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.
- 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 Scholar
- E. Ben-Sasson, P. Harsha, and S. Raskhodnikova. 2005. 3CNF properties are hard to test. SIAM J. Comput. 35, 1 (2005), 1--21.Google Scholar
Digital Library
- 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 Scholar
Cross Ref
- 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 Scholar
- 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 Scholar
- 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 Scholar
Digital Library
- 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 Scholar
- G. Elek. 2006. The combinatorial cost. arXiv:math/0608474. Retrieved from https://arxiv.org/abs/math/0608474.Google Scholar
- G. Elek. 2010. Parameter testing with bounded degree graphs of subexponential growth. Rand. Struct. Algor. 37, 2 (2010), 248--270.Google Scholar
Digital Library
- 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 Scholar
- 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 Scholar
- 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 Scholar
- O. Goldreich. 2008. Computational Complexity: A Conceptual Perspective. Cambridge University Press.Google Scholar
Cross Ref
- O. Goldreich. 2017. Introduction to Property Testing. Cambridge University Press.Google Scholar
- 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 Scholar
Digital Library
- O. Goldreich and D. Ron. 1999. A sublinear bipartite tester for bounded-degree graphs. Combinatorica 19, 3 (1999), 335--373.Google Scholar
Cross Ref
- O. Goldreich and D. Ron. 2002. Property testing in bounded degree graphs. Algorithmica 32, 2 (2002), 302--343.Google Scholar
Digital Library
- O. Goldreich and D. Ron. 2011. On proximity oblivious testing. SIAM J. Comput. 40, 2 (2011), 534--566.Google Scholar
Digital Library
- O. Goldreich and D. Ron. 2018. The subgraph testing model. Electr. Colloq. Comput. Complex. 25, 45 (2018). Technical report: TR18-045.Google Scholar
- M. Gonen and D. Ron. 2010. On the benefit of adaptivity in property testing of dense graphs. Algorithmica 58, 4 (2010), 811--830.Google Scholar
Digital Library
- S. Halevy, O. Lachish, I. Newman, and D. Tsur. 2005. Testing orientation properties. Electr. Colloq. Comput. Complex. 153 (2005).Google Scholar
- 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 Scholar
- L. S. Heath. 1987. Embedding outerplanar graphs in small books. SIAM J. Algebr. Discr. Methods 8, 2 (1987), 198--218.Google Scholar
Digital Library
- 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 Scholar
Digital Library
- 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 Scholar
- R. J. Lipton and R. E. Tarjan. 1979. A separator theorem for planar graphs. SIAM J. Discr. Math. 36, 2 (1979), 177--189.Google Scholar
- 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 Scholar
- I. Newman and C. Sohler. 2013. Every property of hyperfinite graphs is testable. SIAM J. Comput. 42, 3 (2013), 1095--1112.Google Scholar
Cross Ref
- M. Parnas and D. Ron. 2002. Testing the diameter of graphs. Rand. Struct. Algor. 20, 2 (2002), 165--183.Google Scholar
Digital Library
- M. Parnas, D. Ron, and R. Rubinfeld. 2006. Tolerant property testing and distance approximation. J. Comput. Syst. Sci. 72, 6 (2006), 1012--1042.Google Scholar
Digital Library
- E. Petrank. 1994. The hardness of approximation: Gap location. Comput. Complex. 4 (1994), 133--157.Google Scholar
Digital Library
- G. Valiant. 2012. Algorithmic Approaches to Statistical Questions. Ph.D. Dissertation. University of California at Berkeley.Google Scholar
Digital Library
- 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 Scholar
Digital Library
Index Terms
The Subgraph Testing Model
Recommendations
Three theorems regarding testing graph properties
Property testing is a relaxation of decision problems in which it is required to distinguish YES-instances (i.e., objects having a predetermined property) from instances that are far from any YES-instance. We presents three theorems regarding testing ...
Approximating the distance to properties in bounded-degree and general sparse graphs
We address the problem of approximating the distance of bounded-degree and general sparse graphs from having some predetermined graph property P. That is, we are interested in sublinear algorithms for estimating the fraction of edge modifications (...
Every Property of Hyperfinite Graphs Is Testable
† Special Section on the Forty-Second Annual ACM Symposium on Theory of Computing (STOC 2010)A $k$-disc around a vertex $v$ of a graph $G=(V,E)$ is the subgraph induced by all vertices of distance at most $k$ from $v$. We show that the structure of a planar graph on $n$ vertices, and with constant maximum degree $d$, is determined, up to the ...






Comments