Problem 38: Query Complexity of Local Partitioning Oracles

From Open Problems in Sublinear Algorithms
Revision as of 03:13, 17 November 2012 by Krzysztof Onak (talk | contribs) (Created page with "{{Header |title=Query Complexity of Local Partitioning Oracles |source=bertinoro11 |who=Krzysztof Onak }} A local partitioning oracle is defined in the paper of Hassidim, Keln...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
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].