A Nearly-Quadratic Gap between Adaptive and Non-adaptive Property Testers - (Extended 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 = Θ̃(q2−2/t), where q = Õ( −1) is the adaptive query complexity. This resolves the question of how bene cial adaptivity is, in the context of proximity-dependent properties ([GR07]). This also gives evidence that the canonical transformation of Goldreich and Trevisan ([GT03]) is essentially optimal when converting an adaptive property tester into a non-adaptive property tester. To do so, we consider the property of being decomposable into a disjoint union of subgraphs, each of which is a (possibly unbalanced) blow-up of a given base-graph H. In [GR09], Goldreich and Ron proved that when H is a simple t-cycle, the non-adaptive query complexity is Ω( −2+2/t), even under the promise that G has maximum degree O( N). In this thesis, we prove a matching upper bound for the non-adaptive complexity and a tight (up to a polylogarithmic factor) upper bound on the adaptive complexity. Speci cally, we show that for all H, testing whether G is a collection of blow-ups of H and has maximum degree O( N) requires only O( −1 lg −1) adaptive queries or O( −2+1/(δ+2) + −2+2/W ) non-adaptive queries, where δ = ∆(H) is the maximum degree of H and W < |H| is a bound on the size of witnesses against H.