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

Abstract

We show that for all integers t ≥ 8 and arbitrarily small > 0, there exists a graph property Π (which depends on ) such that -testing Π has non-adaptive query complexity Q = Θ(q 2−2/t ), where q = Θ( −1 ) is the adaptive query complexity. This resolves the question of how beneficial adaptivity is, in the context of proximity-dependent properties ([9]). This also gives evidence that the canonical transformation of Goldreich and Trevisan ([8]) 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( N ) and being a blowup collection of an arbitrary base graph H.