Advances in group testing

Abstract

In combinatorial group testing, there are n items; each has an unknown binary status, positive (i.e., defective) or negative (i.e., good), and the number of positives is upper bounded by an integer d. Suppose there is some method to test whether a subset of items contains at least one positive or not. The test result is said to be positive if it indicates that the subset contains at least one positive item; otherwise, the test result is called negative. The problem is to resolve the status of every item using the minimum number of tests. Group testing (GT) algorithms can be adaptive or nonadaptive. An adaptive algorithm conducts the tests one by one and allows to design later tests using the outcome information of all previous tests. A nonadaptive group testing (NGT) algorithm specifies all tests before knowing any test results, and the benefit is that all tests can be performed in parallel. For the above group testing problem, nonadaptive algorithms require inherently more tests than adaptive ones. Though the research of group testing dates back to Dorfman's 1943 paper, a renewed interest in the subject occurred recently mainly due to the applications of group testing to the area of computational molecular biology. In applications of molecular biology, a group testing algorithm is called a pooling design, and the composition of each test is called a pool. While it is still important to minimize the number of tests, there are two other goals. First, in the biological setting, screening one pool at a time is far more expensive than screening many pools in parallel; this strongly encourages the use of nonadaptive algorithms. Second, DNA screening is error prone, so it is desirable to design error-tolerant algorithms, which can detect or correct some errors in the test results. In this monograph, some recent algorithmic, complexity, and mathematical results on nonadaptive group testing (and on pooling design) are presented.