Explicit Non-Adaptive Combinatorial Group Testing Schemes

Group testing is a long studied problem in combinatorics: A small set of $r$ ill people should be identified out of the whole ($n$ people) by using only queries (tests) of the form "Does set X contain an ill human?". In this paper we provide an explicit construction of a testing scheme which is better (smaller) than any known explicit construction. This scheme has \bigT{\min[r^2 \ln n,n]} tests which is as many as the best non-explicit schemes have. In our construction we use a fact that may have a value by its own right: Linear error-correction codes with parameters $[m,k,\delta m]_q$ meeting the Gilbert-Varshamov bound may be constructed quite efficiently, in \bigT{q^km} time.Comment: 15 pages, accepted to ICALP 200

Learning Immune-Defectives Graph through Group Tests

This paper deals with an abstraction of a unified problem of drug discovery and pathogen identification. Pathogen identification involves identification of disease-causing biomolecules. Drug discovery involves finding chemical compounds, called lead compounds, that bind to pathogenic proteins and eventually inhibit the function of the protein. In this paper, the lead compounds are abstracted as inhibitors, pathogenic proteins as defectives, and the mixture of "ineffective" chemical compounds and non-pathogenic proteins as normal items. A defective could be immune to the presence of an inhibitor in a test. So, a test containing a defective is positive iff it does not contain its "associated" inhibitor. The goal of this paper is to identify the defectives, inhibitors, and their "associations" with high probability, or in other words, learn the Immune Defectives Graph (IDG) efficiently through group tests. We propose a probabilistic non-adaptive pooling design, a probabilistic two-stage adaptive pooling design and decoding algorithms for learning the IDG. For the two-stage adaptive-pooling design, we show that the sample complexity of the number of tests required to guarantee recovery of the inhibitors, defectives, and their associations with high probability, i.e., the upper bound, exceeds the proposed lower bound by a logarithmic multiplicative factor in the number of items. For the non-adaptive pooling design too, we show that the upper bound exceeds the proposed lower bound by at most a logarithmic multiplicative factor in the number of items.Comment: Double column, 17 pages. Updated with tighter lower bounds and other minor edit

Constraining the Number of Positive Responses in Adaptive, Non-Adaptive, and Two-Stage Group Testing

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 two-stage 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

Optimal Algorithms for Two Group Testing Problems and New Bounds on Generalized Superimposed Codes

Two variants of the well known group testing problem are considered. In the first variant a finite set of items O and an unknown subset P of O are given, and one wants to identify the set P by asking the least number of questions of the form "Do Q and P share a single element?'', where Q is a subset of O. This problem naturally arises in the design of efficient contention resolution algorithms for certain random multiple-access communication systems [Berger et al. “Random multiple-access communication and group testing,” IEEE Trans. Commun., vol. 32, no. 7, pp. 769–779, 1984]. In the second variant of the problem, the answer to the question "Do Q and P share a single element?'' is correctly YES if Q and P intersect in a single element and NO if Q and P have an empty intersection, and it is left to a (possibly malicious) adversary otherwise. This model was introduced in [Damaschke, “Randomized group testing for mutually obscuring defectives,” Inf. Process. Lett., vol. 67, pp. 131–135, 1998] in the context of chemical compound testing. In this correspondence several algorithms for these group testing problems are presented, trying to optimize different measures of performance: The overall number of tests performed by the algorithm, the number of stages in which tests can be arranged, and the decoding complexity of identifying the elements of P from tests outcomes. Some of the given algorithms are optimal with respect to more than one of the above criteria. Instrumental to the results presented in the correspondence are new and improved bounds on certain generalization of superimposed codes introduced in [Dyachkov and Rykov, “A generalization of superimposed codes and its application to the multiple-access channel,” in Proc. 1984 IEEE Int. Symp. Inf. Theory, pp. 62–64], [De Bonis and Vaccaro, “Constructions of generalized superimposed codes with applications to group testing and conflict resolution in multiple access channels,” Theoretic. Comput. Sci., vol. 306, no. 1–3, pp. 223–243, 2003], a result that it is believed to be of independent interest