NDLI: Every Monotone Graph Property Is Testable

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Every Monotone Graph Property Is Testable

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Shapira, Asaf Alon, Noga
Copyright Year	2008
Abstract	A graph property is called monotone if it is closed under removal of edges and vertices. Many monotone graph properties are some of the most well-studied properties in graph theory, and the abstract family of all monotone graph properties was also extensively studied. Our main result in this paper is that any monotone graph property can be tested with one-sided error, and with query complexity depending only on $\epsilon$. This result unifies several previous results in the area of property testing and also implies the testability of well-studied graph properties that were previously not known to be testable. At the heart of the proof is an application of a variant of Szemerdi's regularity lemma. The main ideas behind this application may be useful in characterizing all testable graph properties and in generally studying graph property testing. As a byproduct of our techniques we also obtain additional results in graph theory and property testing, which are of independent interest. One of these results is that the query complexity of testing testable graph properties with one-sided error may be arbitrarily large. Another result, which significantly extends previous results in extremal graph theory, is that for any monotone graph property ${\cal P}$, any graph that is $\epsilon$-far from satisfying ${\cal P}$ contains a subgraph of size depending on $\epsilon$ only, which does not satisfy ${\cal P}$. Finally, we prove the following compactness statement: If a graph G is $\epsilon$-far from satisfying a (possibly infinite) set of monotone graph properties ${\cal P}$, then it is at least $\delta_{{\cal P}}(\epsilon)$-far from satisfying one of the properties.
Starting Page	505
Ending Page	522
Page Count	18
File Format	PDF
ISSN	00975397
DOI	10.1137/050633445
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	2 (Special Issue Dedicated to the Thirty-Seventh Annual ACM Symposium on Theory of Computing (STOC 2005))
Volume Number	38
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2008-05-23
Access Restriction	Subscribed
Subject Keyword	regularity lemma graphs monotone properties Randomized algorithms Approximation algorithms property testing Graph algorithms
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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