Efficient Decodable Group Testing

The basic group testing problem is to identify the unknown set of " positive items " from a large population of " items " using as few " tests " as possible. A test is a subset of items. A test returns positive if there is a positive item in the subset. The semantics of " positives, " " items, " and " tests " depend on the application. In the original context [3], group testing was invented to solve the problem of identifying syphilis infected blood samples from a large collection of WWII draftees' blood samples. In this case, items are blood samples, which are positive if they are infected. A test is a pool (group) of blood samples. Testing a group of samples at a time will save resources if the test outcome is negative. On the other hand, if the test outcome is positive then all we know is that at least one sample in the pool is positive but we do not know which one(s). In non-adaptive combinatorial group testing (NACGT), we assume that the number of positives is at most d for some fixed integer d, and that all tests have to be specified in advance before any test outcome is known. The NACGT paradigm has found numerous applications in many areas of Mathematics, Computer Science, and Computational Biology [4; 9; 10]. A NACGT strategy with t tests on a universe of N items is represented by a t × N binary matrix M = (m ij), where m ij = 1 iff item j belongs to test i. Let M i and M j denote row i and column j of M, respectively. Abusing notation, we will also use M i (respectively, M j) to denote the set of rows (respectively, columns) corresponding