Breaking the n^k Barrier for Minimum k-cut on Simple Graphs
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In the minimum k-cut problem, we want to find the minimum number of edges whose deletion breaks the input graph into at least k connected components. The classic algorithm of Karger and Stein runs in Õ(n^2k-2) time, and recent, exciting developments have improved the running time to O(n^k). For general, weighted graphs, this is tight assuming popular hardness conjectures. In this work, we show that perhaps surprisingly, O(n^k) is not the right answer for simple, unweighted graphs. We design an algorithm that runs in time O(n^(1-ϵ)k) where ϵ>0 is an absolute constant, breaking the natural n^k barrier. This establishes a separation of the two problems in the unweighted and weighted cases.
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