Parametric polyhedra with at least k lattice points: Their semigroup structure and the k-Frobenius problem

Abstract

Given an integral d × n matrix A, the well-studied affine semigroup $\mathrm{Sg}(A) =\{ b: Ax = b,\ x \in \mathbb{Z}^{n},x \geq 0\}$ can be stratified by the number of lattice points inside the parametric polyhedra P _A (b) = { x: Ax = b, x ≥ 0}. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra, and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset Sg_≥ k (A) of all vectors $b \in \mathrm{Sg}(A)$ such that $P_{A}(b) \cap \mathbb{Z}^{n}$ has at least k solutions. We demonstrate that this set is finitely generated, it is a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computations. Related results can be derived for those right-hand-side vectors b for which $P_{A}(b) \cap \mathbb{Z}^{n}$ has exactly k solutions or fewer than k solutions. (2) A computational complexity theory. We show that, when n, k are fixed natural numbers, one can compute in polynomial time an encoding of $\mathrm{Sg}_{\geq k}(A)$ as a multivariate generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors of bounded norm that have at least k solutions. (3) Applications and computation for the k-Frobenius numbers. Using generating functions we prove that for fixed n, k the k-Frobenius number can be computed in polynomial time. This generalizes a well-known result for k = 1 by R. Kannan. Using some adaptation of dynamic programming we show some practical computations of k-Frobenius numbers and their relatives.