On the Number of Latent Subsets of Intersecting Collections

Not AvailableSupported in part by O.N.R. Contract N00014-67-A-0204-0016 and supported in part by the U.S. Army Research Office (Durham) under contract DAHCO4-70-C-0058

A Note on Chvátal's Conjecture

The main purpose of this thesis is to treat and clarify two results found by Hunter Snevily on the open problem of Chvátal's conjecture. Chvátal's conjecture states that a certain class of sets, called ideals, always have a maximum intersecting family of subsets, where all members contain a common element. Such a family of subsets is called a star. Snevily presents his results in an article, the structure of which will be followed when presenting and proving both his and earlier acquired results, constituting the basis of his conclusions

Efficient Identification of Equivalences in Dynamic Graphs and Pedigree Structures

We propose a new framework for designing test and query functions for complex structures that vary across a given parameter such as genetic marker position. The operations we are interested in include equality testing, set operations, isolating unique states, duplication counting, or finding equivalence classes under identifiability constraints. A motivating application is locating equivalence classes in identity-by-descent (IBD) graphs, graph structures in pedigree analysis that change over genetic marker location. The nodes of these graphs are unlabeled and identified only by their connecting edges, a constraint easily handled by our approach. The general framework introduced is powerful enough to build a range of testing functions for IBD graphs, dynamic populations, and other structures using a minimal set of operations. The theoretical and algorithmic properties of our approach are analyzed and proved. Computational results on several simulations demonstrate the effectiveness of our approach.Comment: Code for paper available at http://www.stat.washington.edu/~hoytak/code/hashreduc