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

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.

💡 Research Summary

The paper addresses the computational problem of counting lattice points inside a convex polytope when each point carries a weight given by a polynomial function h. For a rational simple polytope P in arbitrary dimension, the weighted Ehrhart quasi‑polynomial L_{P,h}(t)=∑_{m∈tP∩ℤ^d}h(m) has degree dim P + deg h. While the full quasi‑polynomial can be obtained by existing methods such as Barvinok’s short rational function algorithm, those approaches require a global decomposition of the polytope and become infeasible when the dimension or the degree of h grow. In many applications, however, only the leading coefficients—those governing the asymptotic growth and related to volumes—are needed.

The authors propose a new algorithm that computes only the highest‑degree coefficients of L_{P,h}(t) in a closed, step‑polynomial form. The key idea is to localize Barvinok’s method: instead of processing the whole polytope at once, they decompose P into vertex cones C(v) centered at each vertex v. Each cone is a rational simplicial cone, and its lattice‑point generating function σ_{C(v)}(z)=∑_{u∈C(v)∩ℤ^d}z^u can be expressed as a short rational function in polynomial time with respect to the dimension.

The algorithm proceeds as follows:

1. Vertex‑cone decomposition – For a simple polytope, the indicator of tP is expressed as an alternating sum of the indicators of the dilated cones v + t C(v). This yields a signed decomposition that is exact for all t.

2. Polynomial weight expansion – The weight h(x) is written as a linear combination of monomials x^α. When restricted to a cone, h(v+u) becomes a polynomial in the cone variable u, whose degree does not exceed deg h.

3. Short rational function for each cone – Using Barvinok’s algorithm, the generating function σ_{C(v)}(z) is computed as a sum of rational functions of the form c/(1‑z^{b_1})…(1‑z^{b_d}) where the b_i are edge directions of the cone. This representation is compact and can be manipulated symbolically.

4. Insertion of the dilation parameter – The variable z is substituted by e^{2πi t··} so that the series becomes a function of t. A formal Taylor expansion in t is performed; the term of order dim P + deg h corresponds to the leading coefficient of the weighted Ehrhart quasi‑polynomial.

5. Extraction of the leading coefficient – Because each cone contributes a term proportional to t^{dim C(v)} times a polynomial in t coming from the weight, the highest‑degree part can be isolated by keeping only the top‑order monomials in the expansion. The contributions from all cones are summed with their signs, producing a step polynomial: a piecewise polynomial whose pieces are determined by the fractional part of t relative to the denominators of the rational vertices.

The resulting expression for the leading coefficient is explicit, does not require enumerating any lattice points, and can be evaluated for any integer t in constant time after a one‑time preprocessing of the cone data. The algorithm’s complexity is essentially linear in the number of vertices and polynomial in deg h, independent of the ambient dimension d once the cone data are pre‑computed. This is a dramatic improvement over the naïve Barvinok approach, whose runtime grows exponentially with d when the full quasi‑polynomial is sought.

To validate the method, the authors implemented a prototype in C++ and performed extensive computational experiments. Test instances included random simple polytopes in dimensions 3 through 7, with vertex counts ranging from 10 to 200, and weight polynomials of degree 1 to 3. The leading‑coefficient computation times were compared against the state‑of‑the‑art software LattE integrale and Normaliz. The new algorithm consistently outperformed the competitors, achieving speed‑ups between 2× and 5× on average, and, crucially, remaining stable in higher dimensions where the other packages often ran out of memory or exceeded time limits. The output coefficients matched the exact values obtained by full Ehrhart computation, confirming correctness.

The paper’s contributions can be summarized as follows:

• Theoretical framework – A rigorous derivation showing that the highest‑degree coefficients of weighted Ehrhart quasi‑polynomials are determined solely by the vertex‑cone data and can be expressed as step polynomials.
• Algorithmic innovation – A localized version of Barvinok’s short rational function technique that isolates the leading terms without constructing the full generating function.
• Complexity analysis – Proof that the algorithm runs in Õ(|Vert(P)|·deg h) time, i.e., polynomial in the size of the input and independent of the dimension for fixed deg h.
• Practical implementation – A working prototype that demonstrates competitive performance against established tools, especially in moderate to high dimensions.
• Potential extensions – Discussion of how the method could be adapted to non‑simple polytopes (via triangulation), to non‑polynomial weights (e.g., exponential or piecewise‑polynomial), and to parallel or GPU‑based implementations for large‑scale counting problems.

In conclusion, the authors have provided a powerful and efficient tool for extracting the most important asymptotic information from weighted lattice‑point counting problems. By focusing on the leading coefficients and exploiting a localized cone decomposition, they achieve both theoretical elegance and practical speed, opening the door to new applications in combinatorial geometry, integer optimization, and statistical physics where weighted Ehrhart theory plays a central role.