Efficient and Near-Optimal Noisy Group Testing: An Information-Theoretic Framework

The group testing problem consists of determining a small set of defective items from a larger set of items based on a number of tests, and is relevant in applications such as medical testing, communication protocols, pattern matching, any many more. In this paper, we revisit a practical algorithm for noisy group testing in which each item is decoded separately (Malyutov and Mateev, 1980), and develop several performance guarantees via a unified information-theoretic framework for general noise models. For the special cases of no noise and symmetric noise, we find that the asymptotic number of tests required for vanishing error probability is within a factor log 2 ≈ 0.7 of the information-theoretic optimum at low sparsity levels, and that with a small number of allowed false positives and false negatives this guarantee extends to all sublinear sparsity levels. In addition, we provide a converse bound showing that if one tries to move slightly beyond this log 2 factor with separate decoding of items and i.i.d. randomized testing, the average number of items decoded incorrectly approaches that of a trivial decoder.