Editing Open Problems:38

{{Header
|source=bertinoro11
|who=Krzysztof Onak
}}
A local partitioning oracle is defined in the paper of Hassidim, Kelner, Nguyen, and Onak {{cite|HassidimKNO-09}}, and an implicit construction of a partitioning oracle is shown in the earlier paper of Benjamini, Schramm, and Shapira {{cite|BenjaminiSS-08}}. Partitioning oracles are a useful abstraction for approximation and testing algorithms in the bounded degree model.

The best known oracle for bounded-degree planar graphs makes at most $d^{\operatorname{poly}(1/\epsilon)}$ queries to the underlying graph to answer each query about the resulting partition, where $d$ is the bound on the maximum vertex degree in the graph. See {{cite|Onak-10}} for a description of the method.

'''Question:''' Can one design an oracle that makes only $\operatorname{poly}(d/\epsilon)$ queries? If so, then among other things, this would lead to a tester for planarity in the bounded-degree model that makes only $\operatorname{poly}(1/\epsilon)$ queries, resolving an open question of Benjamini et al. {{cite|BenjaminiSS-08}}.
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 '''Question:''' Can one design an oracle that makes only $\operatorname{poly}(d/\epsilon)$ queries? If so, then among other things, this would lead to a tester for planarity in the bounded-degree model that makes only $\operatorname{poly}(1/\epsilon)$ queries, resolving an open question of Benjamini et al. {{cite|BenjaminiSS-08}}.
-== Update ==
-Levi and Ron {{cite|LeviR-13}} showed a partitioning oracle for bounded-degree minor-free graphs that makes only $(d/\epsilon)^{O(\log(1/\epsilon))}$ queries to the input graph for each query about the partition.