# Weighted Min-Cut: Sequential, Cut-Query and Streaming Algorithms

Publication
In STOC 2020

Consider the following 2-respecting min-cut problem. Given a weighted graph $G$ and its spanning tree $T$, find the minimum cut among the cuts that contain at most two edges in $T$. This problem is an important subroutine in Karger's celebrated randomized near-linear-time min-cut algorithm [STOC'96]. We present a new approach for this problem which can be easily implemented in many settings, leading to the following randomized min-cut algorithms for weighted graphs.

• An $O\left(m\frac{\log^2 n}{\log\log n} + n\log^6 n\right)$-time sequential algorithm: This improves Karger's long-standing $O(m \log^3 n)$ and $O\left(m\frac{(\log^2 n)\log (n^2/m)}{\log\log n} + n\log^6 n\right)$ bounds when the input graph is not extremely sparse or dense. Improvements over Karger's bounds were previously known only under a rather strong assumption that the input graph is {\em simple} (unweighted without parallel edges) [Henzinger, Rao, Wang, SODA'17; Ghaffari, Nowicki, Thorup, SODA'20]. For unweighted graphs (possibly with parallel edges) and using bit operations, our bound can be further improved to $O\left(m\frac{\log^{1.5} n}{\log\log n} + n\log^6 n\right)$.

• An algorithm that requires $\tilde O(n)$ cut queries to compute the min-cut of a weighted graph: This answers an open problem by Rubinstein, Schramm, and Weinberg [ITCS'18], who obtained a similar bound for simple graphs. Our bound is tight up to polylogarithmic factors.

• A streaming algorithm that requires $\tilde O(n)$ space and $O(\log n)$ passes to compute the min-cut: The only previous non-trivial exact min-cut algorithm in this setting is the 2-pass $\tilde O(n)$-space algorithm on simple graphs [Rubenstein et al. ITCS 2018] observed by Assadi, Chen, and Khanna [STOC'19]).

In contrast to Karger's 2-respecting min-cut algorithm which deploys sophisticated dynamic programming techniques, our approach exploits some cute structural properties so that it only needs to compute the values of $\tilde O(n)$ cuts corresponding to removing $\tilde O(n)$ pairs of tree edges, an operation that can be done quickly in many settings.