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Diversities and the Geometry of Hypergraphs

Abstract : The embedding of finite metrics in 1 has become a fundamental tool for both combinatorial optimization and large-scale data analysis. One important application is to network flow problems as there is close relation between max-flow min-cut theorems and the minimal distortion embeddings of metrics into 1. Here we show that this theory can be generalized to a larger set of combinatorial optimization problems on both graphs and hypergraphs. This theory is not built on metrics and metric embeddings, but on diversities, a type of multi-way metric introduced recently by the authors. We explore diversity embeddings, 1 diversities, and their application to Steiner Tree Packing and Hypergraph Cut problems.
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https://hal.inria.fr/hal-01178648
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David Bryant, Paul Tupper. Diversities and the Geometry of Hypergraphs. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2014, Vol. 16 no. 2 (in progress) (2), pp.1 - 20. ⟨hal-01178648⟩

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