Welfare-Maximizing Pooled Testing

In an epidemic, how should an organization with limited testing resources safely return to in-person activities after a period of lockdown? We study this question in a setting where the population at hand is heterogeneous in both utility for in-person activities and probability of infection. In such a period of re-integration, tests can be used as a certificate of non-infection, whereby those in negative tests are permitted to return to in-person activities for a designated amount of time. Under the assumption that samples can be pooled, the question of how to allocate the limited testing budget to the population in such a way as to maximize the utility of individuals in negative tests (who subsequently return to in-person activities) is non-trivial, with a large space of potential testing allocations. We show that non-overlapping testing allocations, which are both conceptually and (crucially) logistically more simple to implement, are approximately optimal, and we design an efficient greedy algorithm for finding non-overlapping testing allocations with approximately optimal welfare (overall utility). We also provide empirical evidence in support of the efficacy of our approach.