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The Subgraph Testing Model

Published:08 November 2020Publication History
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Abstract

Following Newman (2010), we initiate a study of testing properties of graphs that are presented as subgraphs of a fixed (or an explicitly given) graph. The tester is given free access to a base graph G = ([n], E) and oracle access to a function f : E → {0, 1} that represents a subgraph of G. The tester is required to distinguish between subgraphs that possess a predetermined property and subgraphs that are far from possessing this property.

We focus on bounded-degree base graphs and on the relation between testing graph properties in the subgraph model and testing the same properties in the bounded-degree graph model. We identify cases in which testing is significantly easier in one model than in the other as well as cases in which testing has approximately the same complexity in both models. Our proofs are based on the design and analysis of efficient testers and on the establishment of query-complexity lower bounds.

References

  1. N. Alon, P. D. Seymour, and R. Thomas. 1990. A separator theorem for graphs with an excluded minor and its applications. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing (STOC’90). 293--299.Google ScholarGoogle Scholar
  2. E. Ben-Sasson, P. Harsha, and S. Raskhodnikova. 2005. 3CNF properties are hard to test. SIAM J. Comput. 35, 1 (2005), 1--21.Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. I. Benjamini, O. Schramm, and A. Shapira. 2010. Every minor-closed property of sparse graphs is testable. Adv. Math. 223, 6 (2010), 2200--2218.Google ScholarGoogle ScholarCross RefCross Ref
  4. A. Bogdanov, K. Obata, and L. Trevisan. 2002. A lower bound for testing 3-colorability in bounded-degree graphs. In Proceedings of the 43rd Annual Symposium on Foundations of Computer Science (FOCS’02). 93--102.Google ScholarGoogle Scholar
  5. A. Czumaj, M. Monemizadeh, K. Onak, and C. Sohler. 2019. Planar graphs: Random walks and bipartiteness testing. Rand. Struct. Algor. 55, 1 (2019), 104--124.Google ScholarGoogle Scholar
  6. A. Czumaj, A. Shapira, and C. Sohler. 2009. Testing hereditary properties of nonexpanding bounded-degree graphs. SIAM J. Comput. 38, 6 (2009), 2499--2510.Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. A. Edelman, A. Hassidim, H. N. Nguyen, and K. Onak. 2011. An efficient partitioning oracle for bounded-treewidth graphs. In Proceedings of the 15th International Workshop on Randomization and Computation (RANDOM’11). 530--541.Google ScholarGoogle Scholar
  8. G. Elek. 2006. The combinatorial cost. arXiv:math/0608474. Retrieved from https://arxiv.org/abs/math/0608474.Google ScholarGoogle Scholar
  9. G. Elek. 2010. Parameter testing with bounded degree graphs of subexponential growth. Rand. Struct. Algor. 37, 2 (2010), 248--270.Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. U. Feige and S. Jozeph. 2012. Universal factor graphs. In Proceedings of the 39th International Colloquium on Automata, Languages and Programming (ICALP’12). 339--350.Google ScholarGoogle Scholar
  11. E. Fischer, O. Lachish, A. Matsliah, I. Newman, and O. Yahalom. 2012. On the query complexity of testing orientations for being Eulerian. ACM Trans. Algor. 8, 2 (2012), 15:1--15:41.Google ScholarGoogle Scholar
  12. S. Forster, D. Nanongkai, L. Yang, T. Saranurak, and S. Yingchareonthawornchai. 2020. Computing and testing small connectivity in near-linear time and queries via fast local cut algorithms. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA’20), Shuchi Chawla (Ed.). SIAM, 2046--2065. DOI:https://doi.org/10.1137/1.9781611975994.126Google ScholarGoogle Scholar
  13. O. Goldreich. 2008. Computational Complexity: A Conceptual Perspective. Cambridge University Press.Google ScholarGoogle ScholarCross RefCross Ref
  14. O. Goldreich. 2017. Introduction to Property Testing. Cambridge University Press.Google ScholarGoogle Scholar
  15. O. Goldreich, S. Goldwasser, and D. Ron. 1998. Property testing and its connection to learning and approximation. J. ACM 45, 4 (1998), 653--750.Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. O. Goldreich and D. Ron. 1999. A sublinear bipartite tester for bounded-degree graphs. Combinatorica 19, 3 (1999), 335--373.Google ScholarGoogle ScholarCross RefCross Ref
  17. O. Goldreich and D. Ron. 2002. Property testing in bounded degree graphs. Algorithmica 32, 2 (2002), 302--343.Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. O. Goldreich and D. Ron. 2011. On proximity oblivious testing. SIAM J. Comput. 40, 2 (2011), 534--566.Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. O. Goldreich and D. Ron. 2018. The subgraph testing model. Electr. Colloq. Comput. Complex. 25, 45 (2018). Technical report: TR18-045.Google ScholarGoogle Scholar
  20. M. Gonen and D. Ron. 2010. On the benefit of adaptivity in property testing of dense graphs. Algorithmica 58, 4 (2010), 811--830.Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. S. Halevy, O. Lachish, I. Newman, and D. Tsur. 2005. Testing orientation properties. Electr. Colloq. Comput. Complex. 153 (2005).Google ScholarGoogle Scholar
  22. A. Hassidim, J. Kelner, H. Nguyen, and K. Onak. 2009. Local graph partitions for approximation and testing. In Proceedings of the 50th Annual Symposium on Foundations of Computer Science (FOCS’09). 22--31.Google ScholarGoogle Scholar
  23. L. S. Heath. 1987. Embedding outerplanar graphs in small books. SIAM J. Algebr. Discr. Methods 8, 2 (1987), 198--218.Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. T. Kaufman, M. Krivelevich, and D. Ron. 2004. Tight bounds for testing bipartiteness in general graphs. SIAM J. Comput. 33, 6 (2004), 1441--1483.Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. R. Levi and D. Ron. 2015. A quasi-polynomial time partition oracle for graphs with an excluded minor. ACM Trans. Algor. 11, 3 (2015), 24:1--24:13.Google ScholarGoogle Scholar
  26. R. J. Lipton and R. E. Tarjan. 1979. A separator theorem for planar graphs. SIAM J. Discr. Math. 36, 2 (1979), 177--189.Google ScholarGoogle Scholar
  27. I. Newman. 2010. Property testing of massively parametrized problems—A survey. In Property Testing: Current Research and Surveys, LNCS 6390, O. Goldreich (Ed.). Springer, 142--157.Google ScholarGoogle Scholar
  28. I. Newman and C. Sohler. 2013. Every property of hyperfinite graphs is testable. SIAM J. Comput. 42, 3 (2013), 1095--1112.Google ScholarGoogle ScholarCross RefCross Ref
  29. M. Parnas and D. Ron. 2002. Testing the diameter of graphs. Rand. Struct. Algor. 20, 2 (2002), 165--183.Google ScholarGoogle ScholarDigital LibraryDigital Library
  30. M. Parnas, D. Ron, and R. Rubinfeld. 2006. Tolerant property testing and distance approximation. J. Comput. Syst. Sci. 72, 6 (2006), 1012--1042.Google ScholarGoogle ScholarDigital LibraryDigital Library
  31. E. Petrank. 1994. The hardness of approximation: Gap location. Comput. Complex. 4 (1994), 133--157.Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. G. Valiant. 2012. Algorithmic Approaches to Statistical Questions. Ph.D. Dissertation. University of California at Berkeley.Google ScholarGoogle ScholarDigital LibraryDigital Library
  33. G. Valiant and P. Valiant. 2017. Estimating the unseen: Improved estimators for entropy and other properties. J. ACM 64, 6 (2017), 37:1--37:41.Google ScholarGoogle ScholarDigital LibraryDigital Library

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      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 12, Issue 4
        December 2020
        156 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/3427631
        Issue’s Table of Contents

        Copyright © 2020 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 8 November 2020
        • Accepted: 1 August 2020
        • Revised: 1 June 2020
        • Received: 1 February 2020
        Published in toct Volume 12, Issue 4

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