Fast and Deterministic Approximations for k-Cut

Kent Quanrud, Michael Wagner
2019 International Workshop on Approximation Algorithms for Combinatorial Optimization  
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 NP-Hard [18]. For poly(m, n, k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [37] . Saran and Vazirani [46] showed that a 2 − 2 k -approximately minimum weight k-cut can be
more » ... computed via O(k) minimum cuts, which implies a Õ(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [27] . Nagamochi and Kamidoi [42] showed that a 2 − 2 k -approximately 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 Õ(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut -in Õ(m) randomized time? We give a deterministic approximation algorithm that computes (2 + )-minimum k-cuts in O m log 3 n/ 2 time, via a (1 + )-approximation for an LP relaxation of k-cut.
doi:10.4230/lipics.approx-random.2019.23 dblp:conf/approx/Quanrud19 fatcat:ffcftkfmlrhypb3bxm7oggfthy