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

TL;DR

This paper investigates the complexity of constructing and sampling partite, 3-uniform hypergraphs with specified degree sequences, providing NP-completeness results, efficient algorithms for certain cases, and a novel sampling method tested on synthetic and real data.

Contribution

It establishes NP-completeness for the degree sequence problem, offers a polynomial algorithm for almost-regular sequences, and introduces a Parallel Tempering sampling method for hypergraphs.

Findings

NP-complete decision problem in general

Polynomial algorithm for third almost-regular sequences

Parallel Tempering method effectively samples hypergraphs and improves chi-squared testing sensitivity

Abstract

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 $\chi^2$ testing of contingency tables. We have shown that this hypergraph-based $\chi^2$ test is more sensitive than the standard $\chi^2$ test. The extra sensitivity is especially advantageous on small data sets, where the proposed Parallel Tempering method shows promising performance.