Cut Sparsification and Succinct Representation of Submodular Hypergraphs

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.