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Fast and Deterministic Approximations for k-Cut

Author Kent Quanrud

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Kent Quanrud
  • Department of Computer Science, University of Illinois at Urbana-Champaign, USA

Funding

Work on this paper is partly supported by NSF grant CCF-1526799.

Acknowledgements

The author thanks Chandra Chekuri for introducing him to the problem and providing helpful feedback, including pointers to the literature for rounding the LP. The author thanks Chao Xu for pointers to references. The author thanks the anonymous reviewers for their helpful comments.

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Kent Quanrud. Fast and Deterministic Approximations for k-Cut. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 23:1-23:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.23

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 n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [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 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 O~(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 O~(m) randomized time?
We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Network optimization
  • Theory of computation → Linear programming
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Routing and network design problems
Keywords
  • k-cut
  • multiplicative weight updates

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