Boolean compressed sensing: LP relaxation for group testing

We revisit the well-known problem of boolean group testing which attempts to discover a sparse subset of faulty items in a large set of mostly good items using a small number of pooled (or grouped) tests. This problem originated during the second WorldWar, and has been the subject of active research during the 70´s, and 80´s. Recently, there has been a resurgence of interest due to the striking parallels between group testing and the now highly popular field of compressed sensing. In fact, boolean group testing is nothing but compressed sensing in a different algebra - with boolean `AND´ and `OR´ operations replacing vector space multiplication and addition. In this paper we review existing solutions for non-adaptive (batch) group testing and propose a linear programming relaxation solution, which has a resemblance to the basis pursuit algorithm for sparse recovery in linear models. We compare its performance to alternative methods for group testing.