A note on palindromic δ-vectors for certain rational polytopes

Mathematical Programming, 2018

In this paper, we study the following problem: given a polytope P with Chvátal rank 1, does P contain an integer point? Boyd and Pulleyblank observed that this problem is in the complexity class NP ∩ co-NP, an indication that it is probably not NP-complete. We solve this problem in polynomial time for polytopes arising from the satisfiability problem of a given formula with at least three literals per clause, for simplices whose integer hull can be obtained by adding at most a constant number of Chvátal inequalities, and for rounded polytopes. We prove that any closed convex set whose Chvátal closure is empty has an integer width of at most n, and we give an example showing that this bound is tight within an additive constant of 1. The promise that a polytope has Chvátal rank 1 seems hard to verify though. We prove that deciding emptiness of the Chvátal closure of a given rational polytope P is NP-complete, even when P is contained in the unit hypercube or is a rational simplex, and even when P does not contain any integer point. This has two implications: (i) It is NP-hard to decide whether a given rational polytope P has Chvátal rank 1, even when P is contained in the unit cube or is a rational simplex; (ii) The optimization and separation problems over the Chvátal closure of a given rational polytope contained in the unit hypercube or of a given rational simplex are NP-hard. These results improve earlier complexity results of Cornuéjols and Li and Eisenbrand. Finally, we prove that, for any positive integer k, it is NP-hard to decide whether adding at most k Chvátal inequalities is sufficient to describe the integer hull of a given rational polytope.

Discrete and Computational Geometry, 2004

Given A ∈ Z m×n and b ∈ Z m , we consider the integer program max{c x|Ax = b; x ∈ N n } and provide an equivalent and explicit linear program max{ c q|Mq = r; q ≥ 0}, where M, r, c are easily obtained from A, b, c with no calculation. We also provide an explicit algebraic characterization of the integer hull of the convex polytope P = {x ∈ R n |Ax = b; x ≥ 0}. All strong valid inequalities can be obtained from the generators of a convex cone whose definition is explicit in terms of M.

IOSR Journal of Mathematics, 2012

The aim of this work is to find the Ehrhart polynomial for a certain type of a convex polytope and the Ehrhart polynomial for the dual of these polytopes together with a comparison between them; two theorems that related with the number of lattice points and the volume are also given. Different examples are presented in order to demonstrate our results.

Hibi showed that the polynomials in the numerator of the Ehrhart series of a reflexive polytope are palindromic. We proved that those in the numerator of the Ehrhart series of the the special polytope of completely symmetric reflexive polytope (defined in section2) are palindromic. From this, one of the conjectures (raised in the A205497 of OEIS \cite{[O]}) follows immediately.

2017

i=1 σ i v : v ∈ V (σ) , where V (σ) is the set of vertices v = (x 1 ,. .. , x d) of Π n that satisfy the following property: for each cycle σ j = (j 1 j 2 • • • j r) of σ, x j 1 ,. .. , x jr is a consecutive sequence of integers.

Journal of Symbolic Computation, 2004

This paper discusses algorithms and software for the enumeration of all lattice points inside a rational convex polytope: we describe LattE, a computer package for lattice point enumeration which contains the first implementation of A. Barvinok's algorithm (Math. Oper. Res. 19 (1994)

2012

We describe an algorithm for determining whether two convex polytopes P and Q, embedded in a lattice, are isomorphic with respect to a lattice automorphism. We extend this to a method for determining if P and Q are equivalent, i.e. whether there exists an affine lattice automorphism that sends P to Q. Methods for calculating the automorphism group and affine automorphism group of P are also described. An alternative strategy is to determine a normal form such that P and Q are isomorphic if and only if their normal forms are equal. This is the approach adopted by Kreuzer and Skarke in their PALP software. We describe the Kreuzer–Skarke method in detail, and give an improved algorithm when P has many symmetries. Numerous examples, plus two appendices containing detailed pseudo-code, should help with any future reimplementations of these techniques. We conclude by explaining how to define and calculate the normal form of a Laurent polynomial.

2007

Quasi-period collapse occurs when the Ehrhart quasi-polynomial of a rational polytope has a quasi-period less than the denominator of that polytope. This phenomenon is poorly understood, and all known cases in which it occurs have been proven with ad hoc methods. In this note, we present a conjectural explanation for quasi-period collapse in rational polytopes. We show that this explanation applies to some previous cases appearing in the literature. We also exhibit examples of Ehrhart polynomials of rational polytopes that are not the Ehrhart polynomials of any integral polytope.

Advances in Mathematics, 2009

Let L be a lattice (that is, a Z-module of finite rank), and let L = P(L) denote the family of convex polytopes with vertices in L; here, convexity refers to the underlying rational vector space V = Q ⊗ L. In this paper it is shown that any valuation on L satisfies the inclusion-exclusion principle, in the strong sense that appropriate extension properties of the valuation hold. Indeed, the core result is that the class of a lattice polytope in the abstract group L = P(L) for valuations on L can be identified with its characteristic function in V. In fact, the same arguments are shown to apply to P(M), when M is a module of finite rank over an ordered ring, and more generally to appropriate families of (not necessarily bounded) polyhedra.

European Journal of Combinatorics, 2008

In a recent paper, Karpenkov has classified the lattice polytopes (that is, with vertices in the integer lattice Z d) which are regular with respect to those affinities which preserve the lattice. An alternative approach is adopted in this paper. For each regular polytope P in euclidean space E d , those lattices Λ are classified which are compatible with P, in the sense that some translate of Λ contains the vertices of P, and this translate is preserved by the symmetries of P.