Problem 38: Query Complexity of Local Partitioning Oracles

Suggested by	Krzysztof Onak
Source	Bertinoro 2011
Short link	https://sublinear.info/38

A local partitioning oracle is defined in the paper of Hassidim, Kelner, Nguyen, and Onak [HassidimKNO-09], and an implicit construction of a partitioning oracle is shown in the earlier paper of Benjamini, Schramm, and Shapira [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 [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. [BenjaminiSS-08].

Update

Levi and Ron [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.