Optimal Bounds for the <i>k</i> -cut Problem

In the k -cut problem, we want to find lowest-weight set of edges whose deletion breaks a given (multi)graph into connected components. Algorithms Karger and Stein can solve this in roughly O ( n 2k ) time. However, lower bounds from conjectures about -clique problem imply that ? (1- o (1)) time is likely needed. Recent results Gupta, Lee, Li have new algorithms for general 1.98k + O(1) time, as well specialized with better performance certain classes graphs (e.g., small integer edge weights). work, resolve graphs. We show Contraction Algorithm outputs any fixed weight ? ? probability - ), where denotes minimum weight. This also gives an extremal bound on number -cuts algorithm compute polylog( runtime. Both are tight up lower-order factors, algorithmic assuming hardness max-weight -clique. The first main ingredient our result cuts less than 2 / , using Sunflower lemma. second fine-grained analysis how graph shrinks—and average degree evolves—in process.