Difference between revisions of "Open Problems:90"

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{{Header
|title=Dense Graph Property Testing "Tradeoffs"
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|source=warwick18
|source=online
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|who=Clément Canonne
|who=Clement Canonne
 
 
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In the dense graph model of property testing (where the testing algorithm is granted query access to the adjacency matrix of an unknown $n$-node graph, and the distance measure is the fraction of entries of this matrix that needs to be changed for the graph to satisfy the property), ''testability'' was originally defined as meaning “a number of queries depending only on the proximity parameter $\varepsilon$ (but not $n$).” Many properties, such as graph partition properties, triangle-freeness, and more generally $H$-freeness, are known to be testable with a constant number of queries in this sense. Moreover, we now have a full characterization of the graph properties that admit $O_\varepsilon(1)$-query testers {{Cite|AlonFNS-09}}.
  
In the dense graph model of property testing (where the testing algorithm is granted query access to the adjacency matrix of an unknown $n$-node graph, and the distance measure is the fraction of entries of this matrix that needs to be changed for the graph to satisfy the property), ''testability'' was originally defined as meaning "a number of queries depending only on the proximity parameter $\varepsilon$ (but not $n$)".  
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These kinds of testers are, however, typically obtained via regularity lemmata, such as Szemerédi's Regularity Lemma, and the “constant” query complexity is often superpolynomial in $1/\varepsilon$ (more often than not, it is even a tower of 2's of height $\operatorname{poly}(1/\varepsilon)$). This is not an oversight: triangle freeness, for instance, provably does ''not'' have any tester with query complexity $\operatorname{poly}(1/\varepsilon)$ {{Cite|Alon-02}}. See for instance Section 8.5 of Goldreich's book {{Cite|Goldreich-17}} for a summary of the state of the art in property testing in the dense graph model.
  
Many properties, such as graph partition properties, triangle-freeness, and more generally $H$-freeness, are known to be testable with a constant number of queries (in the above sense). Moreover, we now have a full characterization of the graph properties which admit such $O_\varepsilon(1)$-query testers.
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Nevertheless, nothing ''a priori'' precludes these properties from having testers with query complexity $\sqrt{n}\operatorname{poly}(1/\varepsilon)$, $\operatorname{poly}(\log n, 1/\varepsilon)$, or even $\operatorname{poly}(\log^* n, 1/\varepsilon)$. What is the best possible tradeoff between $n$ and $\varepsilon$? Note that ''every'' property trivially has a tester with query complexity $O(n^2)$, so the question is how far down this dependence on $n$ can be brought while keeping a reasonable dependence on $\varepsilon$.
  
Yet, these results are typically obtained via regularity lemmata, such as Szemerédi's Regularity Lemma, and the "constant" query complexity if often superpolynomial in $1/\varepsilon$ (or, often, even a tower of exponent size $\operatorname{poly}(1/\varepsilon)$). This is not an oversight: triangle freeness, for instance, provably does '''not''' have any tester with query complexity $\operatorname{poly}(1/\varepsilon)$. (
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'''Concrete instance of the problem:''' Is there a $\operatorname{poly}(\log n, 1/\varepsilon)$-query tester for triangle freeness in the dense graph model?
See for instance Section 8.5 of Oded Goldreich's *Introduction to Property Testing* book for a summary of the state-of-the-art in property testing in the dense graph model.)
 
 
 
However, nothing ''a priori'' precludes these properties to have testers with query complexity $\sqrt{n}\operatorname{poly}(1/\varepsilon)$, $\operatorname{poly}(\log n, 1/\varepsilon)$, or even $\operatorname{poly}(\log^* n, 1/\varepsilon)$. What is the best possible tradeoff between $n$ and $\varepsilon$ one can achieve? (Note that ''every'' property trivially has a tester with query complexity $O(n^2)$, so it's only a question of how far down this dependence on $n$ can be brought while keeping a reasonable dependence on $\varepsilon$)
 
 
 
As a concrete instance of the problem:
 
 
 
'''Open problem.''' Is there a $\operatorname{poly}(\log n, 1/\varepsilon)$-query tester for triangle freeness in the dense graph model?
 

Latest revision as of 04:31, 20 August 2019

Suggested by Clément Canonne
Source Warwick 2018
Short link https://sublinear.info/90

In the dense graph model of property testing (where the testing algorithm is granted query access to the adjacency matrix of an unknown $n$-node graph, and the distance measure is the fraction of entries of this matrix that needs to be changed for the graph to satisfy the property), testability was originally defined as meaning “a number of queries depending only on the proximity parameter $\varepsilon$ (but not $n$).” Many properties, such as graph partition properties, triangle-freeness, and more generally $H$-freeness, are known to be testable with a constant number of queries in this sense. Moreover, we now have a full characterization of the graph properties that admit $O_\varepsilon(1)$-query testers [AlonFNS-09].

These kinds of testers are, however, typically obtained via regularity lemmata, such as Szemerédi's Regularity Lemma, and the “constant” query complexity is often superpolynomial in $1/\varepsilon$ (more often than not, it is even a tower of 2's of height $\operatorname{poly}(1/\varepsilon)$). This is not an oversight: triangle freeness, for instance, provably does not have any tester with query complexity $\operatorname{poly}(1/\varepsilon)$ [Alon-02]. See for instance Section 8.5 of Goldreich's book [Goldreich-17] for a summary of the state of the art in property testing in the dense graph model.

Nevertheless, nothing a priori precludes these properties from having testers with query complexity $\sqrt{n}\operatorname{poly}(1/\varepsilon)$, $\operatorname{poly}(\log n, 1/\varepsilon)$, or even $\operatorname{poly}(\log^* n, 1/\varepsilon)$. What is the best possible tradeoff between $n$ and $\varepsilon$? Note that every property trivially has a tester with query complexity $O(n^2)$, so the question is how far down this dependence on $n$ can be brought while keeping a reasonable dependence on $\varepsilon$.

Concrete instance of the problem: Is there a $\operatorname{poly}(\log n, 1/\varepsilon)$-query tester for triangle freeness in the dense graph model?