Short rational generating functions for lattice point problems

Abstract

We prove that for any fixed d the generating function of the projection of the set of integer points in a rational d-dimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice points, notably integer semigroups and (minimal) Hilbert bases of rational cones, have short rational generating functions provided certain parameters (the dimension and the number of generators) are fixed. It follows then that many computational problems for such sets (for example, finding the number of positive integers not representable as a non-negative integer combination of given coprime positive integers a 1 , . . . , a d ) admit polynomial time algorithms. We also discuss a related problem of computing the Hilbert series of a ring generated by monomials.

Integer affine transformations of parametric ℤ-polytopes and applications to loop nest optimization

“…For example, when n = 4, the number of array cells causing a dependency is given as: {4 if m = 4, 10 if m = 5, 18 if m = 6 and 5m − 7 if m ≥ 7}. Barvinok and Woods [2003] proposed an algorithm for computing the integer projection of a polytope that is theoretically polynomial-time for fixed dimension. However, no implementation of this algorithm has been reported till 2008.…”

“…Another algorithm for computing the integer projection of a polytope was first proposed by Barvinok and Woods [2003], and then extended to parametric case by Verdoolaege and Woods [2005]. Even if this algorithm is theoretically polynomial-time (for fixed dimension), we believe that it still remains unusable in practice, because it uses algorithmic flatness theory and iterated Boolean combinations of rational generating functions.…”

“…Z-polytopes [Zhao and Malik 2000], [Ramanujam et al 2001] [Barvinok 1994a], exact one parameter one Z-polytope [De Loera et al 2004] [ Clauss and Loechner 1998], exact yes one Z-polytope [Verdoolaege et al 2004], [Baldoni-Silva et al 2006] [ Barvinok and Woods 2003] exact no image of one Z-polytope [Parker and Chatterjee 2004], exact no image of a union of [Boigelot and Latour 2004], Z-polytopes [Zhu et al 2006] [ Gupta and Rajopadhye 2007] exact yes computing the image of a union of Z-polytopes [Pugh 1994], exact yes image of a union of , Z-polytopes [Lasaruk and Sturm 2007], [Köppe et al 2008] Z-polytopes, in the form of a worst-case exponential union of parametric Z-polytopes. We also compare our implementation with other work that compute the exact result, are parametric (for many parameters) and general (integer transformation of unions of Z-polytopes), and show that it is efficient and can be used to optimize real applications.…”

How to change coins, M&M's, or chicken nuggets: The linear Diophantine problem of Frobenius

“…The case d = 3 is solved algorithmically, i.e., there are efficient algorithms to compute g(a, b, c) [7,9,10], and in form of a semi-explicit formula [8,14]. The Frobenius problem for fixed d ≥ 4 has been proved to be computationally feasible [1,11], but not even an efficient practical algorithm for d = 4 is known.…”

“…Sylvester proved that exactly half of the integers between 1 and (a − 1)(b − 1) are representable (in terms of a and b). In other words, there are exactly 1 2 (a − 1)(b − 1) non-representable integers. We will also outline a proof of Sylvester's Theorem.…”