## Abstract

Given an undirected edge-weighted graph G = (V, E) with m edges and n vertices, the minimum cut problem asks to find a subset of vertices S such that the total weight of all edges between S and V \S is minimized. Karger’s longstanding O(m log^{3} n) time randomized algorithm for this problem was very recently improved in two independent works to O(m log^{2} n) [ICALP’20] and to O(m log^{2} n + n log^{5} n) [STOC’20]. These two algorithms use different approaches and techniques. In particular, while the former is faster, the latter has the advantage that it can be used to obtain efficient algorithms in the cut-query and in the streaming models of computation. In this paper, we show how to simplify and improve the algorithm of [STOC’20] to O(m log^{2} n + n log^{3} n). We obtain this by replacing a randomized algorithm that, given a spanning tree T of G, finds in O(m log n + n log^{4} n) time a minimum cut of G that 2-respects (cuts two edges of) T with a simple O(m log n + n log^{2} n) time deterministic algorithm for the same problem.

Original language | English |
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Title of host publication | 4th Symposium on Simplicity in Algorithms, SOSA 2021 |

Editors | Valerie King, Hung Viet Le |

Publisher | Society for Industrial and Applied Mathematics Publications |

Pages | 74-79 |

Number of pages | 6 |

ISBN (Electronic) | 9781713827122 |

State | Published - 2021 |

Event | 4th Symposium on Simplicity in Algorithms, SOSA 2021, co-located with SODA 2021 - Alexandria, United States Duration: 11 Jan 2021 → 12 Jan 2021 |

### Publication series

Name | 4th Symposium on Simplicity in Algorithms, SOSA 2021 |
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### Conference

Conference | 4th Symposium on Simplicity in Algorithms, SOSA 2021, co-located with SODA 2021 |
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Country/Territory | United States |

City | Alexandria |

Period | 11/01/21 → 12/01/21 |

### Bibliographical note

Publisher Copyright:Copyright © 2021 by SIAM.

## ASJC Scopus subject areas

- Computational Theory and Mathematics
- Computational Mathematics
- Numerical Analysis
- Theoretical Computer Science