A Nearly Optimal Min-Cut Algorithm for Dynamic Streams

Overview & Synopsis

We give an $\tilde{O}(n/\varepsilon)$-space algorithm for the minimum cut problem in dynamic streams, resolving the space complexity of streaming min-cut up to polylogarithmic factors. Ding, Garces, Li, Lin, Nelson, Shah, and Woodruff (ITCS 2025) obtained the same $\tilde{O}(n/\varepsilon)$ space bound for insertion-only streams, while prior to our work the best known bound even for unweighted dynamic streams was $\tilde{O}(n/\varepsilon^2)$, due to Ahn, Guha, and McGregor (PODS 2012). Our main technical contribution is an $\tilde{O}(n/\varepsilon)$-space for-each cut sketch in dynamic streams, which may be of independent interest. The construction applies to weighted graphs under general $\pmΔ$ updates, a stronger model than the standard dynamic-stream setting in which weighted edges are inserted and deleted in their entirety. Despite this stronger guarantee, our algorithm is surprisingly simple and relies on very little machinery. We also obtain applications to minimum $\ell_p$-norm $k$-partitioning for every fixed $k$, weighted correlation-clustering evaluation for two clusters, and for-each cut sketching in balanced directed graphs.