Near-linear Size Hypergraph Cut Sparsifiers

Cuts in graphs are a fundamental object of study, and play a central role in the study of graph algorithms. The problem of sparsifying a graph while approximately preserving its cut structure has been extensively studied and has many applications. In a seminal work, Bencz'ur and Karger (1996) showed that given any n-vertex undirected weighted graph G and a parameter varepsilon ∈ (0,1), there is a near-linear time algorithm that outputs a weighted subgraph G' of G of size tildeO(n/varepsilon2) such that the weight of every cut in G is preserved to within a (1 ± varepsilon)-factor in G'. The graph G' is referred to as a em (1 ± varepsilon)-approximate cut sparsifier of G. A natural question is if such cut-preserving sparsifiers also exist for hypergraphs. Kogan and Krauthgamer (2015) initiated a study of this question and showed that given any weighted hypergraph H where the cardinality of each hyperedge is bounded by r, there is a polynomial-time algorithm to find a (1 ± varepsilon)-approximate cut sparsifier of H of size tildeO(fracnrvarepsilon2). Since r can be as large as n, in general, this gives a hypergraph cut sparsifier of size tildeO(n2/varepsilon2), which is a factor n larger than the Bencz'ur-Karger bound for graphs. It has been an open question whether or not Bencz'ur-Karger bound is achievable on hypergraphs. In this work, we resolve this question in the affirmative by giving a new polynomial-time algorithm for creating hypergraph sparsifiers of size tildeO(n/varepsilon2).