Noise-Resilient Group Testing: Limitations and Constructions
arXiv:0811.2609 · doi:10.1007/978-3-642-03409-1_7
Abstract
We study combinatorial group testing schemes for learning -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 that is known for exact reconstruction of -sparse vectors of length via non-adaptive measurements, by a multiplicative factor . Specifically, we give simple randomized constructions of non-adaptive measurement schemes, with measurements, that allow efficient reconstruction of -sparse vectors up to false positives even in the presence of false positives and false negatives within the measurement outcomes, for any constant . 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 measurements. We also obtain explicit constructions that allow fast reconstruction in time $\poly(m)$, which would be sublinear in for sufficiently sparse vectors. The main tool used in our construction is the list-decoding view of randomness condensers and extractors.
Full version. A preliminary summary of this work appears (under the same title) in proceedings of the 17th International Symposium on Fundamentals of Computation Theory (FCT 2009)