E cient testing of large graphs

Let P be a property of graphs. An -test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modi ed by adding and removing more than n 2 edges to make it satisfy P . The property P is called testable, if for every there exists an -test for P whose total number of queries is independent of the size of the input graph. Goldreich, Goldwasser and Ron [8] showed that certain individual graph properties, like k-colorability admit an -test. In this paper we make a rst step towards a complete logical characterization of all testable graph properties, and show that properties describable by a very general type of coloring problem are testable. We use this theorem to prove that rst order graph properties not containing a quanti er alternation of type \89" are always testable, while we show that some properties containing this alternation are not. Our results are proven using a combinatorial lemma, a special case of which, that may be of independent interest, is the following. A graph H is called -unavoidable in G if all graphs that di er A preliminary version of this paper appeared in the Proceedings of the 40 th Symposium on Foundation of Computer Science (FOCS'99), IEEE Press 1999, 656{666. y Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel, and AT&T Labs{Research, Florham Park, NJ 07932, USA. Email: noga@math.tau.ac.il. Research supported by a USA Israeli BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. z NEC Research Institute, 4 Independence Way, Princeton NJ, 08540, USA; and DIMACS. Research performed while at the Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel, and supported by the Fritz Brann Doctoral Fellowship in Engineering and Exact Sciences. E-mail: scher@research.nj.nec.com x Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: krivelev@math.tau.ac.il. Research was performed while this author was with DIMACS Center, Rutgers University, Piscataway, NJ 08854, USA, and AT&T Labs{Research, Florham Park, NJ 07932, USA. Research supported in part by a DIMACS Postdoctoral Fellowship. { School of Mathematics, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA. E-mail: szegedy@math.ias.edu. Research was performed while this author was with AT&T Labs{Research, Florham Park, NJ 07932, USA. Mathematics Subject Classi cation (2000): 68R10, 05C85, 05C35.