In cut sparsification, all cuts of a hypergraph $H=(V,E,w)$ are approximated within $1\pm\epsilon$ factor by a small hypergraph $H'$. This widely applied method was generalized recently to a setting where the cost of cutting each $e\in E$ is provided by a splitting function, $g_e: 2^e\to\mathbb{R}_+$. This generalization is called a submodular hypergraph when the functions $\{g_e\}_{e\in E}$ are submodular, and it arises in machine learning, combinatorial optimization, and algorithmic game theory. Previous work focused on the setting where $H'$ is a reweighted sub-hypergraph of $H$, and measured size by the number of hyperedges in $H'$. We study such sparsification, and also a more general notion of representing $H$ succinctly, where size is measured in bits. In the sparsification setting, where size is the number of hyperedges, we present three results: (i) all submodular hypergraphs admit sparsifiers of size polynomial in $n=|V|$; (ii) monotone-submodular hypergraphs admit sparsifiers of size $O(\epsilon^{-2} n^3)$; and (iii) we propose a new parameter, called spread, to obtain even smaller sparsifiers in some cases. In the succinct-representation setting, we show that a natural family of splitting functions admits a succinct representation of much smaller size than via reweighted subgraphs (almost by factor $n$). This large gap is surprising because for graphs, the most succinct representation is attained by reweighted subgraphs. Along the way, we introduce the notion of deformation, where $g_e$ is decomposed into a sum of functions of small description, and we provide upper and lower bounds for deformation of common splitting functions.

翻译：在割稀疏化中，超图 $H=(V,E,w)$ 的所有割被一个较小的超图 $H'$ 在 $1\pm\epsilon$ 因子内逼近。这一广泛应用的方法最近被推广到一种场景，其中切割每个 $e\in E$ 的成本由分裂函数 $g_e: 2^e\to\mathbb{R}_+$ 提供。当函数 $\{g_e\}_{e\in E}$ 为子模函数时，这一推广被称为子模超图，它在机器学习、组合优化和算法博弈论中均有应用。先前的工作主要关注 $H'$ 是 $H$ 的加权子超图的情形，并以 $H'$ 中超边的数量衡量规模。我们研究了此类稀疏化，以及一种更广义的用比特衡量规模的超图简洁表示概念。在稀疏化场景（规模以超边数量衡量）中，我们提出三项结果：(i) 所有子模超图均接受规模为 $n=|V|$ 多项式的稀疏化器；(ii) 单调子模超图接受规模为 $O(\epsilon^{-2} n^3)$ 的稀疏化器；(iii) 我们提出一个新参数——扩散度，可在某些情况下获得更小的稀疏化器。在简洁表示场景中，我们展示了一类自然的分裂函数族可以通过远小于加权子图（几乎缩小因子 $n$）的规模实现简洁表示。这一巨大差距令人惊讶，因为对于图而言，最简洁的表示正是通过加权子图实现的。在此过程中，我们引入了变形概念，即将 $g_e$ 分解为若干小描述函数的和，并为常见分裂函数的变形提供了上下界。