Towards Tight Bounds for Spectral Sparsification of Hypergraphs

Cut and spectral sparsification of graphs have numerous applications, including e.g. speeding up algorithms for cuts and Laplacian solvers. These powerful notions have recently been extended to hypergraphs, which are much richer and may offer new applications. However, the current bounds on the size of hypergraph sparsifiers are not as tight as the corresponding bounds for graphs. Our first result is a polynomial-time algorithm that, given a hypergraph on n vertices with maximum hyperedge size r, outputs an epsilon-spectral sparsifier with O* (nr) hyperedges, where O* suppresses (epsilon(-1) log n)(O(1) )factors. This size bound improves the two previous bounds: O*(n(3)) [Soma and Yoshida, SODA'19] and O* (nr(3)) [Bansal, Svensson and Trevisan, FOCS'19]. Our main technical tool is a new method for proving concentration of the nonlinear analogue of the quadratic form of the Laplacians for hypergraph expanders. We complement this with lower bounds on the bit complexity of any compression scheme that (1 + epsilon)-approximates all the cuts in a given hypergraph, and hence also on the bit complexity of every epsilon-cut/spectral sparsifier. These lower bounds are based on Ruzsa-Szemeredi graphs, and a particular instantiation yields an Omega(nr) lower bound on the bit complexity even for fixed constant epsilon. In the case of hypergraph cut sparsifiers, this is tight up to polylogarithmic factors in n, due to recent result of [Chen, Khanna and Nagda, FOCS'20]. For spectral sparsifiers it narrows the gap to O*(r). Finally, for directed hypergraphs, we present an algorithm that computes an c-spectral sparsifier with O*(n(2)r(3)) hyperarcs, where r is the maximum size of a hyperarc. For small r, this improves over O*(n(3)) known from [Soma and Yoshida, SODA'19], and is getting close to the trivial lower bound of Omega(n(2)) hyperarcs.