Noise-resilient group testing: Limitations and constructions

We study combinatorial group testing schemes for learning d-sparse Boolean vectors using highly unreliable disjunctive measurements. We consider an adversarial noise model that only limits the number of false observations, and show that any noise-resilient scheme in this model can only approximately reconstruct the sparse vector. On the positive side, we take this barrier to our advantage and show that approximate reconstruction (within a satisfactory degree of approximation) allows us to break the information theoretic lower bound of @[email protected]?(d^2logn) that is known for exact reconstruction of d-sparse vectors of length n via non-adaptive measurements, by a multiplicative factor @[email protected]?(d). Specifically, we give simple randomized constructions of non-adaptive measurement schemes, with m=O(dlogn) measurements, that allow efficient reconstruction ofd-sparse vectors up to O(d) false positives even in the presence of @dm false positives and O(m/d) false negatives within the measurement outcomes, for any constant @d<1. We show that, information theoretically, none of these parameters can be substantially improved without dramatically affecting the others. Furthermore, we obtain several explicit constructions, in particular one matching the randomized trade-off but using m=O(d^1^+^o^(^1^)logn) measurements. We also obtain explicit constructions that allow fast reconstruction in time poly(m), which would be sublinear in n for sufficiently sparse vectors. The main tool used in our construction is the list-decoding view of randomness condensers and extractors. An immediate consequence of our result is an adaptive scheme that runs in only two non-adaptive rounds and exactly reconstructs any d-sparse vector using a total O(dlogn) measurements, a task that would be impossible in one round and fairly easy in O(log(n/d)) or d rounds.