Concomitant Group Testing
Thach V. Bui Jonathan Scarlett
In this paper, we introduce a variation of the group testing problem
capturing the idea that a positive test requires a combination of multiple
``types'' of item. Specifically, we assume that there are multiple disjoint
\emph{semi-defective sets}, and a test is positive if and only if it contains
at least one item from each of these sets. The goal is to reliably identify all
of the semi-defective sets using as few tests as possible, and we refer to this
problem as \textit{Concomitant Group Testing} (ConcGT). We derive a variety of
algorithms for this task, focusing primarily on the case that there are two
semi-defective sets. Our algorithms are distinguished by (i) whether they are
deterministic (zero-error) or randomized (small-error), and (ii) whether they
are non-adaptive, fully adaptive, or have limited adaptivity (e.g., 2 or 3
stages). Both our deterministic adaptive algorithm and our randomized
algorithms (non-adaptive or limited adaptivity) are order-optimal in broad
scaling regimes of interest, and improve significantly over baseline results
that are based on solving a more general problem as an intermediate step (e.g.,
hypergraph learning).