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

In this short paper, we present an improved algorithm for approximating the minimum cut on distributed (CONGEST) networks. Let $λ$ be the minimum cut. Our algorithm can compute $λ$ exactly in $\tilde{O}((\sqrt{n}+D)\poly(λ))$ 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 $(1+ε)$-approximation $\tilde{O}((\sqrt{n}+D)/\poly(ε))$-time algorithm. The latter result improves over the previous $(2+ε)$-approximation $\tilde{O}((\sqrt{n}+D)/\poly(ε))$-time algorithm of Ghaffari and Kuhn [DISC 2013]. Due to the lower bound of $\tildeΩ(\sqrt{n}+D)$ 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].