Combinatorial Group Testing Research Articles

Optimal superimposed codes and designs for Renyi's search model

Abstract Renyi (Bull. Amer. Math. Soc. 71 (6) (1965) 809) suggested a combinatorial group testing model, in which the size of a testing group was restricted. In this model, Renyi considered the search of one defective element (significant factor) from the finite set of elements (factors). The corresponding optimal search designs were obtained by Katona (J. Combin. Theory 1 (2) (1966) 174). In the present work, we study Renyi's search model of several significant factors. This problem is closely related to the concept of binary superimposed codes, which were introduced by Kautz and Singleton (IEEE Trans. Inform Theory 10 (4) (1964) 363) and were investigated by D'yachkov and Rykov (Problems Control Inform. Theory 12 (4) (1983) 229), Erdos et al. (Israel J. Math. 51 (1–2) (1985) 75), Ruszinko (J. Combin. Theory Ser. A 66 (1994) 302) and Furedi (J. Combin. Theory Ser. A 73 (1996) 172). Our goal is to prove a lower bound on the search length and to construct the optimal superimposed codes and search designs. The preliminary results have been published by D'yachkov and Rykov (Conference on Computer Science & Engineering Technology, Yerevan, Armenia, September 1997, p. 242).

When is Individual Testing Optimal for Nonadaptive Group Testing?

The combinatorial group testing problem is, assuming the existence of up to d defectives among n items, to identify the defectives by as few tests as possible. In this paper, we study the problem for what values of n, given d, individual testing is optimal in nonadaptive group testing. Let N(d) denote the largest n for fixed d for which individual testing is optimal. We will show that N(d)=(d+1)2 under a prevalent constraint in practical nonadaptive algorithms and prove that N(d)=(d+1)2 for d=1, 2, 3, 4 without any constraint.

On the cut-off point for combinatorial group testing

The following problem is known as group testing problem for n objects. Each object can be essential (defective) or non-essential (intact). The problem is to determine the set of essential objects by asking queries adaptively. A query can be identified with a set Q of objects and the query Q is answered by 1 if Q contains at least one essential object and by 0 otherwise. In the statistical setting the objects are essential, independently of each other, with a given probability p < 1 while in the combinatorial setting the number k < n of essential objects is known. The cut-off point of statistical group testing is equal to p ∗ = 1 2 (3 − √5) , i.e., the strategy of testing each object individually minimizes the average number of queries iff p ⩾ p ∗ or n = 1. In the combinatorial setting the worst case number of queries is of interest. It has been conjectured that the cut-off point of combinatorial group testing is equal to x ∗ = 1 3 i.e., the strategy of testing n − 1 objects individually minimizes the worst case number of queries iff k n ⩾ α ∗ and k < n. Some results in favor of this conjecture are proved.

Reconstructing a Hamiltonian cycle by querying the graph: Application to DNA physical mapping

This paper studies four mathematical models of the multiplex PCR method of genome physical mapping described in Sorokin et al. (1996). The models are expressed as combinatorial group testing problems of finding an unknown Hamiltonian cycle in the complete graph by means of queries of different type. For each model, an efficient algorithm is proposed that matches asymptotically the information-theoretic lower bound.