The Karger-Stein algorithm is optimal for k-cut

Anupam Gupta, Euiwoong Lee, Jason Li

Research output: Chapter in Book/Report/Conference proceedingConference contribution


In the k-cut problem, we are given an edge-weighted graph and want to find the least-weight set of edges whose deletion breaks the graph into k connected components. Algorithms due to Karger-Stein and Thorup showed how to find such a minimum k-cut in time approximately O(n2k-2). The best lower bounds come from conjectures about the solvability of the k-clique problem and a reduction from k-clique to k-cut, and show that solving k-cut is likely to require time ω(nk). Our recent results have given special-purpose algorithms that solve the problem in time n1.98k + O(1), and ones that have better performance for special classes of graphs (e.g., for small integer weights). In this work, we resolve the problem for general graphs, by showing that for any fixed k ≥ 2, the Karger-Stein algorithm outputs any fixed minimum k-cut with probability at least O(n-k), where O(·) hides a 2O(lnlnn)2 factor. This also gives an extremal bound of O(nk) on the number of minimum k-cuts in an n-vertex graph and an algorithm to compute a minimum k-cut in similar runtime. Both are tight up to O(1) factors. The first main ingredient in our result is a fine-grained analysis of how the graph shrinks - and how the average degree evolves - under the Karger-Stein process. The second ingredient is an extremal result bounding the number of cuts of size at most (2-) OPT/k, using the Sunflower lemma.

Original languageEnglish (US)
Title of host publicationSTOC 2020 - Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing
EditorsKonstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, Julia Chuzhoy
PublisherAssociation for Computing Machinery
Number of pages12
ISBN (Electronic)9781450369794
StatePublished - Jun 8 2020
Event52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020 - Chicago, United States
Duration: Jun 22 2020Jun 26 2020

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017


Conference52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020
Country/TerritoryUnited States


  • Graph Algorithms
  • Minimum Cut
  • Randomized Algorithms

ASJC Scopus subject areas

  • Software


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