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 Ω~(d2logn) that is known for exact reconstruction of
d-sparse vectors of length n via non-adaptive measurements, by a
multiplicative factor Ω~(d).
Specifically, we give simple randomized constructions of non-adaptive
measurement schemes, with m=O(dlogn) measurements, that allow efficient
reconstruction of d-sparse vectors up to O(d) false positives even in the
presence of δm false positives and O(m/d) false negatives within the
measurement outcomes, for any constant δ<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(d1+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.Comment: 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