Improved Bounds For Efficiently Decodable Probabilistic Group Testing With Unreliable Items

This work uses non-adaptive probabilistic group testing to find a set of L defective items out of n items. In contrast to traditional group testing, in the considered setup each item can hide itself (or become inactive) during any given test with probability 1−α and is active with probability α. The authors of [Cheraghchi et al.] proposed an efficiently decodable probabilistic group testing scheme which requires $O\left( {\frac{{L\log (n)}}{{{\alpha ^3}}}} \right)$ tests for the per-instance scenario (where the group testing matrix works for any arbitrary, but fixed, set of L defective items) and $O\left( {\frac{{{L^2}\log (n/L)}}{{{\alpha ^3}}}} \right)$ tests for the universal scenario (where the same group testing matrix works for all possible defective sets of L items). The contribution of this work is two-fold: (i) with a slight modification in the construction of the group testing matrix proposed by [Cheraghchi et al.], the corresponding bounds on the number of sufficient tests are improved to $O\left( {\frac{{L\log (n)}}{{{\alpha ^2}}}} \right)$ and $O\left( {\frac{{{L^2}\log (n/L)}}{{{\alpha ^2}}}} \right)$ for the per-instance and universal scenarios respectively, while still using their efficient decoding method; and (ii) it is shown that the same bounds also hold for the fixed pool-size probabilistic group testing scenario, where in every test a fixed number of items are included for testing.