Sublinear Time Hypergraph Sparsification via Cut and Edge Sampling Queries

TL;DR

This paper introduces the first sublinear time algorithm for hypergraph cut sparsification, enabling efficient approximation of hypergraph cuts with query access, independent of the hypergraph's size.

Contribution

It presents a novel sublinear time algorithm for hypergraph cut sparsification that operates independently of the hypergraph's size, a significant advancement over previous methods.

Findings

Achieves hypergraph sparsification in sublinear time.

Operates with query access, independent of hypergraph size.

Extends classical graph sparsification results to hypergraphs.

Abstract

The problem of sparsifying a graph or a hypergraph 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 $\varepsilon \in (0,1)$, there is a near-linear time algorithm that outputs a weighted subgraph $G'$ of $G$ of size $\tilde{O}(n/\varepsilon^2)$ such that the weight of every cut in $G$ is preserved to within a $(1 \pm \varepsilon)$-factor in $G'$. The graph $G'$ is referred to as a {\em $(1 \pm \varepsilon)$-approximate cut sparsifier} of $G$. Subsequent recent work has obtained a similar result for the more general problem of hypergraph cut sparsifiers. However, all known sparsification algorithms require $\Omega(n + m)$ time where $n$ denotes the number of vertices and $m$ denotes the number of hyperedges in the hypergraph. Since $m$ can be exponentially large in $n$, a natural question is if it is possible to create a hypergraph cut sparsifier in time polynomial in $n$, {\em independent of the number of edges}. We resolve this question in the affirmative, giving the first sublinear time algorithm for this problem, given appropriate query access to the hypergraph.