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 e of objects and the query e is answered by I if p 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* = i(3 &), i.e., the strategy of testing each object individually minimizes the average number of queries iff p > p* or n = I. 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 r* = f. i.e., the strategy of testing II I objects individually minimizes the worst case number of queries iff k/n b T* and k < II. Some results in favor of this conjecture are proved.