Computing minimum cuts in hypergraphs

TL;DR

This paper advances the understanding of hypergraph connectivity by developing new algorithms for minimum cuts, including sparsification, approximation, and hypercactus representation, improving efficiency and extending graph results.

Contribution

The paper introduces novel algorithms for hypergraph minimum cuts, including a hypergraph sparsifier, a $(2+ ext{epsilon})$-approximation method, and a hypercactus representation, generalizing and improving upon graph algorithms.

Findings

Developed a $O(p)$ time algorithm for hypergraph sparsification preserving edge-connectivities.

Created a $O(rac{1}{ ext{epsilon}} (p ext{log} n + n ext{log}^2 n))$ time approximation algorithm for minimum cuts.

Established a $O(np + n^2 ext{log} n)$ time algorithm for hypercactus representation of all minimum cuts.

Abstract

We study algorithmic and structural aspects of connectivity in hypergraphs. Given a hypergraph $H=(V,E)$ with $n = |V|$, $m = |E|$ and $p = \sum_{e \in E} |e|$ the best known algorithm to compute a global minimum cut in $H$ runs in time $O(np)$ for the uncapacitated case and in $O(np + n^2 \log n)$ time for the capacitated case. We show the following new results. 1. Given an uncapacitated hypergraph $H$ and an integer $k$ we describe an algorithm that runs in $O(p)$ time to find a subhypergraph $H'$ with sum of degrees $O(kn)$ that preserves all edge-connectivities up to $k$ (a $k$-sparsifier). This generalizes the corresponding result of Nagamochi and Ibaraki from graphs to hypergraphs. Using this sparsification we obtain an $O(p + \lambda n^2)$ time algorithm for computing a global minimum cut of $H$ where $\lambda$ is the minimum cut value. 2. We generalize Matula's argument for graphs to hypergraphs and obtain a $(2+\epsilon)$-approximation to the global minimum cut in a capacitated hypergraph in $O(\frac{1}{\epsilon} (p \log n + n \log^2 n))$ time. 3. We show that a hypercactus representation of all the global minimum cuts of a capacitated hypergraph can be computed in $O(np + n^2 \log n)$ time and $O(p)$ space. We utilize vertex ordering based ideas to obtain our results. Unlike graphs we observe that there are several different orderings for hypergraphs which yield different insights.