An integer point in a polyhedron is called irreducible iff it is not the midpoint of two other integer points in the polyhedron. We prove that the number of irreducible integer points in $n$-dimensional polytope with radius $k$ given by a system of $m$ linear inequalities is at most $O(m^{\lfloor\frac{n}{2}\rfloor}\log^{n-1} k)$ if $n$ is fixed. Using this result we prove the hypothesis asserting that the teaching dimension in the class of threshold functions of $k$-valued logic in $n$ variables is $Θ(\log^{n-2} k)$ for any fixed $n\ge 2$.
This paper focuses on vertices of the master corner polyhedra $P(G,g_0),$ the core of the group-theoretical approach to integer linear programming. We introduce two combinatorial operations that transform each vertex of $P(G,g_0)$ to adjacent ones. This implies that for any $P(G,g_0),$ there exists a subset of basic vertices, we call them support vertices, from which all others can be built. The class of support vertices is proved to be invariant under the automorphism group of $G,$ so this basis can be further reduced to a subset of pairwise non-equivalent support vertices. Among other result…
Several recent papers have explored families of rational polyhedra whose integer points are in bijection with certain families of numerical semigroups. One such family, first introduced by Kunz, has integer points in bijection with numerical semigroups of fixed multiplicity, and another, introduced by Hellus and Waldi, has integer points corresponding to oversemigroups of numerical semigroups with two generators. In this paper, we provide a combinatorial framework from which to study both families of polyhedra. We introduce a new family of polyhedra called group cones, each constructed from so…
In this expository article we give an introduction to Ehrhart theory, i.e., the theory of integer points in polyhedra, and take a tour through its applications in enumerative combinatorics. Topics include geometric modeling in combinatorics, Ehrhart's method for proving that a couting function is a polynomial, the connection between polyhedral cones, rational functions and quasisymmetric functions, methods for bounding coefficients, combinatorial reciprocity theorems, algorithms for counting integer points in polyhedra and computing rational function representations, as well as visualizations …
We give an optimal upper bound for the maximum-norm distance from a vertex of a knapsack polyhedron to its nearest feasible lattice point. In a randomised setting, we show that the upper bound can be significantly improved on average. As a corollary, we obtain an optimal upper bound for the additive integrality gap of integer knapsack problems and show that the integrality gap of a "typical" knapsack problem is drastically smaller than the integrality gap that occurs in a worst case scenario. We also prove that, in a generic case, the integer programming gap admits a natural optimal lower boun…
A D-polyhedron is a polyhedron $P$ such that if $x,y$ are in $P$ then so are their componentwise max and min. In other words, the point set of a D-polyhedron forms a distributive lattice with the dominance order. We provide a full characterization of the bounding hyperplanes of D-polyhedra. Aside from being a nice combination of geometric and order theoretic concepts, D-polyhedra are a unifying generalization of several distributive lattices which arise from graphs. In fact every D-polyhedron corresponds to a directed graph with arc-parameters, such that every point in the polyhedron correspon…
We prove that the problem of minimizing the number of integer points inparallel translations of a rational convex polytope in $\mathbb{R}^6$ is NP-hard. We apply this result to show that given a rational convex polytope $P \subset \mathbb{R}^6$, finding the largest integer $t$ s.t. the expansion $tP$ contains fewer than $k$ integer points is also NP-hard. We conclude that the Ehrhart quasi-polynomials of rational polytopes can have arbitrary fluctuations.
We extend the Barvinok-Woods algorithm for enumerating projections of integer points in polytopes to unbounded polyhedra. For this, we obtain a new structural result on projections of semilinear subsets of the integer lattice. We extend the results to general formulas in Presburger Arithmetic. We also give an application to the k-Frobenius problem.
This paper is a study of the interaction between the combinatorics of boundaries of convex polytopes in arbitrary dimension and their metric geometry. Let S be the boundary of a convex polytope of dimension d+1, or more generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a polyhedral nonoverlapping unfolding into R^d, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R^d by identifying pairs of boundary faces isometrically. Our existence proof exploits geodesic flow away from a source point v in S, which is the exponential map to S …
The celebrated upper bound theorem of McMullen determines the maximal number of extreme points of a polyhedron in terms of its dimension and the number of constraints which define it, showing that the maximum is attained by the polar of the cyclic polytope. We show that the same bound is valid in the tropical setting, up to a trivial modification. Then, we study the natural candidates to be the maximizing polyhedra, which are the polars of a family of cyclic polytopes equipped with a sign pattern. We construct bijections between the extreme points of these polars and lattice paths depending on…
Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern--Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.
Lattice polyhedra $Q_1$ and $Q_2$ with the same tail cone are said to be normally located if every lattice point in the Minkowski sum $Q_1+Q_2$ is the sum of lattice points from $Q_1$ and $Q_2$, respectively. We prove that if the normal fan of $Q_1$ refines the normal fan of $Q_2$, then there is a positive integer $k$ such that for any positive integer $s$ the polyhedra $skQ_1$ and $skQ_2$ are normally located. This result is based on an interpretation of the problem in terms of graded algebras and earlier results on surjectivity of the multiplicaiton map on homogeneous components. Also we pro…
We study higher-dimensional analogs of the Dedekind-Carlitz polynomials c(u,v;a,b) := sum_{k=1..b-1} u^[ka/b] v^(k-1), where u and v are indeterminates and a and b are positive integers. Carlitz proved that these polynomials satisfy the reciprocity law (v-1) c(u,v;a,b) + (u-1) c(v,u;b,a) = u^(a-1) v^(b-1) - 1, from which one easily deduces many classical reciprocity theorems for the Dedekind sum and its generalizations. We illustrate that Dedekind-Carlitz polynomials appear naturally in generating functions of rational cones and use this fact to give geometric proofs of the Carlitz reciprocity…
We prove that the sequence of the characters of the Kirillov-Reshetikhin (KR) modules $W_{m}^{(a)}, m\in \mathbb{Z}_{m\geq 0}$ associated to a node $a$ of the Dynkin diagram of a complex simple Lie algebra $\mathfrak{g}$ satisfies a linear recurrence relation except for some cases in types $E_7$ and $E_8$. To this end we use the $Q$-system and the existing lattice point summation formula for the decomposition of KR modules, known as domino removal rules when $\mathfrak{g}$ is of classical type. As an application, we show how to reduce some unproven lattice point summation formulas in exception…
In the article, a series of neigbourly polyhedra is constructed. They have $N=2d+4$ vertices and are embedded in $\mathbb R^{2d}$. Their (affine) Gale diagrams in $\mathbb R^2$ have $d+3$ black points that form a convex polygon. These Gale diagams can be enumerated using 3-trees (trees with some additional structure). Given $d$ and $m$, each of the constructed polyhedra in $\mathbb R^{2d}$ has a fixed number of faces of dimension $m$ that contain a vertex $A$. (This number depends on $d$ and $m$ does not depend on the polyhedron and the vertex $A$).
How much of the combinatorial structure of a pointed polyhedron is contained in its vertex-facet incidences? Not too much, in general, as we demonstrate by examples. However, one can tell from the incidence data whether the polyhedron is bounded. In the case of a polyhedron that is simple and "simplicial," i.e., a d-dimensional polyhedron that has d facets through each vertex and d vertices on each facet, we derive from the structure of the vertex-facet incidence matrix that the polyhedron is necessarily bounded. In particular, this yields a characterization of those polyhedra that have circul…
A convex set with nonempty interior is maximal lattice-free if it is inclusion-maximal with respect to the property of not containing integer points in its interior. Maximal lattice-free convex sets are known to be polyhedra. The precision of a rational polyhedron $P$ in $\mathbb{R}^d$ is the smallest integer $s$ such that $sP$ is an integral polyhedron. In this paper we show that, up to affine mappings preserving $\mathbb{Z}^d$, the number of maximal lattice-free rational polyhedra of a given precision $s$ is finite. Furthermore, we present the complete list of all maximal lattice-free integr…
The integer hull of a polyhedron is the convex hull of the integer points contained in it. We show that the vertices of the integer hulls of a rational family of polyhedra of size O(n) have quasipolynomial coordinates. As a corollary, we show that the stable commutator length of elements in a surgery family is a ratio of quasipolynomials, and that unit balls in the scl norm quasi-converge in finite dimensional surgery families.
Given a system of linear equations $\ell_i(x)=β_i$ in an $n$-vector $x$ of 0-1 variables, we compute the expectation of $\exp\left\{- \sum_i γ_i \left(\ell_i(x) - β_i\right)^2\right\}$, where $x$ is a vector of independent Bernoulli random variables and $γ_i >0$ are constants. The algorithm runs in quasi-polynomial $n^{O(\ln n)}$ time under some sparseness condition on the matrix of the system. The result is based on the absence of the zeros of the analytic continuation of the expectation for complex probabilities, which can also be interpreted as the absence of a phase transition in the Ising…
This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algorithm for computing the highest degree coefficients of the weighted Ehrhart quasi-polynomial for a rational simple polytope in varying dimension, when the weights of the lattice points are given by a polynomial function h. Our technique is based on a refinement of an algorithm of A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), pp. 1449--1466] in the u…
Given an integral $d \times n$ matrix $A$, the well-studied affine semigroup $\mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\}$ can be stratified by the number of lattice points inside the parametric polyhedra $P_A(b)=\{x: Ax=b, x\geq0\}$. 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 $\mbox{ Sg}_{\geq k}(A)$ of all vectors $b \in \mbox{ Sg}(A)$ such that $P_A(b) \cap {\mathbb Z}^n $ has at least $k$ solutions. …
We propose to take a look at a new approach to the study of integral polyhedra. The main idea is to give an integral representation, or matrix model representation, for the key combinatorial characteristics of integral polytopes. Based on the well-known geometric interpretations of matrix model digram techniques, we construct a new model that enumerates triangulations, subdivisions, and numbers of integral points of integral polygons. This approach allows us to look at their combinatorics from a new perspective, motivated by knowledge about matrix models and their integrability. We show how an…
