A Nearly-Quadratic Gap Between Adaptive and Non-Adaptive Property Testers

• Main Author: Hurwitz, Jeremy Scott
• Format: Others
• Published: 2012
• Online Access: https://thesis.library.caltech.edu/6741/1/thesis.pdf; Hurwitz, Jeremy Scott (2012) A Nearly-Quadratic Gap Between Adaptive and Non-Adaptive Property Testers. Master's thesis, California Institute of Technology. doi:10.7907/W178-HP57. https://resolver.caltech.edu/CaltechTHESIS:11302011-091414252 <https://resolver.caltech.edu/CaltechTHESIS:11302011-091414252>

<p>We show that for all integers t ≥ 8 and arbitrarily small ε &#62; 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 = Õ(ε^-1) is the adaptive query complexity. This resolves the question of how beneficial 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.</p> <p>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(ε^-2+2/t, even under the promise that G has maximum degree O(εN). 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.</p> <p>Specifically, we show that for all H, testing whether G is a collection of blow-ups of H and has maximum degree O(εN) requires only O(ε^-1lg³ε^-1) adaptive queries or O(ε^-2+1/(δ+2)+ε^-2+2/W) non-adaptive queries, where δ = Δ(H) is the maximum degree of H and W&#60; |H|² is a bound on the size of witnesses against H.</p>