GSF-Locality Is Not Sufficient For Proximity-Oblivious Testing

Authors Isolde Adler, Noleen Köhler, Pan Peng

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Author Details

Isolde Adler
  • School of Computing, University of Leeds, UK
Noleen Köhler
  • School of Computing, University of Leeds, UK
Pan Peng
  • Department of Computer Science, University of Sheffield, UK

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Isolde Adler, Noleen Köhler, and Pan Peng. GSF-Locality Is Not Sufficient For Proximity-Oblivious Testing. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 34:1-34:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In Property Testing, proximity-oblivious testers (POTs) form a class of particularly simple testing algorithms, where a basic test is performed a number of times that may depend on the proximity parameter, but the basic test itself is independent of the proximity parameter. In their seminal work, Goldreich and Ron [STOC 2009; SICOMP 2011] show that the graph properties that allow constant-query proximity-oblivious testing in the bounded-degree model are precisely the properties that can be expressed as a generalised subgraph freeness (GSF) property that satisfies the non-propagation condition. It is left open whether the non-propagation condition is necessary. Indeed, calling properties expressible as a generalised subgraph freeness property GSF-local properties, they ask whether all GSF-local properties are non-propagating. We give a negative answer by exhibiting a property of graphs that is GSF-local and propagating. Hence in particular, our property does not admit a POT, despite being GSF-local. We prove our result by exploiting a recent work of the authors which constructed a first-order (FO) property that is not testable [SODA 2021], and a new connection between FO properties and GSF-local properties via neighbourhood profiles.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Finite Model Theory
  • Property testing
  • proximity-oblivous testing
  • locality
  • first-order logic
  • lower bound


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  1. Isolde Adler, Noleen Köhler, and Pan Peng. On testability of first-order properties in bounded-degree graphs. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1578-1597. SIAM, 2021. Google Scholar
  2. Noga Alon, Eldar Fischer, Ilan Newman, and Asaf Shapira. A combinatorial characterization of the testable graph properties: it’s all about regularity. SIAM Journal on Computing, 39(1):143-167, 2009. Google Scholar
  3. Itai Benjamini, Oded Schramm, and Asaf Shapira. Every minor-closed property of sparse graphs is testable. Advances in mathematics, 223(6):2200-2218, 2010. Google Scholar
  4. Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/correcting with applications to numerical problems. Journal of computer and system sciences, 47(3):549-595, 1993. Google Scholar
  5. Heinz-Dieter Ebbinghaus and Jörg Flum. Finite model theory. Perspectives in Mathematical Logic. Springer, 1995. Google Scholar
  6. Hendrik Fichtenberger, Pan Peng, and Christian Sohler. Every testable (infinite) property of bounded-degree graphs contains an infinite hyperfinite subproperty. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 714-726. Society for Industrial and Applied Mathematics, 2019. Google Scholar
  7. Sebastian Forster, Danupon Nanongkai, Thatchaphol Saranurak, Liu Yang, and Sorrachai Yingchareonthawornchai. Computing and testing small connectivity in near-linear time and queries via fast local cut algorithms. SODA, 2020. Google Scholar
  8. Haim Gaifman. On local and non-local properties. In Studies in Logic and the Foundations of Mathematics, volume 107, pages 105-135. Elsevier, 1982. Google Scholar
  9. Oded Goldreich. Introduction to property testing. Cambridge University Press, 2017. Google Scholar
  10. Oded Goldreich, Shari Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. Journal of the ACM (JACM), 45(4):653-750, 1998. Google Scholar
  11. Oded Goldreich and Tali Kaufman. Proximity oblivious testing and the role of invariances. In Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation, pages 173-190. Springer, 2011. Google Scholar
  12. Oded Goldreich and Dana Ron. Property testing in bounded degree graphs. Algorithmica, 32(2):302-343, 2002. URL:
  13. Oded Goldreich and Dana Ron. On proximity-oblivious testing. SIAM Journal on Computing, 40(2):534-566, 2011. Preliminary version appeared at Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC 2009). Google Scholar
  14. Oded Goldreich and Igor Shinkar. Two-sided error proximity oblivious testing. Random Structures & Algorithms, 48(2):341-383, 2016. Google Scholar
  15. William Hanf. The Theory of Models, chapter Model-theoretic methods in the study of elementary logic, pages 132-145. North Holland, 1965. Google Scholar
  16. Avinatan Hassidim, Jonathan A Kelner, Huy N Nguyen, and Krzysztof Onak. Local graph partitions for approximation and testing. In 2009 50th Annual IEEE Symposium on Foundations of Computer Science, pages 22-31. IEEE, 2009. Google Scholar
  17. Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. BULL. AMER. MATH. SOC., 43(4):439-561, 2006. Google Scholar
  18. Hiro Ito, Areej Khoury, and Ilan Newman. On the characterization of 1-sided error strongly testable graph properties for bounded-degree graphs. Computational Complexity, 29(1):1-45, 2020. Google Scholar
  19. Ken-ichi Kawarabayashi and Yuichi Yoshida. Testing subdivision-freeness: property testing meets structural graph theory. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pages 437-446. ACM, 2013. Google Scholar
  20. Akash Kumar, C Seshadhri, and Andrew Stolman. Random walks and forbidden minors ii: a poly (d ε-1)-query tester for minor-closed properties of bounded degree graphs. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 559-567, 2019. Google Scholar
  21. László Lovász. Large Networks and Graph Limits, volume 60 of Colloquium Publications. American Mathematical Society, 2012. Google Scholar
  22. Ilan Newman and Christian Sohler. Every property of hyperfinite graphs is testable. SIAM Journal on Computing, 42(3):1095-1112, 2013. Google Scholar
  23. Omer Reingold, Salil Vadhan, and Avi Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders. Annals of mathematics, pages 157-187, 2002. Google Scholar
  24. Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing, 25(2):252-271, 1996. Google Scholar
  25. Yuichi Yoshida and Hiro Ito. Property testing on k-vertex-connectivity of graphs. Algorithmica, 62(3-4):701-712, 2012. Google Scholar
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