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标题：Fast and Deterministic Approximations for k-Cut
本地全文：下载
作者：Kent Quanrud
期刊名称：Theory of Computing
印刷版ISSN：1557-2862
电子版ISSN：1557-2862
出版年度：2022
卷号：18
页码：1-24
DOI：10.4086/toc.2022.v018a007
语种：English
出版社：University of Chicago
摘要：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 n O(k) time, but when k is treated as part of the input, computing the minimum-weight k-cut is NPhard [Goldschmidt 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 via O(k) minimum cuts, which implies a O˜(km) randomized running time via the nearly linear-time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 − 2/k)-approximately minimum-weight k-cut can be computed deterministically in O(mn + n 2 logn) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate minimum k-cut, matching the randomized running time of O˜(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate minimum k-cut be computed as fast as the minimum cut, in O˜(m) randomized time?
关键词：k-cut;multiplicative weight updates