A Nearly-Quadratic Gap Between Adaptive and Non-Adaptive Property Testers
<p>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<sup>2-2/t</sup>), where q = Õ(ε<sup>-1</sup>) is the adaptive query complexit...
| Summary: | <p>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<sup>2-2/t</sup>), where q = Õ(ε<sup>-1</sup>) 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<sup>-2+2/t</sup>, 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<sup>-1</sup>lg<sup>3</sup>ε<sup>-1</sup>) adaptive queries or O(ε<sup>-2+1/(δ+2)</sup>+ε<sup>-2+2/W</sup>) non-adaptive queries, where δ = Δ(H) is the maximum degree of H and W< |H|<sup>2</sup> is a bound on the size of witnesses against H.</p> |
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