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 hyperedge $e$ 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 studied the setting where $H'$ is a reweighted sub-hypergraph of $H$, and measured the size of $H'$ by the number of hyperedges in it. In this setting, we present two results: (i) all submodular hypergraphs admit sparsifiers of size polynomial in $n=|V|$ and $\epsilon^{-1}$; (ii) we propose a new parameter, called spread, and use it to obtain smaller sparsifiers in some cases. We also show that for a natural family of splitting functions, relaxing the requirement that $H'$ be a reweighted sub-hypergraph of $H$ yields a substantially smaller encoding of the cuts of $H$ (almost a factor $n$ in the number of bits). This is in contrast to graphs, where the most succinct representation is attained by reweighted subgraphs. A new tool in our construction of succinct representation is the notion of deformation, where a splitting function $g_e$ 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：在图割稀疏化中，超图$H=(V,E,w)$的所有割被一个较小的超图$H'$以$1\pm\epsilon$因子近似。这一广泛应用的方法最近被推广到每个超边$e$的切割代价由分裂函数$g_e: 2^e\to\mathbb{R}_+$表示的设定。当函数$\{g_e\}_{e\in E}$为子模函数时，这种推广被称为子模超图，并出现在机器学习、组合优化和算法博弈论中。先前的工作研究了$H'$是$H$的加权子超图的情形，并通过超边数量衡量$H'$的大小。在此设定下，我们给出两个结果：(i) 所有子模超图均存在大小为$n=|V|$和$\epsilon^{-1}$的多项式稀疏化；(ii) 我们提出一个称为扩展度的新参数，并在某些情况下利用它获得更小的稀疏化。我们还证明，对于一类自然的分裂函数，放宽$H'$必须是$H$的加权子超图的要求，可实现对$H$割的显著更简洁编码（比特数接近因子$n$）。这与图的情形形成对比，在图中最简洁的表示是通过加权子图实现的。我们构建简洁表示的新工具是形变概念，其中分裂函数$g_e$被分解为小描述函数的和，并给出了常见分裂函数形变的上界和下界。