Efficient Algorithms for Noisy Group Testing

Group-testing refers to the problem of identifying (with high probability) a (small) subset of D defectives from a (large) set of N items via a “small” number of “pooled” tests (i.e., tests that have a positive outcome if at least one of the items being tested in the pool is defective, else have a negative outcome). For ease of presentation in this paper, we focus on the regime when D = O(N
^1-δ
) for some δ > 0. The tests may be noiseless or noisy, and the testing procedure may be adaptive (the pool defining a test may depend on the outcome of a previous test), or non-adaptive (each test is performed independent of the outcome of other tests). A rich body of the literature demonstrates that θ(D log(N)) tests are information-theoretically necessary and sufficient for the group-testing problem, and provides algorithms that achieve this performance. However, it is only recently that reconstruction algorithms with computational complexities that are sub-linear in N have started being investigated. In the scenario with adaptive tests with noisy outcomes, we present the first scheme that is simultaneously order-optimal (up to small constant factors) in both the number of tests and the decoding complexity (O (D log(N)) in both the performance metrics). The total number of stages of our adaptive algorithm is “small” (O (log(D))). Similarly, in the scenario with nonadaptive tests with noisy outcomes, we present the first scheme that is simultaneously near-optimal in both the number of tests and the decoding complexity (via an algorithm that requires O (D log(D) log(N)) tests and has a decoding complexity of O(D(log N +log
²
D)). Finally, we present an adaptive algorithm that only requires two stages, and for which both the number of tests and the decoding complexity scale as O(D(log N +log
²
D)). For all three settings, the probability of error of our algorithms scales as O (1/(poly(D)). For each of the statements mentioned earlier about the order of the number of measurements, decoding complexity, and probability of error, we provide explicitly computed “small” universal factors in our theorem statements.