Brief Announcement: Almost-Tight Approximation Distributed Algorithm for Minimum Cut

TL;DR

This paper introduces an almost-tightly optimal distributed algorithm for approximating the minimum cut in networks, achieving near-lower-bound running times with a simple combination of existing techniques.

Contribution

It presents a new distributed algorithm that computes exact or approximate minimum cuts efficiently, improving over previous methods and matching known lower bounds up to polylogarithmic factors.

Findings

Exact minimum cut can be computed in near-optimal time.

A $(1+psilon)$-approximation algorithm is achieved with improved running time.

The algorithm's simplicity stems from combining established theorems and algorithms.

Abstract

In this short paper, we present an improved algorithm for approximating the minimum cut on distributed (CONGEST) networks. Let $\lambda$ be the minimum cut. Our algorithm can compute $\lambda$ exactly in $\tilde{O}((\sqrt{n}+D)\poly(\lambda))$ time, where $n$ is the number of nodes (processors) in the network, $D$ is the network diameter, and $\tilde{O}$ hides $\poly\log n$. By a standard reduction, we can convert this algorithm into a $(1+\epsilon)$-approximation $\tilde{O}((\sqrt{n}+D)/\poly(\epsilon))$-time algorithm. The latter result improves over the previous $(2+\epsilon)$-approximation $\tilde{O}((\sqrt{n}+D)/\poly(\epsilon))$-time algorithm of Ghaffari and Kuhn [DISC 2013]. Due to the lower bound of $\tilde{\Omega}(\sqrt{n}+D)$ by Das Sarma et al. [SICOMP 2013], this running time is {\em tight} up to a $\poly\log n$ factor. Our algorithm is an extremely simple combination of Thorup's tree packing theorem [Combinatorica 2007], Kutten and Peleg's tree partitioning algorithm [J. Algorithms 1998], and Karger's dynamic programming [JACM 2000].