Problem 90: Dense Graph Property Testing “Tradeoffs”
| Suggested by | Clement Canonne |
|---|---|
| Source | submitted online |
| 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 the above sense). Moreover, we now have a full characterization of the graph properties which admit such $O_\varepsilon(1)$-query testers.
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 $\operatorname{poly}(1/\varepsilon)$. ( 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 $\sqrt{n}\operatorname{poly}(\log n, 1/\varepsilon)$, $\operatorname{poly}(\log n, 1/\varepsilon)$, 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?
