The Review of Socionetwork Strategies

, Volume 13, Issue 2, pp 101–121 | Cite as

What Graph Properties Are Constant-Time Testable?

Dense Graphs, Sparse Graphs, and Complex Networks
  • Hiro ItoEmail author


In this paper, we survey what graph properties have been found to be constant-time testable at present. How to handle big data is a very important issue in computer science. Developing efficient algorithms for problems on big data is an urgent task. For this purpose, constant-time algorithms are powerful tools, since they run by reading only a constant-sized part of each input. In other words, the running time is invariant regardless of the size of the input. The idea of constant-time algorithms appeared in the 1990s and spread quickly especially in this century. Research on graph algorithms is one of the best studied areas in theoretical computer science. When we study constant-time algorithms on graphs, we have three models that differ in the way that the graphs are represented: the dense-graph model, the bounded-degree model, and the general model. The first one is used to treat properties on dense graphs, and the other two are used to treat properties on sparse graphs. In this paper, we survey one by one what properties have been found to be constant-time testable in each of the three models.


Constant-time algorithms Property testing Graphs Regularity lemma Hyperfinite Universal testers Complex networks 



The author thanks Professor Ilan Newman of the University of Haifa, Israel, Associate Professor Yuichi Yoshida of the National Institute of Informatics, Japan, and Associate Professor Suguru Tamaki of University of Hyogo, Japan for their helpful advice.


  1. 1.
    Albert, R., & Barabási, A.-L. (2002). Statistical mechanics of complex networks. Review of Modern Physics, 74, 47–97.CrossRefGoogle Scholar
  2. 2.
    Alon, N., Fischer, E., Newman, I., & Shapira, A. (2008). A combinatorial characterization of the testable graph properties: It’s all about regularity. SIAM Journal on Computing, 37(6), 1703–1727.CrossRefGoogle Scholar
  3. 3.
    Alon, N., Seymour, P., & Thomas, R. (1990). Separator theorem for graphs with an excluded minor and its applications. Proceedings STOC, 1990, 293–299.CrossRefGoogle Scholar
  4. 4.
    Alon, N., & Shapira, A. (2005). Every monotone graph property is testable. In Proceedings of STOC 2005 (pp. 128–138). (Journal version: SIAM J. Comput., Vol. 38, No. 6, pp. 1703–1727, (2008)).Google Scholar
  5. 5.
    Alon, N., & Shapira, A. (2005). A characterization of the (natural) graph properties testable with one-sided error. In Proceedings of FOCS 2005 (pp. 429–438) (Journal version: SIAM J. Comput., Vol. 37, No. 6, pp. 1703–1727, (2008)).Google Scholar
  6. 6.
    Babu, J., Khoury, A., & Newman, I. (2016). Every property of outerplanar graphs is testable. In Proceedings of RANDOM 2016. LIPICS (pp. 1–21:19).Google Scholar
  7. 7.
    Benjamini, I., Schramm, O., & Shapira, A. (2008). Every minor-closed property of sparse graphs is testable. In Proceedings of STOC 2008. ACM (pp. 393–402).Google Scholar
  8. 8.
    Bottreau, A., & Métivier, Y. (1998). Some remarks on the Kronecker product of graphs. In Information Processing Letters (vol. 68, pp. 55–61). Amsterdam: Elsevier.Google Scholar
  9. 9.
    Broder, A. Z., Kumar, S. R., Maghoul, F., Raghavan, R., Rajagoplalan, S., Stata, R., et al. (2000). Graph structure in the Web. Computer Networks, 33, 309–320.CrossRefGoogle Scholar
  10. 10.
    Cohen-Steiner, D., Kong, W., Sohler, C., & Valiant, G. (2018). Approximating the spectrum of a graph. Proceedings of SIGKDD, 2018, 1263–1271.Google Scholar
  11. 11.
    Cooper, C., & Uehara, R. (2010). Scale free properties of random \(k\)-trees. Mathematics in Computer Science, 3, 489–496.CrossRefGoogle Scholar
  12. 12.
    Diestel, R. (2016). Graph Theory (5th ed.). Berlin: Springer.Google Scholar
  13. 13.
    Elek, G. (2008). \(L^2\)-spectral invariants and convergent sequence of finite graphs. Journal of Functional Analysis, 254(10), 2667–2689.CrossRefGoogle Scholar
  14. 14.
    Fichtenberger, H., Peng, P., & Sohler, C. (2018). Every testable (infinite) property of bounded-degree graphs contains an infinite hyperfinite subproperty. arXiv: 1811.02937 (also appeared in SODA2019).
  15. 15.
    Gao, Y. (2009). The degree distribution of random \(k\)-trees. Theoretical Computer Science, 410, 688–695.CrossRefGoogle Scholar
  16. 16.
    Goldreich, O. (2017). Introduction to Property Testing. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  17. 17.
    Goldreich, O., & Ron, D. (1997). Property testing in bounded degree graphs: Proc. STOC, 1997, 406–415.CrossRefGoogle Scholar
  18. 18.
    Goldreich, O., & Ron, D. (2011). On proximity-oblivious testing. SIAM Journal on Computing, 40(02), 534–566.CrossRefGoogle Scholar
  19. 19.
    Goldreich, O., & Trevisan, L. (2001). Three theorems regarding testing graph properties. In Proceedings FOCS 2001 (pp. 460–469) (Journal version: Random Structures & Algorithms, Wiley, Vol. 23, Issue 1, pp. 23–57, (2003)).Google Scholar
  20. 20.
    Hassidim, A., Kelner, J.A., Nguyen, H.N., & Onak, K. (2009). Local graph partitions for approximation and testing. In: Proceedings FOCS 2009 (pp. 22–31). IEEE.Google Scholar
  21. 21.
    Ito, H. (2015). Every property is testable on a natural class of scale-free multigraphs, Cornell University (pp. 1–12). arXiv:1504.00766
  22. 22.
    Ito, H. (2016). Every property is testable on a natural class of scale-free multigraphs. In: Proceedings of ESA 2016, LIPICS (ISBN 978-3-95977-005-7) (vol. 49, pp. 21:2–21:15).Google Scholar
  23. 23.
    Ito, H., & Iwama, K. (2009). Enumeration of isolated cliques and pseudo-cliques. In ACM Transactions on Algorithms (vol. 5, Issue 4, Article 40, pp. 1–13).Google Scholar
  24. 24.
    Ito, H., Iwama, K., & Osumi, T. (2005). Linear-time enumeration of isolated cliques. In Proceedings ESA2005, LNCS (vol. 3669, pp. 119–130). Springer.Google Scholar
  25. 25.
    Kawarabayashi, K., & Reed, B. (2010). A separator theorem in minor-closed classes. Proceedings of FOCS, 2010, 153–162.Google Scholar
  26. 26.
    Kleinberg, J., & Lawrence, S. (2001). The structure of the web. Science, 294, 1894–1895.CrossRefGoogle Scholar
  27. 27.
    Kusumoto, M., & Yoshida, Y. (2014). Testing forest-isomorphizm in the adjacency list model. In Proceedings of ICALP2014 (1), LNSC 8572 (pp. 763–774).Google Scholar
  28. 28.
    Levi, R., & Ron, D. (2013). A quasi-polynomial time partition oracle for graphs with an excluded mino. In Proceedings of ICALP 2013 (1), LNCS, 7965 (pp. 709–720). Springer. (Journal version: ACM Transactions on Algorithms, Vol. 11, No. 3, Article 24, pp. 1–13, (2014)).Google Scholar
  29. 29.
    Mahdian, M., & Xu, Y. (2007). Stochastic Kronecker graphs. In Proc. WAW 2007, LNCS, 4863. (pp. 179–186). Springer.Google Scholar
  30. 30.
    Marko, S., & Ron, D. (2009). Approximating the distance to properties in bounded-degree and general sparse graphs. In ACM Transactions on Algorithms (Vol. 5, Issue. 2, Article No. 22).Google Scholar
  31. 31.
    Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Review, 45, 167–256.Google Scholar
  32. 32.
    Newman, I., & Sohler, C. (2011). Every property of hyperfinite graphs is testable. In PROC. STOC 2011, ACM (pp. 675–784) (Journal version: SIAM J. Comput., Vol. 42, No. 3, pp. 1095–1112, (2013)).Google Scholar
  33. 33.
    Robertson, N., & Seymour, P.D. (1983–2012). Graph Minors I–XXII. Journal of Combinatorial Theory, Series B. Google Scholar
  34. 34.
    Shigezumi, T., Uno, Y., & Watanabe, O. (2011). A new model for a scale-free hierarchical structure of isolated cliques. Journal of Graph Algorithms and Applications, 15(5), 661–682.CrossRefGoogle Scholar
  35. 35.
    Szemerédi, E. (1978). Regular partitions of graphs. In J. C. Bermond, J. C. Fournier, M. Las Vergnas, & D. Sotteau (Eds.) Colloq. Internat. CNRS, Paris (pp. 399–401).Google Scholar
  36. 36.
    Sárközy, G.N., Song, F., Szemerédi, E., & Trivedi, S. (2012). A practical regularity partitioning algorithm and its applications in clustering. (pp. 1–13) Cornell University, Sept. 28. arXiv:1209.6540v1.
  37. 37.
    Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature, 393, 440–442.CrossRefGoogle Scholar
  38. 38.
    Yoshida, Y., & Ito, H. (2008). Property testing on k-vertex connectivity of graphs. In: Proc. ICALP 2008, (1), LNCS, #5125 (pp. 539–550). Springer (Journal version: Algorithmica, Vol. 62, No. 3–4, pp. 701–712, (2012)).Google Scholar
  39. 39.
    Yoshida, Y., Yamamoto, M., & Ito, H. (2009). An improved constant-time approximation algorithm for maximum matchings. In Proceedings of STOC 2009 (pp. 225–234). (Journal version: SIAM J. Comput., Vol. 41, No. 4, pp. 1074–1093, (2012)).Google Scholar
  40. 40.
    Zhang, Z., Rong, L., & Comellas, F. (2006). High-dimensional random apollonian networks. Physica A: Statistical Mechanics and its Applications, 364, 610–618.CrossRefGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Informatics and EngineeringThe University of Electro-CommunicationsTokyoJapan

Personalised recommendations