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Tolerant Testers for Subgraph-Freeness

Authors Reut Levi , Jonathan Meiri

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Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Tolerant Testing
  • Property Testing
  • Subgraph freeness
  • distance approximation
  • arboricity

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Abstract

In this paper we study the problem of tolerantly testing the property of being H-free (which also implies distance approximation from being H-free).
In the general-graphs model, we show that for tolerant K_k-freeness testing can be achieved with query complexity that is polynomial in the arboricity of the input graph G, arb(G), and independent of the size of G (for graphs in which the average degree is Ω(1)). 
Specifically for triangles, our algorithm distinguished graphs which are ε-close to being triangle-free from graphs that 3ε(1+η)-far from being triangle-free with expected query complexity which is Õ(arb³(G)) (for constant η and ε).
For general k-cliques our algorithm distinguishes graphs which are ε-close to being K_k-free from graphs which are binom(k,2)ε(1+η)-far from being K_k-free with expected query complexity which is polynomial in k, ε, γ and arb(G). 
We then generalize our result and provide a similar result for any motif H which is 2-connected of radius 1. This includes for example the wheel-graph. 
Finally, we show that our tester can be applied to the bounded-degree model for tolerantly testing H-freeness for any motif H. The query complexity of the algorithm is polynomial in the degree bound, d, improving the previous state-of-the-art by Marko and Ron (TALG 2009) that obtained quasi-polynomial query complexity in d.

Cite As Get BibTex

Reut Levi and Jonathan Meiri. Tolerant Testers for Subgraph-Freeness. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 77:1-77:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.77

Author Details

Reut Levi
  • Efi Arazi School of Computer Science, Reichman University, Herzliya, Israel
Jonathan Meiri
  • Efi Arazi School of Computer Science, Reichman University, Herzliya, Israel

Funding

  • Levi, Reut: The author was supported by the Israel Science Foundation under Grant 1867/20.
  • Meiri, Jonathan: The author was supported by the Israel Science Foundation under Grant 1867/20.

References

  1. Noga Alon, Tali Kaufman, Michael Krivelevich, and Dana Ron. Testing triangle-freeness in general graphs. SIAM Journal on Discrete Mathematics, 22(2):786-819, 2008. URL: https://doi.org/10.1137/07067917X.
  2. Andrej Bogdanov, Kenji Obata, and Luca Trevisan. A lower bound for testing 3-colorability in bounded-degree graphs. In 43rd Symposium on Foundations of Computer Science (FOCS 2002), 16-19 November 2002, Vancouver, BC, Canada, Proceedings, pages 93-102. IEEE Computer Society, 2002. URL: https://doi.org/10.1109/SFCS.2002.1181886.
  3. Talya Eden, Reut Levi, and Dana Ron. Testing c_k-freeness in bounded-arboricity graphs. In Karl Bringmann, Martin Grohe, Gabriele Puppis, and Ola Svensson, editors, 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024, July 8-12, 2024, Tallinn, Estonia, volume 297 of LIPIcs, pages 60:1-60:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.ICALP.2024.60.
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  5. Talya Eden, Dana Ron, and Will Rosenbaum. Almost optimal bounds for sublinear-time sampling of k-cliques in bounded arboricity graphs. In 49th International Colloquium on Automata, Languages, and Programming, ICALP, pages 56:1-56:19, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.56.
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  7. Talya Eden, Dana Ron, and C. Seshadhri. Faster sublinear approximation of the number of k-cliques in low-arboricity graphs. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 1467-1478, 2020. URL: https://doi.org/10.1137/1.9781611975994.89.
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