Subquadratic Non-adaptive Threshold Group Testing

Abstract

We consider threshold group testing – a generalization of a well known and thoroughly examined problem of combinatorial group testing. In the classical setting, the goal is to identify a set of positive individuals in a population, by performing tests on pools of elements. The output of each test is an answer to the question: is there at least one positive element inside a query set Q ? The threshold group testing is a natural generalization of this classical setting which arises when the answer to a test is positive if at least $t>0$ elements under test are positive.

We show that there exists a testing strategy for the threshold group testing consisting of $O(d^{3/2}\log (N/d))$ tests, for d positive items in a population of size N. For any value of the threshold t, we also provide a lower bound of order $\varOmega \left( \min \left\{ \left( \frac{d}{t}\right) ^2,\frac{N}{t}\right\} \right)$. Our subquadratic bound shows a complexity separation with the classical group testing (which corresponds to $t = 1$) where $\varOmega (d^2 \log _d N)$ tests are needed [25].

Next, we introduce a further generalization, the multi-threshold group testing problem. In this setting, we have a set of $s > 0$ thresholds, $t_1,t_2, \ldots , t_s$. The output of each test is an integer between 0 and s which corresponds to which thresholds get passed by the number of positives in the queried pool. Here, one may be interested in minimizing not only the number of tests, but also the number of thresholds which is related to the accuracy of the tests. We show the existence of two strategies for this problem. The first one of size $O(d^{3/2}\log (N/d))$ is an extension of the above-mentioned result. The second strategy is more general and works for a range of parameters. As a consequence, we show that $O(\frac{d^2}{t}\log (N/d))$ tests are sufficient for $t\le d/2$. Both strategies use respectively $O(\sqrt{d})$ and $O(\sqrt{t})$ thresholds.

This work was supported by the Polish National Science Centre grants DEC-2012/06/M/ST6/00459 and 2014/13/N/ST6/01850.