Constraining the number of positive responses in adaptive, non-adaptive, and two-stage group testing

Abstract

Group testing is a well known search problem that consists in detecting the defective members of a set of objects O by performing tests on properly chosen subsets (pools) of the given set O. In classical group testing the goal is to find all defectives by using as few tests as possible. We consider a variant of classical group testing in which one is concerned not only with minimizing the total number of tests but aims also at reducing the number of tests involving defective elements. The rationale behind this search model is that in many practical applications the devices used for the tests are subject to deterioration due to exposure to or interaction with the defective elements. In this paper we consider adaptive, non-adaptive and two-stage group testing. For all three considered scenarios, we derive upper and lower bounds on the number of "yes" responses that must be admitted by any strategy performing at most a certain number t of tests. In particular, for the adaptive case we provide an algorithm that uses a number of "yes" responses that exceeds the given lower bound by a small constant. Interestingly, this bound can be asymptotically attained also by our twostage algorithm, which is a phenomenon analogous to the one occurring in classical group testing. For the non-adaptive scenario we give almost matching upper and lower bounds on the number of "yes" responses. In particular, we give two constructions both achieving the same asymptotic bound. An interesting feature of one of these constructions is that it is an explicit construction. The bounds for the non-adaptive and the two-stage cases follow from the bounds on the optimal sizes of new variants of d-cover free families and (p, d)-cover free families introduced in this paper, which we believe may be of interest also in other contexts.

Hypothesis Test for Bounds on the Size of Random Defective Set

“…In the zero-error case τ -WDR requires a special condition on a disjunctive s-code X: for any subset S of size |S| = s, the weight |x(S)| of the response vector is at most τ N , and, for any subset S of size |S | = s + 1, the weight |x(S )| of the response vector is at least τ N + 1. A similar model of specific disjunctive s-codes was considered in [21], where a disjunctive s-code is supplied with a weaker additional condition: the weight |x(S)| of the response vector for any subset S, S ⊆ [t], |S| ≤ s, is at most T . In [21] the authors motivate their group testing model by a risk for the safety of the persons who perform tests, in some contexts, when the number of positive test results is too large.…”

Threshold Disjunctive Codes

“…Remark 2. A similar model of special disjunctive s-codes was considered in [4], where the conventional disjunctive s-code is supplied with an additional condition: the weight |x (S)| of the response vector of any subset S, S ⊂ [t], |S| ≤ s, is at most T . Note that these codes have a weaker condition that our threshold disjunctive s-codes.…”

“…Note that these codes have a weaker condition that our threshold disjunctive s-codes. In [4] authors motivate their group testing model with bounded weight of the response vector by a risk for the safety of the persons, that perform test, in some contexts, when the number of positive test results is too big.…”

“…As already mentioned in Remark 2 the paper [4] considers the family of disjunctive s-codes with a weaker condition than the definition of threshold disjunctive s T -codes. These codes given as Definition 3.…”

“…These codes given as Definition 3. [4]. A binary code X of length N and size t is said to be a disjunctive s ≤Tcode if the disjunctive sum of any s codewords of X has weight ≤ T and covers those and only those codewords of X which are the terms of the given disjunctive sum.…”