Computational determination of the largest lattice polytope diameter

A lattice (d, k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Let {\delta}(d, k) be the largest diameter over all lattice (d, k)-polytopes. We develop a computational framework to determine {\delta}(d, k) for small instances. We show that {\delta}(3, 4) = 7 and {\delta}(3, 5) = 9; that is, we verify for (d, k) = (3, 4) and (3, 5) the conjecture whereby {\delta}(d, k) is at most (k + 1)d/2 and is achieved, up to translation, by a Minkowski sum of lattice vectors

Berge Sorting

In 1966, Claude Berge proposed the following sorting problem. Given a string of $n$ alternating white and black pegs on a one-dimensional board consisting of an unlimited number of empty holes, rearrange the pegs into a string consisting of $\lceil\frac{n}{2}\rceil$ white pegs followed immediately by $\lfloor\frac{n}{2}\rfloor$ black pegs (or vice versa) using only moves which take 2 adjacent pegs to 2 vacant adjacent holes. Avis and Deza proved that the alternating string can be sorted in $\lceil\frac{n}{2}\rceil$ such {\em Berge 2-moves} for $n\geq 5$. Extending Berge's original problem, we consider the same sorting problem using {\em Berge $k$-moves}, i.e., moves which take $k$ adjacent pegs to $k$ vacant adjacent holes. We prove that the alternating string can be sorted in $\lceil\frac{n}{2}\rceil$ Berge 3-moves for $n\not\equiv 0\pmod{4}$ and in $\lceil\frac{n}{2}\rceil+1$ Berge 3-moves for $n\equiv 0\pmod{4}$, for $n\geq 5$. In general, we conjecture that, for any $k$ and large enough $n$, the alternating string can be sorted in $\lceil\frac{n}{2}\rceil$ Berge $k$-moves. This estimate is tight as $\lceil\frac{n}{2}\rceil$ is a lower bound for the minimum number of required Berge $k$-moves for $k\geq 2$ and $n\geq 5$.Comment: 10 pages, 2 figure

Hyperplane Arrangements with Large Average Diameter

The largest possible average diameter of a bounded cell of a simple hyperplane arrangement is conjectured to be not greater than the dimension. We prove that this conjecture holds in dimension 2, and is asymptotically tight in fixed dimension. We give the exact value of the largest possible average diameter for all simple arrangements in dimension 2, for arrangements having at most the dimension plus 2 hyperplanes, and for arrangements having 6 hyperplanes in dimension 3. In dimension 3, we give lower and upper bounds which are both asymptotically equal to the dimension

How many double squares can a string contain?

Counting the types of squares rather than their occurrences, we consider the problem of bounding the number of distinct squares in a string. Fraenkel and Simpson showed in 1998 that a string of length n contains at most 2n distinct squares. Ilie presented in 2007 an asymptotic upper bound of 2n - Theta(log n). We show that a string of length n contains at most 5n/3 distinct squares. This new upper bound is obtained by investigating the combinatorial structure of double squares and showing that a string of length n contains at most 2n/3 double squares. In addition, the established structural properties provide a novel proof of Fraenkel and Simpson's result.Comment: 29 pages, 20 figure

A primal-simplex based Tardos' algorithm

In the mid-eighties Tardos proposed a strongly polynomial algorithm for solving linear programming problems for which the size of the coefficient matrix is polynomially bounded by the dimension. Combining Orlin's primal-based modification and Mizuno's use of the simplex method, we introduce a modification of Tardos' algorithm considering only the primal problem and using simplex method to solve the auxiliary problems. The proposed algorithm is strongly polynomial if the coefficient matrix is totally unimodular and the auxiliary problems are non-degenerate.Comment: 7 page

Primitive Zonotopes

We introduce and study a family of polytopes which can be seen as a generalization of the permutahedron of type $B_d$. We highlight connections with the largest possible diameter of the convex hull of a set of points in dimension $d$ whose coordinates are integers between $0$ and $k$, and with the computational complexity of multicriteria matroid optimization.Comment: The title was slightly modified, and the determination of the computational complexity of multicriteria matroid optimization was adde

On Skeletons, Diameters and Volumes of Metric Polyhedra

We survey and present new geometric and combinatorial properties of some polyhedra with application in combinatorial optimization, for example, the max-cut and multicommodity flow problems. Namely we consider the volume, symmetry group, facets, vertices, face lattice, diameter, adjacency and incidence relations and connectivity of the metric polytope and its relatives. In particular, using its large symmetry group, we completely describe all the 13 orbits which form the 275 840 vertices of the 21-dimensional metric polytope on 7 nodes and their incidence and adjacency relations. The edge connectivity, the i-skeletons and a lifting procedure valid for a large class of vertices of the metric polytope are also given. Finally, we present an ordering of the facets of a polytope, based on their adjacency relations, for the enumeration of its vertices by the double description method