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Tighter Bounds for Weighted and Unweighted Shortest Cycle Approximation

Authors Avi Kadria , Liam Roditty , Virginia Vassilevska Williams

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Shortest paths
  • Theory of computation → Complexity classes
  • Mathematics of computing → Graph algorithms
Keywords
  • Fine-grained complexity
  • Graph algorithms
  • shortest cycle
  • girth approximations

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Abstract

We study the problem of approximating the length of a shortest cycle in a given graph, known as the girth of the graph. The state-of-the-art approximation algorithms for unweighted graphs by Kadria et al. [SODA'22] and Roditty and Trabelsi [arXiv'25] achieve the following trade-off: for every integer k ≥ 2, there is an Õ(n^{1+2/k}) time algorithm that achieves a (2k/3)-approximation for the girth in unweighted n-node graphs. The first result of this paper is to achieve the same trade-off for m-edge, n-node graphs with non-negative real edge weights: a 2k/3-approximation algorithm running in Õ(m+n^{1+2/k}) time. The dependence on m is unavoidable in weighted graphs. Our result improves on the work of Kadria et al. [SODA'23] and Ducoffe [ICALP'19 and SIDMA'21], who were only able to achieve such a trade-off for some values of k. We also prove new fine-grained lower bounds for girth approximation and related problems in unweighted graphs.

Cite As Get BibTex

Avi Kadria, Liam Roditty, and Virginia Vassilevska Williams. Tighter Bounds for Weighted and Unweighted Shortest Cycle Approximation. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 82:1-82:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.82

Author Details

Avi Kadria
  • Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
Liam Roditty
  • Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
Virginia Vassilevska Williams
  • Department of Electrical Engineering and Computer Science and CSAIL, MIT, Cambridge, MA, USA

Funding

  • Roditty, Liam: Supported in part by BSF grants 2016365 and 2020356.
  • Vassilevska Williams, Virginia: Supported in part by NSF CAREER Award 1651838, NSF Grants CCF-1909429 and CCF-2129139, BSF grants 2016365 and 2020356, a Google Research Fellowship, and a Sloan Research Fellowship.

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