Fast and Deterministic Approximations for k-Cut
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In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in O(n^2(k-1)) randomized time [Karger and Stein 1996] and O(n^2k + O(1)) deterministic time [Thorup 2008], but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Holdschmidt and Hochbaum 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed by O(k) minimum cuts, which implies an Õ(mk) randomized running time and a Õ(k m n) deterministic running time. These results prompt two basic questions. First, is there a deterministic algorithm for 2-approximate k-cuts in Õ(mk) time? Second, can 2-approximate k-cuts be computed as fast as the (exact) minimum cut - in Õ(m) randomized time or Õ(mn) deterministic time? We make progress on both of these questions with a deterministic approximation algorithm that computes (2 + ϵ)-minimum k-cuts in O(m ^3(n) / ϵ^2) time, via a (1 + ϵ)-approximate for an LP relaxation of k-cut.
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