Near-linear Size Hypergraph Cut Sparsifiers

Abstract

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, Benczr and Karger (1996) showed that given any n-vertex undirected weighted graph G and a parameter ε ∈ (0, 1), there is a near-linear time algorithm that outputs a weighted subgraph G ′ of G of size Õ(n/ε 2 ) such that the weight of every cut in G is preserved to within a (1±ε)-factor in G ′ . The graph G ′ is referred to as a (1 ± ε)-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 polynomialtime algorithm to find a (1 ± ε)-approximate cut sparsifier of H of size Õ( nr ε 2 ). Since r can be as large as n, in general, this gives a hypergraph cut sparsifier of size Õ(n 2 /ε 2 ), which is a factor n larger than the Benczr-Karger bound for graphs. It has been an open question whether or not Benczr-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 Õ(n/ε 2 ).