Efficient Algorithms for Noisy Group Testing

Group-testing refers to the problem of identifying (with high probability) a (small) subset of <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula> defectives from a (large) set of <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$D = \mathcal {O}(N^{1- {\delta }})$ </tex-math></inline-formula> for some <inline-formula> <tex-math notation="LaTeX">$ {\delta }> 0$ </tex-math></inline-formula>. The tests may be <italic>noiseless</italic> or <italic>noisy</italic>, 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 <inline-formula> <tex-math notation="LaTeX">$\Theta (D\log (N))$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> 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 <italic>both</italic> the number of tests and the decoding complexity (<inline-formula> <tex-math notation="LaTeX">$ \mathcal {O}\left ({D\log (N)}\right )$ </tex-math></inline-formula> in both the performance metrics). The total number of <italic>stages</italic> of our adaptive algorithm is “small” (<inline-formula> <tex-math notation="LaTeX">$ \mathcal {O}\left ({\log (D)}\right )$ </tex-math></inline-formula>). Similarly, in the scenario with non-adaptive 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 <inline-formula> <tex-math notation="LaTeX">$ \mathcal {O}\left ({D\log (D)\log (N)}\right )$ </tex-math></inline-formula> tests and has a decoding complexity of <inline-formula> <tex-math notation="LaTeX">${\mathcal{ O}}(D(\log N+\log ^{2}D))$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">${\mathcal{ O}}(D(\log N+\log ^{2}D))$ </tex-math></inline-formula>. For all three settings, the probability of error of our algorithms scales as <inline-formula> <tex-math notation="LaTeX">$ \mathcal {O}\left ({1/(poly(D)}\right )$ </tex-math></inline-formula>. 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.