Testing versus estimation of graph properties

The topic of tolerant property testing, that of distinguishing input instances that are far from satisfying a property from those that are close enough to satisfying it (as opposed to distinguishing the far instances only from the satisfying instances), has recently become an active topic of research in the field of combinatorial property testing [13]. In the general setting, there exist properties that are testable but not tolerantly testable [10]. However, we show here that in the setting of the dense graph model, all testable properties are not only tolerantly testable, but also admit a constant query size algorithm that estimates the distance from the property up to any fixed additive constant. In the course of the construction of this algorithm we develop a framework for extending Szemerédi's Regularity Lemma, both as a prerequisite for formulating what kind of information about the input graph will provide us with the correct estimation, and as the means for efficiently gathering this information. This work is also connected to the question of finding a combinatorial characterization of the testable graph properties, and to the question of efficiently finding a regular partition.