Distributed Weighted Min-Cut in Nearly-Optimal Time

Abstract: Minimum-weight cut (min-cut) is a basic measure of a network's connectivity strength. While the min-cut can be computed efficiently in the sequential setting [Karger STOC'96], there was no efficient way for a distributed network to compute its own min-cut without limiting the input structure or dropping the output quality: In the standard CONGEST model, existing algorithms with nearly-optimal time (e.g. [Ghaffari, Kuhn, DISC'13; Nanongkai, Su, DISC'14]) can guarantee a solution that is $(1+\epsilon)$-approximation at best while the exact $\tilde O(n^{0.8}D^{0.2} + n^{0.9})$-time algorithm [Ghaffari, Nowicki, Thorup, SODA'20] works only on $\textit{simple}$ networks (no weights and no parallel edges). Throughout, $n$ and $D$ denote the network's number of vertices and hop-diameter, respectively. For the weighted case, the best bound was $\tilde O(n)$ [Daga et al. STOC'19]. In this paper, we provide an $\textit{exact}$ $\tilde O(\sqrt n + D)$-time algorithm for computing min-cut on $\textit{weighted}$ networks. Our result improves even the previous algorithm that works only on simple networks. Its time complexity matches the known lower bound up to polylogarithmic factors. At the heart of our algorithm are a combination of two kinds of tree-decompositions and a novel structural theorem that generalizes a theorem in Mukhopadhyay-Nanongkai [STOC'20] and, in turn, helps simplify their algorithms.