## Abstract

Given an edge-weighted directed graph G, the Negative-k-Cycle problem asks whether G contains a negative-weight cycle with at most k edges. For k = 3 the problem is known as the NegativeTriangle problem and is equivalent to all-pairs shortest paths (and to min-plus matrix multiplication) and solvable in O(n^{3}) time. In this paper, we consider the case of directed planar graphs. We show that the Negative-k-Cycle problem can be solved in min{O(nk^{2} log n), O(n^{2})} time. Assuming the min-plus convolution conjecture, we then show, for k > n^{1}/^{3} that there is no algorithm polynomially faster than O(n^{1.5 √}k), and for k ≤ n^{1}/^{3} that our O(nk^{2} log n) upper bound is essentially tight. The latter gives the first non-trivial tight bounds for a planar graph problem in P. Our lower bounds are obtained by introducing a natural problem on matrices that generalizes both min-plus matrix multiplication and min-plus convolution, and whose complexity lies between the complexities of these two problems.

Original language | English |
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Title of host publication | ACM-SIAM Symposium on Discrete Algorithms, SODA 2021 |

Editors | Daniel Marx |

Publisher | Association for Computing Machinery |

Pages | 2717-2724 |

Number of pages | 8 |

ISBN (Electronic) | 9781611976465 |

State | Published - 2021 |

Event | 32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021 - Alexandria, Virtual, United States Duration: 10 Jan 2021 → 13 Jan 2021 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Conference

Conference | 32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021 |
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Country/Territory | United States |

City | Alexandria, Virtual |

Period | 10/01/21 → 13/01/21 |

### Bibliographical note

Publisher Copyright:© 2021 by SIAM

## ASJC Scopus subject areas

- Software
- General Mathematics