Noise-Resilient Group Testing: Limitations and Constructions

We study combinatorial group testing schemes for learning dsparse 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 give a general framework for construction of highly noise-resilient group testing schemes using randomness condensers. Simple randomized instantiations of this construction give non-adaptive measurement schemes, with m = O(d log n) measurements, that allow efficient reconstruction of d-sparse vectors up to O(d) false positives even in the presence of δm false positives and Ω(m/d) false negatives within the measurement outcomes, for any constant δ < 1. None of these parameters can be substantially improved without dramatically affecting the others. Furthermore, we obtain several explicit (and incomparable) constructions, in particular one matching the randomized trade-off but using m = O(d log n) measurements. We also obtain explicit constructions that allow fast reconstruction in time poly(m), which would be sublinear in n for sufficiently sparse vectors.