Subquadratic non-adaptive threshold group testing

We consider threshold group testing – a generalization of group testing, which asks to identify a set of positive individuals in a population, by performing tests on pools of elements. Each test is represented by a subset Q of individuals and its output is yes if Q contains at least one positive element and no otherwise. Threshold group testing is the natural generalization, introduced by P. Damaschke in 2005, arising when we are given a threshold t>0 and the answer to a test Q is yes if Q contains at least t positives and no otherwise. We give upper and lower bounds for this general problem, showing a complexity separation with the classical group testing. Next, we introduce a further generalization in which the goal is minimizing not only the number of tests, but also the number of thresholds which is related to the accuracy of the tests.