The complexity of determinacy problem on group testing

Group testing was first discovered by Dorfman [l] when he did blood testing during World War II. Since then, one has found many applications in statistics, , combinatorics and computer science. The general form of the problem is as follows: Let N be a set of n items denoted by 1,2, . . . , n; some items are defective. We need to find out all defective items by a given kind of tests. What is the best way in some sense to do so? All defective items form a set that is called a sample. The sample space consists of all possible samples. For every set S, let ISI denote the number of elements of S, we will study the following sample spaces: The space pn consists of all subsets S of N, the space q,,d consists of all subsets S of N with /SI = d and the space %, <d consists of all subsets S of N with JS 15 d. Each test in group testing problems is on a subset T of N. For simplicity, we denote by the same symbol T the test and the subset on which one tests. For a sample S, ANS,(T) is the result of the test T and is called the answer function. There are different definitions in various models. In this paper, we consider the following definition: