A note on approximate strengths of edges in a hypergraph

TL;DR

This paper extends an edge strength approximation algorithm from graphs to hypergraphs, enabling faster near-linear time algorithms for approximate mincut problems in hypergraphs of small rank.

Contribution

It introduces a near-linear time algorithm to compute approximate edge strengths in hypergraphs, building on previous sparsification results.

Findings

Provides a near-linear time algorithm for hypergraph edge strength approximation

Enables faster $(1+)$-approximate mincut algorithms for small-rank hypergraphs

Builds on recent hypergraph sparsification techniques

Abstract

Let $H=(V,E)$ be an edge-weighted hypergraph of rank $r$. Kogan and Krauthgamer extended Bencz\'{u}r and Karger's random sampling scheme for cut sparsification from graphs to hypergraphs. The sampling requires an algorithm for computing the approximate strengths of edges. In this note we extend the algorithm for graphs to hypergraphs and describe a near-linear time algorithm to compute approximate strengths of edges; we build on a sparsification result for hypergraphs from our recent work. Combined with prior results we obtain faster algorithms for finding $(1+\epsilon)$-approximate mincuts when the rank of the hypergraph is small.