Constructing and sampling partite, 3-uniform hypergraphs with given degree sequence

by András Hubai, Tamás Róbert Mezei, Ferenc Béres, András Benczúr, István Miklós

Partite, 3-uniform hypergraphs are 3-uniform hypergraphs in which each hyperedge contains exactly one point from each of the 3 disjoint vertex classes. We consider the degree sequence problem of partite, 3-uniform hypergraphs, that is, to decide if such a hypergraph with prescribed degree sequences exists. We prove that this decision problem is NP-complete in general, and give a polynomial running time algorithm for third almost-regular degree sequences, that is, when each degree in one of the vertex classes is k or k − 1 for some fixed k, and there is no restriction for the other two vertex classes. We also consider the sampling problem, that is, to uniformly sample partite, 3-uniform hypergraphs with prescribed degree sequences. We propose a Parallel Tempering method, where the hypothetical energy of the hypergraphs measures the deviation from the prescribed degree sequence. The method has been implemented and tested on synthetic and real data. It can also be applied for χ² testing of contingency tables. We have shown that this hypergraph-based χ² test is more sensitive than the standard χ² test. The extra sensitivity is especially advantageous on small data sets, where the proposed Parallel Tempering method shows promising performance.