Quantitative group testing in the sublinear regime

The quantitative group testing (QGT) problem deals with efficiently identifying a small number of infected individuals among a large population. To this end, we can test groups of individuals where each test returns the total number of infected individuals in the tested pool. For the regime where the number of infected individuals is sublinear in the total population we derive the information-theoretic threshold for the minimum number of tests required to identify the infected individuals with high probability. Such a threshold was so far only known for the case where the infected individuals are a constant fraction of the population (Alaoui et al. 2019, Scarlett & Cevher 2017). Moreover, we propose and analyze an efficient greedy reconstruction algorithm that outperforms the best known algorithm (Karimi et al. 2019) for certain sparsity regimes

Quantitative Group Testing and the rank of random matrices

Given a random Bernoulli matrix $A\in \{0,1\}^{m\times n}$, an integer $0< k < n$ and the vector $y:=Ax$, where $x \in \{0,1\}^n$ is of Hamming weight $k$, the objective in the {\em Quantitative Group Testing} (QGT) problem is to recover $x$. This problem is more difficult the smaller $m$ is. For parameter ranges of interest to us, known polynomial time algorithms require values of $m$ that are much larger than $k$. In this work, we define a seemingly easier problem that we refer to as {\em Subset Select}. Given the same input as in QGT, the objective in Subset Select is to return a subset $S \subseteq [n]$ of cardinality $m$, such that for all $i\in [n]$, if $x_i = 1$ then $i\in S$. We show that if the square submatrix of $A$ defined by the columns indexed by $S$ has nearly full rank, then from the solution of the Subset Select problem we can recover in polynomial-time the solution $x$ to the QGT problem. We conjecture that for every polynomial time Subset Select algorithm, the resulting output matrix will satisfy the desired rank condition. We prove the conjecture for some classes of algorithms. Using this reduction, we provide some examples of how to improve known QGT algorithms. Using theoretical analysis and simulations, we demonstrate that the modified algorithms solve the QGT problem for values of $m$ that are smaller than those required for the original algorithms