Computational Combinatorics

Month: March, 2013

3-Uniform Friendship Hypergraphs

A very brief welcome to EXCILL2 participants. Thanks for visiting!

Today we discuss On a question of Sós about 3-uniform friendship hypergraphs by Hartke and Vandenbussche. These authors presented several new examples of friendship hypergraphs exhibiting a property that was not known to be possible. Their method applies careful use of integer programming as a black box. First, a careful construction of an integer linear program discovered some examples (and non-existence of examples) for small orders. Then, by making some symmetry assumptions, they constructed some larger integer programs that found larger examples, hinting towards an infinite family. The existence of larger examples remains open, although Lia, van Reesa, Seoa, and Singhi recently proved some structure theorems that apply to the examples found by Hartke and Vandenbussche.
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A Branch-and-Cut Strategy for the Manickam-Miklós-Singhi Conjecture

Stephen Hartke and I recently uploaded our paper, A Branch-and-Cut Strategy for the Manickam-Miklós-Singhi Conjecture to the arXiv. We build a computational method to verify a conjecture involving the number of nonnegative k-sums in a nonnegative sum of n real numbers. With this method, we proved a stronger statement than the conjecture for all k\leq 7 and formed a stronger conjecture.

The method makes use of linear programming techniques and software, so begins our discussion of linear and integer programming methods. Read on to discover the method and also to figure out what is going on with the picture below.
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