that the number of equalities is fixed, we show that integer/rational Ehrhart's quasi-polynomials of a polytope can be computed by FPT-algorithms, parameterized by sub-determinants of $A$ or its elements; 3) Our representation of $E_P$ is more efficient than other known approaches, if $A$ has bounded elements, especially if it is sparse in addition. Additionally, we provide a discussion about possible applications in the area of compiler optimization. In some "natural" assumptions on a program code, our approach has the fastest complexity bounds.