On the extension complexity of combinatorial polytopes

In this paper we extend recent results of Fiorini et al. on the extension complexity of the cut polytope and related polyhedra. We first describe a lifting argument to show exponential extension complexity for a number of NP-complete problems including subset-sum and three dimensional matching. We then obtain a relationship between the extension complexity of the cut polytope of a graph and that of its graph minors. Using this we are able to show exponential extension complexity for the cut polytope of a large number of graphs, including those used in quantum information and suspensions of cubic planar graphs.Comment: 15 pages, 3 figures, 2 table

Drawing graphs with vertices and edges in convex position

A graph has strong convex dimension $2$, if it admits a straight-line drawing in the plane such that its vertices are in convex position and the midpoints of its edges are also in convex position. Halman, Onn, and Rothblum conjectured that graphs of strong convex dimension $2$ are planar and therefore have at most $3n-6$ edges. We prove that all such graphs have at most $2n-3$ edges while on the other hand we present a class of non-planar graphs of strong convex dimension $2$. We also give lower bounds on the maximum number of edges a graph of strong convex dimension $2$ can have and discuss variants of this graph class. We apply our results to questions about large convexly independent sets in Minkowski sums of planar point sets, that have been of interest in recent years.Comment: 15 pages, 12 figures, improved expositio

Extension complexities of Cartesian products involving a pyramid

It is an open question whether the linear extension complexity of the Cartesian product of two polytopes P, Q is the sum of the extension complexities of P and Q. We give an affirmative answer to this question for the case that one of the two polytopes is a pyramid.Comment: 5 page