A combinatorial characterization of the testable graph properties: it's all about regularity

A common thread in all the recent results concerning the testing of dense graphs is the use of Szemer?edi's regularity lemma. In this paper we show that in some sense this is not a coincidence. Our first result is that the property defined by having any given Szemer?edi-partition is testable with a constant number of queries. Our second and main result is a purely combinatorial characterization of the graph properties that are testable with a constant number of queries. This characterization (roughly) says that a graph property P can be tested with a constant number of queries if and only if testing P can be reduced to testing the property of satisfying one of finitely many Szemer?edi-partitions. This means that in some sense, testing for Szemer?edi-partitions is as hard as testing any testable graph property. We thus resolve one of the main open problems in the area of property-testing, which was first raised in the 1996 paper of Goldreich, Goldwasser and Ron [27] that initiated...

• Type: Conference
• Year: 2006
• Where: STOC
• Authors: Noga Alon, Eldar Fischer, Ilan Newman, Asaf Shapira
• Keywords: Algorithms, Constant Number, Graph Property, Main Open Problems, STOC 2006