20 research outputs found
The unreasonable ubiquitousness of quasi-polynomials
A function , with domain the natural numbers, is a quasi-polynomial if there exists a period and polynomials such that for . Quasi-polynomials classically – and ``reasonably'' – appear in Ehrhart theory and in other contexts where one examines a family of polyhedra, parametrized by a variable t, and defined by linear inequalities of the form . Recent results of Chen, Li, Sam; Calegari, Walker; and Roune, Woods show a quasi-polynomial structure in several problems where the are also allowed to vary with . We discuss these ``unreasonable'' results and conjecture a general class of sets that exhibit various (eventual) quasi-polynomial behaviors: sets that are defined with quantifiers , boolean operations (and, or, not), and statements of the form , where and are polynomials in . These sets are a generalization of sets defined in the Presburger arithmetic. We prove several relationships between our conjectures, and we prove several special cases of the conjectures
Eventual Quasi-Linearity of The Minkowski Length
The Minkowski length of a lattice polytope PP is a natural generalization of the lattice diameter of PP. It can be defined as the largest number of lattice segments whose Minkowski sum is contained in PP. The famous Ehrhart theorem states that the number of lattice points in the positive integer dilates tPtP of a lattice polytope PP behaves polynomially in t∈Nt∈N. In this paper we prove that for any lattice polytope PP, the Minkowski length of tPtP for t∈Nt∈N is eventually a quasi-polynomial with linear constituents. We also give a formula for the Minkowski length of coordinates boxes, degree one polytopes, and dilates of unimodular simplices. In addition, we give a new bound for the Minkowski length of lattice polygons and show that the Minkowski length of a lattice triangle coincides with its lattice diameter
Parametric Presburger Arithmetic: Complexity of Counting and Quantifier Elimination
We consider an expansion of Presburger arithmetic which allows multiplication
by parameters . A formula in this language defines a
parametric set as varies
in , and we examine the counting function as a
function of . For a single parameter, it is known that can
be expressed as an eventual quasi-polynomial (there is a period such that,
for sufficiently large , the function is polynomial on each of the residue
classes mod ). We show that such a nice expression is impossible with 2 or
more parameters. Indeed (assuming \textbf{P} \textbf{NP}) we construct a
parametric set such that is not even
polynomial-time computable on input . In contrast, for parametric
sets with arbitrarily many parameters,
defined in a similar language without the ordering relation, we show that
is always polynomial-time computable in the size of
, and in fact can be represented using the gcd and similar
functions.Comment: 14 pages, 1 figur
Parametric Presburger arithmetic: complexity of counting and quantifier elimination
We consider an expansion of Presburger arithmetic which allows multiplication by k parameters t1,…,tk. A formula in this language defines a parametric set St⊆Zd as t varies in Zk, and we examine the counting function |St| as a function of t. For a single parameter, it is known that |St| can be expressed as an eventual quasi‐polynomial (there is a period m such that, for sufficiently large t, the function is polynomial on each of the residue classes mod m). We show that such a nice expression is impossible with 2 or more parameters. Indeed (assuming P≠NP) we construct a parametric set St1,t2 such that |St1,t2| is not even polynomial‐time computable on input (t1,t2). In contrast, for parametric sets St⊆Zd with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that |St| is always polynomial‐time computable in the size of t, and in fact can be represented using the gcd and similar functions.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/151911/1/malq201800068_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/151911/2/malq201800068.pd