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• W2883647592 abstract "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 $O(n^{O(k)})$ time, but when $k$ is treated as part of the input, computing the minimum weight $k$-cut is NP-Hard [Holdschmidt and Hochbaum 1994]. For $operatorname{poly}(m,n,k)$-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 $tilde{O}(mk)$ randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that the minimum weight $k$-cut can be computed deterministically in $O(mn + n^2 log n)$ time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate $k$-cuts matching the randomized running time of $tilde{O}(mk)$? The second question qualitatively compares minimum cut to 2-approximate minimum $k$-cut. Can 2-approximate $k$-cuts be computed as fast as the (exact) minimum cut - in $tilde{O}(m)$ randomized time? We make progress on these questions with a deterministic approximation algorithm that computes $(2 + epsilon)$-minimum $k$-cuts in $O(m log^3(n) / epsilon^2)$ time, via a $(1 + epsilon)$-approximate for an LP relaxation of $k$-cut." @default.
• W2883647592 created "2018-08-03" @default.
• W2883647592 creator A5019690129 @default.
• W2883647592 date "2018-07-18" @default.
• W2883647592 modified "2023-09-27" @default.
• W2883647592 title "Fast and Deterministic Approximations for $k$-Cut" @default.
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