304 research outputs found
Antimatroids and Balanced Pairs
We generalize the 1/3-2/3 conjecture from partially ordered sets to
antimatroids: we conjecture that any antimatroid has a pair of elements x,y
such that x has probability between 1/3 and 2/3 of appearing earlier than y in
a uniformly random basic word of the antimatroid. We prove the conjecture for
antimatroids of convex dimension two (the antimatroid-theoretic analogue of
partial orders of width two), for antimatroids of height two, for antimatroids
with an independent element, and for the perfect elimination antimatroids and
node search antimatroids of several classes of graphs. A computer search shows
that the conjecture is true for all antimatroids with at most six elements.Comment: 16 pages, 5 figure
A Localization Approach to Improve Iterative Proportional Scaling in Gaussian Graphical Models
We discuss an efficient implementation of the iterative proportional scaling
procedure in the multivariate Gaussian graphical models. We show that the
computational cost can be reduced by localization of the update procedure in
each iterative step by using the structure of a decomposable model obtained by
triangulation of the graph associated with the model. Some numerical
experiments demonstrate the competitive performance of the proposed algorithm.Comment: 12 page
Simultaneous Representation of Proper and Unit Interval Graphs
In a confluence of combinatorics and geometry, simultaneous representations provide a way to realize combinatorial objects that share common structure. A standard case in the study of simultaneous representations is the sunflower case where all objects share the same common structure. While the recognition problem for general simultaneous interval graphs - the simultaneous version of arguably one of the most well-studied graph classes - is NP-complete, the complexity of the sunflower case for three or more simultaneous interval graphs is currently open. In this work we settle this question for proper interval graphs. We give an algorithm to recognize simultaneous proper interval graphs in linear time in the sunflower case where we allow any number of simultaneous graphs. Simultaneous unit interval graphs are much more "rigid" and therefore have less freedom in their representation. We show they can be recognized in time O(|V|*|E|) for any number of simultaneous graphs in the sunflower case where G=(V,E) is the union of the simultaneous graphs. We further show that both recognition problems are in general NP-complete if the number of simultaneous graphs is not fixed. The restriction to the sunflower case is in this sense necessary
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