Abstract
We consider natural graph property tests, which act entirely independently of the size of the graph being tested. We introduce the notion of properties being inflatable — closed under taking (balanced) blowups — and show that the query complexity of natural tests for a property is related to the degree to which it is approximately hereditary and inflatable. Specifically, we show that for properties which are almost hereditary and almost inflatable, any test can be made natural, with a polynomial increase in the number of queries. The naturalization can be considered as an extension of the canonicalization due to [15], so that natural canonical tests can be described as strongly canonical.
Using the technique for naturalization, we restore in part the claim in [15] regarding testing hereditary properties by ensuring that a small random subgraph itself satisfies the property. This allows us to generalize the triangle-freeness lower bound result of [5]: Any lower bound, not only the currently established quasi-polynomial one, on one-sided testing for triangle-freeness holds essentially for two-sided testing as well. We also explore the relations of the notion of inflatability and other already-studied features of properties and property tests, such as one-sidedness, heredity, and proximity-oblivion.
Eldar Fischer’s research supported in part by ERC-2007-StG grant no. 202405 and by an ISF grant no. 1101/06.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alon, N.: Private communication (1999)
Alon, N.: Testing subgraphs in large graphs. Rnd. Str. & Alg. 21(3-4), 359–370 (2002)
Alon, N., Fischer, E., Krivelevich, M., Szegedy, M.: Efficient testing of large graphs. Combinatorica 20, 451–476 (2000)
Alon, N., Fischer, E., Newman, I., Shapira, A.: A combinatorial characterization of the testable graph properties. SIAM J. Comp. 39(1), 143–167 (2009)
Alon, N., Shapira, A.: A characterization of easily testable induced subgraphs. Combinatorics, Probability and Computing 15(6), 791–805 (2006)
Alon, N., Shapira, A.: A characterization of the (natural) graph properties testable with one-sided error. SIAM J. Comp. 37(6), 1703–1727 (2008)
Borgs, C., Chayes, J., Lovász, L., Sós, V.T., Szegedy, B., Vesztergombi, K.: Graph limits and parameter testing. In: 38th STOC, pp. 261–270. ACM Press, New York (2006)
Fischer, E.: The art of uninformed decisions: A primer to property testing. In: Paun, G., Rozenberg, G., Salomaa, A. (eds.) Current Trends in Theoretical Computer Science, vol. 1, pp. 229–264. World Scientific Publishing, Singapore (2004)
Fischer, E., Newman, I.: Testing versus estimation of graph properties. SIAM J. Comp. 37(2), 482–501 (2007)
Fischer, E., Rozenberg, E.: Inflatable graph properties and natural property tests, http://www.cs.technion.ac.il/~eyalroz/publications/FR2011.pdf
Fox, J.: A new proof of the graph removal lemma. Ann. Math (to appear)
Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. Journal of the ACM 45(4), 653–750 (1998)
Goldreich, O., Krivelevich, M., Newman, I., Rozenberg, E.: Hierarchy theorems for property testing. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX 2009. LNCS, vol. 5687, pp. 504–519. Springer, Heidelberg (2009)
Goldreich, O., Ron, D.: Algorithmic aspects of property testing in the dense graphs model. In: Goldreich, O. (ed.) Property Testing. LNCS, vol. 6390, pp. 295–305. Springer, Heidelberg (2010)
Goldreich, O., Trevisan, L.: Three theorems regarding testing graph properties. Rnd. Str. & Alg. 23(1), 23–57 (2003)
Goldreich, O., Trevisan, L.: Errata for [15] (2005), http://www.wisdom.weizmann.ac.il/~oded/PS/tt-rr.ps
Yao, A.C.C.: Probabilistic computations: Toward a unified measure of complexity (extended abstract). In: 18th FOCS, pp. 222–227 (1977)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fischer, E., Rozenberg, E. (2011). Inflatable Graph Properties and Natural Property Tests. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2011 2011. Lecture Notes in Computer Science, vol 6845. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22935-0_46
Download citation
DOI: https://doi.org/10.1007/978-3-642-22935-0_46
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22934-3
Online ISBN: 978-3-642-22935-0
eBook Packages: Computer ScienceComputer Science (R0)