A Nearly-Quadratic Gap between Adaptive and Non-adaptive Property Testers

We show that for all integers t ≥ 8 and arbitrarily small ε > 0, there exists a graph property Π (which depends on ε) such that ε-testing Π has non-adaptive query complexity Q=Θ~(q^(2−2/t)), where q=Θ~(ϵ⁻¹) is the adaptive query complexity. This resolves the question of how beneficial adaptivity is, in the context of proximity-dependent properties. This also gives evidence that the canonical transformation of Goldreich and Trevisan is essentially optimal when converting an adaptive property tester to a non-adaptive property tester. To do so, we provide optimal adaptive and non-adaptive testers for the combined property of having maximum degree O(εN) and being a blow-up collection of an arbitrary base graph H.