A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing

Motivated by an open problem from graph drawing, we study several partitioning problems for line and hyperplane arrangements. We prove a ham-sandwich cut theorem: given two sets of n lines in R^2, there is a line l such that in both line sets, for both halfplanes delimited by l, there are n^{1/2} lines which pairwise intersect in that halfplane, and this bound is tight; a centerpoint theorem: for any set of n lines there is a point such that for any halfplane containing that point there are (n/3)^{1/2} of the lines which pairwise intersect in that halfplane. We generalize those results in higher dimension and obtain a center transversal theorem, a same-type lemma, and a positive portion Erdos-Szekeres theorem for hyperplane arrangements. This is done by formulating a generalization of the center transversal theorem which applies to set functions that are much more general than measures. Back to Graph Drawing (and in the plane), we completely solve the open problem that motivated our search: there is no set of n labelled lines that are universal for all n-vertex labelled planar graphs. As a side note, we prove that every set of n (unlabelled) lines is universal for all n-vertex (unlabelled) planar graphs

Design, implementation and testing of an integrated branch and bound algorithm for piecewise linear and discrete programming problems within an LP framework

A number of discrete variable representations are well accepted and find regular use within LP systems. These are Binary variables, General Integer variables, Variable Upper Bounds or Semi Continuous variables, Special Ordered Sets of type One and type Two. The FortLP system has been extended to include these representations. A Branch and Bound algorithm is designed in which the choice of sub-problems and branching variables are kept general. This provides considerable scope of experimentation with tree development heuristics and the tree search can then be guided by search parameters specified by user subroutines. The data structures for representing the variables and the definition of the branch and bound tree are described. The results of experimental investigation for a few test problems are reported

The capacity of multilevel threshold functions

Lower and upper bounds for the capacity of multilevel threshold elements are estimated, using two essentially different enumeration techniques. It is demonstrated that the exact number of multilevel threshold functions depends strongly on the relative topology of the input set. The results correct a previously published estimate and indicate that adding threshold levels enhances the capacity more than adding variables

Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement

We report new results and generalizations of our work on unextendible product bases (UPB), uncompletable product bases and bound entanglement. We present a new construction for bound entangled states based on product bases which are only completable in a locally extended Hilbert space. We introduce a very useful representation of a product basis, an orthogonality graph. Using this representation we give a complete characterization of unextendible product bases for two qutrits. We present several generalizations of UPBs to arbitrary high dimensions and multipartite systems. We present a sufficient condition for sets of orthogonal product states to be distinguishable by separable superoperators. We prove that bound entangled states cannot help increase the distillable entanglement of a state beyond its regularized entanglement of formation assisted by bound entanglement.Comment: 24 pages RevTex, 15 figures; appendix removed, several small corrections, to appear in Comm. Math. Phy