Subgraph counting is a fundamental and well-studied problem whose computational complexity is well understood. Quite surprisingly, the hypergraph version of subgraph counting has been almost ignored. In this work, we address this gap by investigating the most basic sub-hypergraph counting problem: given a (small) hypergraph $H$ and a (large) hypergraph $G$, compute the number of sub-hypergraphs of $G$ isomorphic to $H$. Formally, for a family $\mathcal{H}$ of hypergraphs, let #Sub($\mathcal{H}$) be the restriction of the problem to $H \in \mathcal{H}$; the induced variant #IndSub($\mathcal{H}$) is defined analogously. Our main contribution is a complete classification of the complexity of these problems. Assuming the Exponential Time Hypothesis, we prove that #Sub($\mathcal{H}$) is fixed-parameter tractable if and only if $\mathcal{H}$ has bounded fractional co-independent edge-cover number, a novel graph parameter we introduce. Moreover, #IndSub($\mathcal{H}$) is fixed-parameter tractable if and only if $\mathcal{H}$ has bounded fractional edge-cover number. Both results subsume pre-existing results for graphs as special cases. We also show that the fixed-parameter tractable cases of #Sub($\mathcal{H}$) and #IndSub($\mathcal{H}$) are unlikely to be in polynomial time, unless respectively #P = P and Graph Isomorphism $\in$ P. This shows a separation with the special case of graphs, where the fixed-parameter tractable cases are known to actually be in polynomial time.

翻译：子图计数是一个基础且被充分研究的问题，其计算复杂性已得到深入理解。然而令人惊讶的是，超图版本的子图计数问题几乎未被关注。在本工作中，我们通过探究最基本的子超图计数问题来填补这一空白：给定一个（小）超图$H$和一个（大）超图$G$，计算$G$中与$H$同构的子超图数目。形式化地，对于超图族$\mathcal{H}$，令#Sub($\mathcal{H}$)表示该问题限制在$H \in \mathcal{H}$的形式；其导出变形#IndSub($\mathcal{H}$)定义类似。我们的主要贡献在于给出了这些问题复杂性的完整分类。在指数时间假设下，我们证明#Sub($\mathcal{H}$)是固定参数易处理的当且仅当$\mathcal{H}$具有有界分数余独立边覆盖数——这是我们引入的一个全新图参数。此外，#IndSub($\mathcal{H}$)是固定参数易处理的当且仅当$\mathcal{H}$具有有界分数边覆盖数。这两个结果分别将图论中的现有结论作为特例纳入其中。我们还证明，#Sub($\mathcal{H}$)和#IndSub($\mathcal{H}$)的固定参数易处理情形不太可能具有多项式时间复杂度，除非分别有#P = P和图同构问题∈ P。这表明与图论特例存在分化——在图的特例中，固定参数易处理情形已知实际上可在多项式时间内求解。