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.

翻译：$3$-均匀部分超图是一类特殊的$3$-均匀超图，其中每条超边恰好包含来自$3$个互不相交的顶点类中各一个顶点。我们研究了$3$-均匀部分超图的度序列问题，即判断是否存在具有指定度序列的此类超图。我们证明该判定问题通常是NP完全的，并针对第三近似正则度序列（即某一顶点类中所有顶点的度均为固定整数$k$或$k-1$，而其他两个顶点类无限制）给出了多项式时间算法。我们还研究了采样问题，即从具有指定度序列的$3$-均匀部分超图中进行均匀采样。我们提出了一种并行回火方法，其中超图的假设性能量衡量了与指定度序列的偏差。该方法已在合成数据和真实数据上实现并测试，也可应用于列联表的$\chi^2$检验。我们证明了这种基于超图的$\chi^2$检验比标准$\chi^2$检验更敏感，这种额外敏感性在小数据集上尤其有利，而所提出的并行回火方法在此类场景中表现出良好的性能。