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Listing 4-Cycles

Authors Amir Abboud, Seri Khoury, Oree Leibowitz, Ron Safier

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Author Details

Amir Abboud
  • Weizmann Institute of Science, Rehovot, Israel
Seri Khoury
  • UC Berkeley, CA, USA
Oree Leibowitz
  • Weizmann Institute of Science, Rehovot, Israel
Ron Safier
  • Weizmann Institute of Science, Rehovot, Israel

Funding

  • Abboud, Amir: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon Europe research and innovation programme (grant agreement No 101078482). Additionally, Amir Abboud is supported by an Alon scholarship and a research grant from the Center for New Scientists at the Weizmann Institute of Science.

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Amir Abboud, Seri Khoury, Oree Leibowitz, and Ron Safier. Listing 4-Cycles. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 25:1-25:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.FSTTCS.2023.25

Abstract

We study the fine-grained complexity of listing all 4-cycles in a graph on n nodes, m edges, and t such 4-cycles. The main result is an Õ(min(n²,m^{4/3})+t) upper bound, which is best-possible up to log factors unless the long-standing O(min(n²,m^{4/3})) upper bound for detecting a 4-cycle can be broken. Moreover, it almost-matches recent 3-SUM-based lower bounds for the problem by Abboud, Bringmann, and Fischer (STOC 2023) and independently by Jin and Xu (STOC 2023). Notably, our result separates 4-cycle listing from the closely related triangle listing for which higher conditional lower bounds exist, and rule out such a "detection plus t" bound. We also show by simple arguments that our bound cannot be extended to mild generalizations of the problem such as reporting all pairs of nodes that participate in a 4-cycle. [Independent work: Jin and Xu [Ce Jin and Yinzhan Xu, 2023] also present an algorithm with the same time bound.]

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Graph algorithms
  • cycles listing
  • subgraph detection
  • fine-grained complexity

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