Faster connectivity in low-rank hypergraphs via expander decomposition

TL;DR

This paper introduces a faster algorithm for computing connectivity in low-rank hypergraphs by leveraging a new structural understanding of min-cuts and extending expander decomposition techniques to hypergraphs.

Contribution

It presents a novel algorithm with improved runtime for hypergraph connectivity and a structural theorem on min-cuts, extending expander decomposition to hypergraphs.

Findings

Algorithm runs faster than existing methods for constant rank and high connectivity.

Structural theorem generalizes known graph results to hypergraphs.

Constructive proof enables practical algorithm implementation.

Abstract

We design an algorithm for computing connectivity in hypergraphs which runs in time $\hat O_r(p + \min\{\lambda^{\frac{r-3}{r-1}} n^2, n^r/\lambda^{r/(r-1)}\})$ (the $\hat O_r(\cdot)$ hides the terms subpolynomial in the main parameter and terms that depend only on $r$) where $p$ is the size, $n$ is the number of vertices, and $r$ is the rank of the hypergraph. Our algorithm is faster than existing algorithms when the the rank is constant and the connectivity $\lambda$ is $\omega(1)$. At the heart of our algorithm is a structural result regarding min-cuts in simple hypergraphs. We show a trade-off between the number of hyperedges taking part in all min-cuts and the size of the smaller side of the min-cut. This structural result can be viewed as a generalization of a well-known structural theorem for simple graphs [Kawarabayashi-Thorup, JACM 19]. We extend the framework of expander decomposition to simple hypergraphs in order to prove this structural result. We also make the proof of the structural result constructive to obtain our faster hypergraph connectivity algorithm.