Let us denote by Ωn the Birkhoff polytope of n× n doubly stochastic matrices. As the Birkhoff–von Neumann theorem famously states, the vertex set of Ωn coincides with the set of all n× n permutation matrices. Here we consider a higherdimensional analog of this basic fact. Let Ω n be the polytope which consists of all tristochastic arrays of order n. These are n×n×n arrays with nonnegative entri...

Each group G of permutation matrices gives rise to a permutation polytope P(G) = cony(G) c Re×d, and for any x ~ W, an orbit polytope P(G, x) = conv(G, x). A broad subclass is formed by the Young permutation polytopes, which correspond bijectively to partitions 2 = (21, ..., 2k)~-n of positive integers, and arise from the Young representations of the symmetric group. Young polytopes provide a f...

In this paper we study enumeration problems for polytopes arising from combinatorial optimization problems. While these polytopes turn out to be quickly intractable for enumeration algorithms designed for general polytopes, tailor-made algorithms using their rich combinatorial features can exhibit strong performances. The main engine of these combinatorial algorithms is the use of the large sym...

Let $\pi:{\mathbb R}^n \to {\mathbb R}^d$ be any linear projection, let $A$ the image of standard basis. Motivated by Postnikov's study postitive Grassmannians via plabic graphs and Galashin's connection to slices zonotopal tilings 3-dimensional cyclic zonotopes, we poset subdivisions induced restriction $\pi$ $k$-th hypersimplex, for $k=1,\dots,n-1$. We show that: - For arbitrary $k\le d+1$, c...

The metric polytope metn is the polyhedron associated with all semimetrics on n nodes and defined by the triangle inequalities xij − xik − xjk ≤ 0 and xij + xik + xjk ≤ 2 for all triples i, j, k of {1, . . . , n}. In 1992 Monique Laurent and Svatopluk Poljak conjectured that every fractional vertex of the metric polytope is adjacent to some integral vertex. The conjecture holds for n ≤ 8 and, i...

From a rational convex polytope of dimension r ≥ 2 J.P. Hansen constructed an error correcting code of length n = (q−1)r over the finite field Fq. A rational convex polytope is the same datum as a normal toric variety and a Cartier divisor. The code is obtained evaluating rational functions of the toric variety defined by the polytope at the algebraic torus, and it is an evaluation code in the ...

Sherali and Adams (1990), Lovász and Schrijver (1991) and, recently, Lasserre (2001) have constructed hierarchies of successive linear or semidefinite relaxations of a 0− 1 polytope P ⊆ Rn converging to P in n steps. Lasserre’s approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments. We present the three methods in a common elementar...

We study the Lovász-Schrijver SDP-operator applied to the fractional stable set polytope of graphs. The problem of obtaining a combinatorial characterization of graphs for which the SDP-operator generates the stable set polytope in one step has been open since 1990. In an earlier publication, we named these graphs N+-perfect. In the current contribution, we propose a conjecture on combinatorial...

It is proved that a certain symmetric sequence (h0, h1, . . . , hd) of nonnegative integers arising in the enumeration of magic squares of given size n by row sums or, equivalently, in the generating function of the Ehrhart polynomial of the polytope of doubly stochastic n × n matrices, is equal to the h-vector of a simplicial polytope and hence that it satisfies the conditions of the g-theorem...

The metric polytope metn is the polyhedron associated with all semimetrics on n nodes and defined by the triangle inequalities xij − xik − xjk ≤ 0 and xij + xik + xjk ≤ 2 for all triples i, j, k of {1, . . . , n}. In 1992 Monique Laurent and Svatopluk Poljak conjectured that every fractional vertex of the metric polytope is adjacent to some integral vertex. The conjecture holds for n ≤ 8 and, i...