Problem 99: Vertex-Distribution-Free Graph Testing

Suggested by	Oded Goldreich
Source	WOLA 2019
Short link	https://sublinear.info/99

The graph query model where one gets to query vertices uniformly at random, as mentioned in the previous open problem, may seem unrealistic in some cases. Thus, one may advocate alternative models, especially in the context of graph property testing, akin to the "distribution-free" model of property testing (for functions) and the PAC model (for learning). In this Vertex-Distribution-Free (VDF) model of testing suggested in a recent paper [Goldreich-19] (note: this model was also briefly discussed in Section 10.1 of [GoldreichGR-98]), one gets i.i.d. vertices sampled from an arbitrary distribution $\mathcal{D}$ over the vertex set, and the goal is to test w.r.t. to the (pseudo) distance induced by $\mathcal{D}$.

Question:Perform a systematic study of property testing, both in the bounded-degree and dense graph models, in this VDF setting.

Question: (Suggested by C. Seshadhri) Can one define, motivate, and prove non-trivial results in an Edge-Distribution-Free model, analogous to the VDF one but with regard to sampling random edges? (Note: This type of variant was also briefly evoked in Section 10.1.4 of [GoldreichGR-98], where it was shown that Bipartiteness is not testable in such an EDF model.)