40 research outputs found
Extended Formulations for Packing and Partitioning Orbitopes
We give compact extended formulations for the packing and partitioning
orbitopes (with respect to the full symmetric group) described and analyzed in
(Kaibel and Pfetsch, 2008). These polytopes are the convex hulls of all
0/1-matrices with lexicographically sorted columns and at most, resp. exactly,
one 1-entry per row. They are important objects for symmetry reduction in
certain integer programs.
Using the extended formulations, we also derive a rather simple proof of the
fact that basically shifted-column inequalities suffice in order to describe
those orbitopes linearly.Comment: 16 page
On largest volume simplices and sub-determinants
We show that the problem of finding the simplex of largest volume in the
convex hull of points in can be approximated with a factor
of in polynomial time. This improves upon the previously best
known approximation guarantee of by Khachiyan. On the other hand,
we show that there exists a constant such that this problem cannot be
approximated with a factor of , unless . % This improves over the
inapproximability that was previously known. Our hardness result holds
even if , in which case there exists a \bar c\,^{d}-approximation
algorithm that relies on recent sampling techniques, where is again a
constant. We show that similar results hold for the problem of finding the
largest absolute value of a subdeterminant of a matrix
Extension complexity of stable set polytopes of bipartite graphs
The extension complexity of a polytope is the minimum
number of facets of a polytope that affinely projects to . Let be a
bipartite graph with vertices, edges, and no isolated vertices. Let
be the convex hull of the stable sets of . It is easy to
see that . We improve
both of these bounds. For the upper bound, we show that is , which is an improvement when
has quadratically many edges. For the lower bound, we prove that
is when is the
incidence graph of a finite projective plane. We also provide examples of
-regular bipartite graphs such that the edge vs stable set matrix of
has a fooling set of size .Comment: 13 pages, 2 figure