48 research outputs found
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 , 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 are planar and therefore have at
most edges. We prove that all such graphs have at most edges
while on the other hand we present a class of non-planar graphs of strong
convex dimension . We also give lower bounds on the maximum number of edges
a graph of strong convex dimension 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
Extension Complexity, MSO Logic, and Treewidth
We consider the convex hull P_phi(G) of all satisfying assignments of a given MSO_2 formula phi on a given graph G. We show that there exists an extended formulation of the polytope P_phi(G) that can be described by f(|phi|,tau)*n inequalities, where n is the number of vertices in G, tau is the treewidth of G and f is a computable function depending only on phi and tau.
In other words, we prove that the extension complexity of P_phi(G) is linear in the size of the graph G, with a constant depending on the treewidth of G and the formula phi. This provides a very general yet very simple meta-theorem about the extension complexity of polytopes related to a wide class of problems and graphs
On Computing the Vertex Centroid of a Polyhedron
Let be an -polytope in with vertex
set . The vertex centroid is defined as the average of the vertices in .
We prove that computing the vertex centroid of an -polytope is
#P-hard. Moreover, we show that even just checking whether the vertex centroid
lies in a given halfspace is already #P-hard for -polytopes. We
also consider the problem of approximating the vertex centroid by finding a
point within an distance from it and prove this problem to be
#P-easy by showing that given an oracle for counting the number of vertices of
an -polytope, one can approximate the vertex centroid in
polynomial time. We also show that any algorithm approximating the vertex
centroid to \emph{any} ``sufficiently'' non-trivial (for example constant)
distance, can be used to construct a fully polynomial approximation scheme for
approximating the centroid and also an output-sensitive polynomial algorithm
for the Vertex Enumeration problem. Finally, we show that for unbounded
polyhedra the vertex centroid can not be approximated to a distance of
for any fixed constant