Cut Sparsification and Succinct Representation of Submodular Hypergraphs

In cut sparsification, all cuts of a hypergraph H = (V, E, w) are approximated within 1 ± ϵ 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 ge : 2^e → R+. This generalization is called a submodular hypergraph when the functions {ge}e∈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 ϵ⁻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 ge is decomposed into a sum of functions of small description, and we provide upper and lower bounds for deformation of common splitting functions.