Planar negative k-cycle

Paweł Gawrychowski, Shay Mozes, Oren Weimann

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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(n3) 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(nk2 log n), O(n2)} time. Assuming the min-plus convolution conjecture, we then show, for k > n1/3 that there is no algorithm polynomially faster than O(n1.5 √k), and for k ≤ n1/3 that our O(nk2 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 languageEnglish
Title of host publicationACM-SIAM Symposium on Discrete Algorithms, SODA 2021
EditorsDaniel Marx
PublisherAssociation for Computing Machinery
Number of pages8
ISBN (Electronic)9781611976465
StatePublished - 2021
Event32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021 - Alexandria, Virtual, United States
Duration: 10 Jan 202113 Jan 2021

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021
Country/TerritoryUnited States
CityAlexandria, Virtual

Bibliographical note

Publisher Copyright:
© 2021 by SIAM

ASJC Scopus subject areas

  • Software
  • General Mathematics


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