Testing C_k-Freeness in Bounded-Arboricity Graphs

Authors Talya Eden , Reut Levi , Dana Ron



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Talya Eden
  • Department of Computer Science, Bar-Ilan University, Ramat-Gan, Israel
Reut Levi
  • Efi Arazi School of Computer Science, Reichman University, Herzliya, Israel
Dana Ron
  • School of Electrical Engineering, Tel Aviv University, Israel

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Talya Eden, Reut Levi, and Dana Ron. Testing C_k-Freeness in Bounded-Arboricity Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 60:1-60:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.60

Abstract

We study the problem of testing C_k-freeness (k-cycle-freeness) for fixed constant k > 3 in graphs with bounded arboricity (but unbounded degrees). In particular, we are interested in one-sided error algorithms, so that they must detect a copy of C_k with high constant probability when the graph is ε-far from C_k-free. We next state our results for constant arboricity and constant ε with a focus on the dependence on the number of graph vertices, n. The query complexity of all our algorithms grows polynomially with 1/ε. 1) As opposed to the case of k = 3, where the complexity of testing C₃-freeness grows with the arboricity of the graph but not with the size of the graph (Levi, ICALP 2021) this is no longer the case already for k = 4. We show that Ω(n^{1/4}) queries are necessary for testing C₄-freeness, and that Õ(n^{1/4}) are sufficient. The same bounds hold for C₅. 2) For every fixed k ≥ 6, any one-sided error algorithm for testing C_k-freeness must perform Ω(n^{1/3}) queries. 3) For k = 6 we give a testing algorithm whose query complexity is Õ(n^{1/2}). 4) For any fixed k, the query complexity of testing C_k-freeness is upper bounded by {O}(n^{1-1/⌊k/2⌋}). The last upper bound builds on another result in which we show that for any fixed subgraph F, the query complexity of testing F-freeness is upper bounded by O(n^{1-1/𝓁(F)}), where 𝓁(F) is a parameter of F that is always upper bounded by the number of vertices in F (and in particular is k/2 in C_k for even k). We extend some of our results to bounded (non-constant) arboricity, where in particular, we obtain sublinear upper bounds for all k. Our Ω(n^{1/4}) lower bound for testing C₄-freeness in constant arboricity graphs provides a negative answer to an open problem posed by (Goldreich, 2021).

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Property Testing
  • Cycle-Freeness
  • Bounded Arboricity

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References

  1. Noga Alon. Testing subgraphs in large graphs. Random Struct. Algorithms, 21(3-4):359-370, 2002. URL: https://doi.org/10.1002/rsa.10056.
  2. Noga Alon, Eldar Fischer, Michael Krivelevich, and Mario Szegedy. Efficient testing of large graphs. Combinatorica, 20(4):451-476, 2000. URL: https://doi.org/10.1007/s004930070001.
  3. 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. Google Scholar
  4. Noga Alon, Raphael Yuster, and Uri Zwick. Color coding. Journal of the ACM, 42(4):844-856, 1995. Google Scholar
  5. Uri Alon. Network motifs: theory and experimental approaches. Nature Reviews Genetics, 8(6):450-461, 2007. Google Scholar
  6. Jonathan W. Berry, Bruce Hendrickson, Randall A. LaViolette, and Cynthia A. Phillips. Tolerating the Community Detection Resolution Limit with Edge Weighting. Physical Review E, 83(5):056119, May 2011. Google Scholar
  7. Ronald S. Burt. Structural holes and good ideas. American journal of sociology, 110(2):349-399, 2004. Google Scholar
  8. James S. Coleman. Social capital in the creation of human capital. American Journal of Sociology, 94:S95-S120, 1988. URL: http://www.jstor.org/stable/2780243.
  9. Artur Czumaj, Oded Goldreich, Dana Ron, C. Seshadhri, Asaf Shapira, and Christian Sohler. Finding cycles and trees in sublinear time. Random Structures and Algorithms, 45(2):139-184, 2014. Google Scholar
  10. Artur Czumaj and Christian Sohler. A characterization of graph properties testable for general planar graphs with one-sided error (it’s all about forbidden subgraphs). In 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS, pages 1525-1548, 2019. URL: https://doi.org/10.1109/FOCS.2019.00089.
  11. Jean-Pierre Eckmann and Elisha Moses. Curvature of co-links uncovers hidden thematic layers in the World Wide Web. Proceedings of the National Academy of Sciences, 99(9):5825-5829, 2002. URL: https://doi.org/10.1073/pnas.032093399.
  12. Talya Eden, Dana Ron, and Will Rosenbaum. The arboricity captures the complexity of sampling edges. In 46th International Colloquium on Automata, Languages, and Programming, ICALP, pages 52:1-52:14, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.52.
  13. 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.
  14. Talya Eden, Dana Ron, and C. Seshadhri. Sublinear time estimation of degree distribution moments: The arboricity connection. SIAM J. Discret. Math., 33(4):2267-2285, 2019. URL: https://doi.org/10.1137/17M1159014.
  15. 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.
  16. Paul Erdős. Extremal problems in graph theory. In In Proc. Symp. Theory of Graphs and its Applications, pages 29-36, 1963. Google Scholar
  17. Guy Even, Orr Fischer, Pierre Fraigniaud, Tzlil Gonen, Reut Levi, Moti Medina, Pedro Montealegre, Dennis Olivetti, Rotem Oshman, Ivan Rapaport, and Ioan Todinca. Three notes on distributed property testing. In 31st International Symposium on Distributed Computing, DISC, pages 15:1-15:30, 2017. URL: https://doi.org/10.4230/LIPIcs.DISC.2017.15.
  18. Brooke Foucault Welles, Anne Van Devender, and Noshir Contractor. Is a friend a friend?: Investigating the structure of friendship networks in virtual worlds. In CHI Extended Abstracts on Human Factors in Computing Systems, pages 4027-4032, 2010. Google Scholar
  19. Oded Goldreich. Open problems in property testing of graphs. In Electron. Colloquium Comput. Complex., volume 28, page 88, 2021. Google Scholar
  20. Oded Goldreich and Dana Ron. Property testing in bounded degree graphs. Algorithmica, 32(2):302-343, 2002. Google Scholar
  21. Paul W. Holland and Samuel Leinhardt. A method for detecting structure in sociometric data. American Journal of Sociology, 76:492-513, 1970. Google Scholar
  22. Kazuo Iwama and Yuichi Yoshida. Parameterized testability. ACM Trans. Comput. Theory, 9(4):16:1-16:16, 2018. URL: https://doi.org/10.1145/3155294.
  23. Matthew O. Jackson, Tomas Rodriguez-Barraquer, and Xu Tan. Social capital and social quilts: Network patterns of favor exchange. American Economic Review, 102(5):1857-1897, 2012. Google Scholar
  24. Tali Kaufman, Michael Krivelevich, and Dana Ron. Tight bounds for testing bipartiteness in general graphs. SIAM Journal on Computing, 33(6):1441-1483, 2004. URL: https://doi.org/10.1137/S0097539703436424.
  25. Akash Kumar, C. Seshadhri, and Andrew Stolman. Random walks and forbidden minors III: poly(dε^-1)-time partition oracles for minor-free graph classes. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS, pages 257-268, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00034.
  26. Reut Levi. Testing triangle freeness in the general model in graphs with arboricity o(√n). In 48th International Colloquium on Automata, Languages, and Programming, ICALP, volume 198, pages 93:1-93:13, 2021. Google Scholar
  27. Reut Levi and Nadav Shoshan. Testing Hamiltonicity (and other problems) in minor-free graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM, pages 61:1-61:23, 2021. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.61.
  28. Ron Milo, Shai Shen-Orr, Shalev Itzkovitz, Nadav Kashtan, Dmitri Chklovskii, and Uri Alon. Network motifs: simple building blocks of complex networks. Science, 298(5594):824-827, 2002. Google Scholar
  29. Ilan Newman and Christian Sohler. Every property of hyperfinite graphs is testable. SIAM Journal on Computing, 42(3):1095-1112, 2013. URL: https://doi.org/10.1137/120890946.
  30. Michal Parnas and Dana Ron. Testing the diameter of graphs. Random Struct. Algorithms, 20(2):165-183, 2002. URL: https://doi.org/10.1002/rsa.10013.
  31. Alejandro Portes. Social capital: Its origins and applications in modern sociology. In Eric L. Lesser, editor, Knowledge and Social Capital, pages 43-67. Butterworth-Heinemann, 2000. URL: https://doi.org/10.1016/B978-0-7506-7222-1.50006-4.
  32. C. Seshadhri, Tamara G. Kolda, and Ali Pinar. Community structure and scale-free collections of Erdös-Rényi graphs. Physical Review E, 85(5):056109, May 2012. URL: https://doi.org/10.1103/PhysRevE.85.056109.