Online versus Offline Adversaries in Property Testing
Esty Kelman, Ephraim Linder, Sofya Raskhodnikova
TL;DR
The paper investigates how adversarial input manipulation—offline erasures/corruptions versus online erasures/corruptions—affects property testing. It introduces a lifting technique that encodes properties with repetition to simulate online adversaries without increasing query complexity, enabling transfers of offline/standard-model separations to the online setting. The authors demonstrate a query-complexity incomparability and, importantly, an exponential randomness-complexity separation between online and offline models, quantified through the τ-Distinct-Elements property and a randomness-reduction framework. The results clarify the robustness limits of property testers under adversarial conditions and provide a toolkit (lifting lemma, repetition codes) for future derandomization and resilience analyses.
Abstract
We study property testing with incomplete or noisy inputs. The models we consider allow for adversarial manipulation of the input, but differ in whether the manipulation can be done only offline, i.e., before the execution of the algorithm, or online, i.e., as the algorithm runs. The manipulations by an adversary can come in the form of erasures or corruptions. We compare the query complexity and the randomness complexity of property testing in the offline and online models. Kalemaj, Raskhodnikova, and Varma (Theory Comput `23) provide properties that can be tested with a small number of queries with offline erasures, but cannot be tested at all with online erasures. We demonstrate that the two models are incomparable in terms of query complexity: we construct properties that can be tested with a constant number of queries in the online corruption model, but require querying a significant fraction of the input in the offline erasure model. We also construct properties that exhibit a strong separation between the randomness complexity of testing in the presence of offline and online adversaries: testing these properties in the online model requires exponentially more random bits than in the offline model, even when they are tested with nearly the same number of queries in both models. Our randomness separation relies on a novel reduction from randomness-efficient testers in the adversarial online model to query-efficient testers in the standard model.