Optimal Bounds for the k-cut Problem
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In the k-cut problem, we want to find the smallest set of edges whose deletion breaks a given (multi)graph into k connected components. Algorithms of Karger Stein and Thorup showed how to find such a minimum k-cut in time approximately O(n^2k). The best lower bounds come from conjectures about the solvability of the k-clique problem and a reduction from k-clique to k-cut, and show that solving k-cut is likely to require time Ω(n^k). Recent results of Gupta, Lee Li have given special-purpose algorithms that solve the problem in time n^1.98k + O(1), and ones that have better performance for special classes of graphs (e.g., for small integer weights). In this work, we resolve the problem for general graphs, by showing that for any fixed k ≥ 2, the Karger-Stein algorithm outputs any fixed k-cut of weight αλ_k with probability at least O_k(n^-α k), where λ_k denotes the minimum k-cut size. This also gives an extremal bound of O_k(n^k) on the number of minimum k-cuts in an n-vertex graph and an algorithm to compute a minimum k-cut in similar runtime. Both are tight up to lower-order factors, with the algorithmic lower bound assuming hardness of max-weight k-clique. The first main ingredient in our result is a fine-grained analysis of how the graph shrinks—and how the average degree evolves—under the Karger-Stein process. The second ingredient is an extremal result bounding the number of cuts of size less than 2 λ_k/k, using the Sunflower lemma.
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