Intermediate Sums on Polyhedra: Computation and Real Ehrhart Theory

Abstract

We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvi-nok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449–1466]. For a given semi-rational polytope p and a rational subspace L, we integrate a given polyno-mial function h over all lattice slices of the polytope p parallel to the subspace L and sum up the integrals. We first develop an al-gorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the result as a function of a dilation of the polytope. We provide an algorithm to compute the resulting Ehrhart quasi-polynomials in the form of explicit step polynomi-als. These formulas are naturally valid for real (not just integer) dilations and thus provide a direct approach to real Ehrhart theory

Three Ehrhart quasi-polynomials

“…Such polynomial time algorithms were the initial motivation for the study of these intermediate sums in our papers. However, in the present paper, in contrast to our previous papers [2,3,5], we suppress detailed statements of such algorithms and their complexity.…”

“…When L = V , S L (p, h) is just the integral I(p, h), while for L = {0}, we recover the discrete sum S(p, h). In the present study, we generalize Barvinok's ideas in several ways, building on our previous work in [3,5,4].…”

“…The present article is the culmination of a study based on [4,5]. These two articles were devoted to the properties of intermediate generating functions only for polyhedral cones.…”

Centerpoints: A Link between Optimization and Convex Geometry

“…All our algorithms discussed in this section are under the standard Turing machine model of computation. We say that x ∈ S is an ǫ-centerpoint for S, µ, if f µ (x) ≥ F (S, µ) − ǫ where F (S, µ) is defined in (2) and f µ is defined in (3).…”

“…For n = 0, SAMPLE can be implemented for the uniform measure on polytopes using well-studied techniques, e.g., see Vempala's survey [43]. For n ≥ 1, SAMPLE can be implemented for the uniform measure on mixed-integer points in a polytope by adapting a result of Igor Pak [37] on d = 0 to d ≥ 1 and using results on computing mixed-integer volumes in polynomial time for fixed dimensions [3].…”