Group Testing on General Set-Systems

Abstract: Group testing is one of the fundamental problems in coding theory and combinatorics in which one is to identify a subset of contaminated items from a given ground set. There has been renewed interest in group testing recently due to its applications in diagnostic virology, including pool testing for the novel coronavirus. The majority of existing works on group testing focus on the \emph{uniform} setting in which any subset of size $d$ from a ground set $V$ of size $n$ is potentially contaminated. In this work, we consider a {\em generalized} version of group testing with an arbitrary set-system of potentially contaminated sets. The generalized problem is characterized by a hypergraph $H=(V,E)$, where $V$ represents the ground set and edges $e\in E$ represent potentially contaminated sets. The problem of generalized group testing is motivated by practical settings in which not all subsets of a given size $d$ may be potentially contaminated, rather, due to social dynamics, geographical limitations, or other considerations, there exist subsets that can be readily ruled out. For example, in the context of pool testing, the edge set $E$ may consist of families, work teams, or students in a classroom, i.e., subsets likely to be mutually contaminated. The goal in studying the generalized setting is to leverage the additional knowledge characterized by $H=(V,E)$ to significantly reduce the number of required tests. The paper considers both adaptive and non-adaptive group testing and makes the following contributions. First, for the non-adaptive setting, we show that finding an optimal solution for the generalized version of group testing is NP-hard. For this setting, we present a solution that requires $O(d\log{|E|})$ tests, where $d$ is the maximum size of a set $e \in E$. Our solutions generalize those given for the traditional setting and are shown to be of order-optimal size $O(\log{|E|})$ for hypergraphs with edges that have ``large'' symmetric differences. For the adaptive setting, when edges in $E$ are of size exactly $d$, we present a solution of size $O(\log{|E|}+d\log^2{d})$ that comes close to the lower bound of $\Omega(\log{|E|} + d)$.