A Characterization of the (natural) Graph Properties Testable with One-Sided Error

Abstract

The problem of characterizing all the testable graph properties is considered by many to be the most important open problem in the area of property-testing. Our main result in this paper is a solution of an important special case of this general problem; Call a property tester oblivious if its decisions are independent of the size of the input graph. We show that a graph property P has an oblivious one-sided error tester, if and only if P is (almost) hereditary. We stress that any "natural" property that can be tested (either with one-sided or with two-sided error) can be tested by an oblivious tester. In particular, all the testers studied thus far in the literature were oblivious. Our main result can thus be considered as a precise characterization of the "natural" graph properties, which are testable with one-sided error.One of the main technical contributions of this paper is in showing that any hereditary graph property can be tested with one-sided error. This general result contains as a special case all the previous results about testing graph properties with one-sided error. These include the results of [20] and [5] about testing k-colorability, the characterization of [21] of the graph-partitioning problems that are testable with one-sided error, the induced vertex colorability properties of [3], the induced edge colorability properties of [14], a transformation from two-sided to one-sided error testing [21], as well as a recent result about testing monotone graph properties [10]. More importantly, as a special case of our main result, we infer that some of the most well studied graph properties, both in graph theory and computer science, are testable with one-sided error. Some of these properties are the well known graph properties of being Perfect, Chordal, Interval, Comparability, Permutation and more. None of these properties was previously known to be testable.

What is the furthest graph from a hereditary property?

“…Alon, Fischer, Krivelevich and Szegedy proved in [2] that for any fixed graph H, the property P * H is testable. In [3], Alon and Shapira extended this result, showing it applies to all hereditary graph properties. The technique used in [3] lays in the core of the proof of Theorem 1.1.…”

“…In [3], Alon and Shapira extended this result, showing it applies to all hereditary graph properties. The technique used in [3] lays in the core of the proof of Theorem 1.1. Similar results on testing of hereditary properties were obtained by Lovász and Szegedy in [18] using convergent graph sequences (see also [13]).…”

“…The proof of the main result follows a method used by Alon and Shapira in [3], which is based on a strengthened version of the Szemerédi Regularity Lemma (proved in [2]). Using this method, we prove the following theorem: Theorem 1.1.…”

Fast Distributed Algorithms for Testing Graph Properties

“…A result by Alon and Shapira [4] states that all graph properties closed under induced subgraphs are testable in a number of queries that depends only on ǫ −1 . We note that, except for certain specific properties for which there are ad-hoc proofs, the dependence is usually a tower function in ǫ −1 or worse (asymptotically larger).…”

Untestable Properties Expressible with Four First-Order Quantifiers

“…We are interested in proving untestability, and our results hold even in the non-uniform case. In oblivious testing (see Alon and Shapira [2]), the testers are not given access to the size of the universe. Again, our results hold in the more general case where the testers may make decisions based on the size of the universe.…”