Sparse Combinatorial Group Testing

In combinatorial group testing, the primary objective is to fully identify the set of at most d defective items from a pool of n items using as few tests as possible. The celebrated result for the combinatorial group testing problem is that the number of tests, denoted by t , can be made logarithmic in n when {d} = {O}(\text {poly}(\log {n})) . However, state-of-the-art group testing codes require the items to be tested {w} = \Omega \left ({\frac {{d} \log {n}}{\log {d} + \log \log {n}} }\right) times and tests to include \rho = \Omega \left ({\frac {{n}}{{d} \log _{{d}} {n}}}\right) items. In many emerging applications, items can only participate in a limited number of tests and tests are constrained to include a limited number of items. In this paper, we study the "sparse" regime for the group testing problem where we restrict the number of tests each item can participate in by {w}_{\max } or the number of items each test can include by \rho _{\max } in both noiseless and noisy settings. These constraints lead to a largely unexplored regime where t is a fractional power of n , rather than logarithmic in n as in the classical setting. Our results characterize the number of tests t needed in this regime as a function of {w}_{\max } or \rho _{\max } and show, for example, that t decreases drastically when {w}_{\max } is increased beyond a bare minimum. In particular, in the noiseless case it can be shown that if {w}_{\max } \leq {d} , then we must have {t}={n} , i.e., testing every item individually is optimal. We show that if {w}_{\max }={d}+1 , the number of tests decreases suddenly from