Welfare-Maximizing Pooled Testing
Pooled testing increases the reach of scarce diagnostic resources, but optimally composing pools for individuals differing in infection risk and the utility they derive from a negative test is combinatorially challenging. We study the problem of maximizing the expected welfare of individuals cleared by a negative result, given a testing budget. Assigning a sample to several pools can raise welfare but is operationally burdensome; we show the restriction to non-overlapping allocations costs at most a factor of two for any budget or population, less under a pool-size cap at high health probabilities, and nothing when no health probability exceeds one-half. Welfare decomposes across non-overlapping pools, whereas evaluating overlapping allocations is #P-hard for pools of three or more. Finding optimal allocations is NP-hard and admits no FPTAS, with or without overlap, unless P = NP. We provide single-test routines and greedy algorithms with constant-factor guarantees. On real-world data, greedy achieves over 99% of optimal non-overlapping welfare in milliseconds, against hours for exact benchmarks. In a randomized field experiment at a Mexican research institute, our mechanism conditioned campus access on negative qPCR results. Relative to unrestricted access, we found no statistical evidence of adverse effects on participants' performance, learning, or mental health.