We deterministically construct quasi-polynomial weights in quasi-polynomial time, such that in a given polytope with totally unimodular constraints, one vertex is isolated, i.e., there is a unique minimum weight vertex. More precisely, the property that we need is that every face of the polytope lies in an affine space defined by a totally unimodular matrix. This derandomizes the famous Isolati...

We consider three well-studied polyhedral relaxations for the maximum cut problem: the metric polytope of the complete graph, the metric polytope of a general graph, and the relaxation of the bipartite subgraph polytope. The metric polytope of the complete graph can be described with a polynomial number of inequalities, while the latter two may require exponentially many constraints. We give an...

This paper describes the polytope Pk;N of i-star counts, for all i 6 k, for graphs on N nodes. The vertices correspond to graphs that are regular or as regular as possible. For even N the polytope is a cyclic polytope, and for odd N the polytope is well-approximated by a cyclic polytope. As N goes to infinity, Pk;N approaches the convex hull of the moment curve. The affine symmetry group of Pk;...

Given a set N of items and a capacity b 2 IN, and let N j be the set of items with weight j, 1 j b. The 0/1 knapsack polytope is the convex hull of all 0/1 vectors that satisfy the inequality b X j=1 X i2N j jx i b: In this paper we rst present a complete linear description of the 0/1 knapsack polytope for two special cases: (a) N j = ; for all 1 < j b b 2 c and (b) N j = ; for all 1 < j b b 3 ...

Given an undirected node-weighted graph and a positiveinteger k, the maximum k-colorable subgraph problem is to select ak-colorable induced subgraph of largest weight. The natural integerprogramming formulation for this problem exhibits a high degree ofsymmetry which arises by permuting the color classes. It is well knownthat such symmetry has negative effects on the perform...

The max-cut problem and the associated cut polytope on complete graphs have been extensively studied over the last 25 years. However, little research has been conducted for the cut polytope on arbitrary graphs. In this study we describe new separation and lifting procedures for the cut polytope on such graphs. These procedures exploit algorithmic and structural results known for the cut polytop...

A sweep-plane algorithm of Lawrence for convex polytope computation is adapted to generate random tuples on simple polytopes. In our method an affine hyperplane is swept through the given polytope until a random fraction (sampled from a proper univariate distribution) of the volume of the polytope is covered. Then the intersection of the plane with the polytope is a simple polytope with smaller...

In this paper we study the structure of the k-assignment polytope, whose vertices are the m× n (0,1)-matrices with exactly k 1:s and at most one 1 in each row and each column. This is a natural generalisation of the Birkhoff polytope and many of the known properties of the Birkhoff polytope are generalised. A representation of the faces by certain bipartite graphs is given. This tool is used to...

In Rothvoÿ [2011] it was shown that there exists a 0/1 polytope (a polytope whose vertices are in {0, 1}) such that any higherdimensional polytope projecting to it must have 2 facets, i.e., its linear extension complexity is exponential. The question whether there exists a 0/1 polytope with high PSD extension complexity was left open. We answer this question in the a rmative by showing that the...