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Node-Weighted Triangles: Faster and Simpler

Authors Shyan Akmal , Nick Fischer

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • fine-grained complexity
  • triangle detection
  • node-weighted triangle

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Abstract

Weighted variants of triangle detection are an important object of study because of their prominence in fine-grained complexity. We revisit the Node-Weighted Triangle problem, where the goal is to decide if a vertex-weighted graph contains a triangle whose node weights sum to zero. This problem has been the focus of a celebrated line of work, beginning with a subcubic-time algorithm [Vassilevska, Williams; STOC '06], and culminating in algorithms running almost in matrix multiplication time, O(MM(n) + n²⋅2^O(√{log n})) [Czumaj, Lingas; SODA '07], [Vassilevska W., Williams; STOC '09]. This runtime is almost-optimal, since even detecting an unweighted triangle is conjectured to require matrix multiplication time MM(n). However, the superpolylogarithmic 2^Ω(√{log n}) overhead persists in a world where near-optimal matrix multiplication is possible (i.e., MM(n) ≤ n²poly(log n)).
In this paper, we present a new algorithm solving Node-Weighted Triangle in O(MM(n)) time, closing the gap to unweighted triangle detection completely. Remarkably, our algorithm is much simpler than previous approaches, which use involved recursion schemes and communication protocols.

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Shyan Akmal and Nick Fischer. Node-Weighted Triangles: Faster and Simpler. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 10:1-10:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.10

Author Details

Shyan Akmal
  • Max Planck Institute for Informatics, Saarbrücken, Germany
Nick Fischer
  • Max Planck Institute for Informatics, Saarbrücken, Germany

Funding

  • Akmal, Shyan: This work received funding from the Klaus Tschira Boost Fund, a joint initiative of GSO Guidance, Skills & Opportunities e.V. and the Klaus Tschira Stiftung.

Acknowledgements

We thank the anonymous reviewers for their helpful comments.

References

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