The random simplex algorithm for linear programming proceeds as follows: at each step, it moves from a vertex u of the polytope to a randomly chosen neighbor of u, the random choice being made from those neighbors of u that improve the objective function. We exhibit a polytope defined by n constraints in three dimensions with height O(log n), for which the expected running time of the random si...

We ask several questions on the structure of the polytope of doubly stochastic n n matrices Pn, known as a Birkhoo polytope. We discuss the volume of Pn, the work of the simplex method, and the mixing of random walks Pn.

We present a multivariate generating function for all n × n nonnegative integral matrices with all row and column sums equal to a positive integer t , the so called semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope Bn of n×n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for t...

Let K be a convex body in Rd and Kt its floating bodies. There is a polytope that satisfies Kt ⊂ Pn ⊂ K and has at most n vertices, where n ≤ e vold(K \Kt) t vold(B d 2 ) . Let Kt be the illumination bodies of K and Qn a polytope that contains K and has at most n (d−1)-dimensional faces. Then vold(K t \K) ≤ cd vold(Qn \K), where n ≤ c dt vold(K t \K).

We study the quadratic assignment problem (with n variables) from a polyhedral point of view by considering the quadratic assignment polytope that is defined as the convex hull of the solutions of the linearized problem (with n + 2 n 2 n −1 ( ) variables). We give the dimension of the polytope and a minimal description of its affine hull. We also propose a family of facets with a separation alg...

We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a strict Cayley polytope if n ≥ 2d+ 1. This gives a sharp answer, for this class of polytopes, to a question raised by V. V. Batyrev and B. Nill.

The polytope of pointed pseudo triangulations was described in [RSS01]. This polytope is a combinatorial tool to observe all possible pseudo triangulations of a certain point set. Each point of the polytope refers to one possible pseudo triangulation of the point set. To polytope vertices are connected by an edge if their pseudo triangulations just differ in one edge-flip. Once we have a PPT-po...

Following the seminal work of Padberg on the Boolean quadric polytope BQP and its LP relaxation BQPLP , we consider a natural extension: SATP and SATPLP polytopes, with BQPLP being projection of the SATPLP face (and BQP projection of the SATP face). We consider the problem of integer recognition: determine whether the maximum of a linear objective function is achieved at an integral vertex of a...

The Birkhoff polytope is defined to be the convex hull of permutation matrices, Pσ ∀σ ∈ Sn. We define a second-order permutation matrix P [2] σ in R ×n corresponding to a permutation σ as (P [2] σ )ij,kl = (Pσ)ij(Pσ)kl. We call the convex hull of the second-order permutation matrices, the second-order Birkhoff polytope and denote it by B. It can be seen that B is isomorphic to the QAP-polytope,...

We propose a simple formula for the coordinates of the vertices of the Stasheff polytope (associahedron) and we compare it to the permutohedron. Introduction. The Stasheff polytope Kn, also called associahedron, appeared in the sixties in the work of Jim Stasheff [St1] on the recognition of loop spaces. It is a convex polytope of dimension n with one vertex for each planar binary tree with n + ...