Quantitative Group Testing with Tunable Adaptation

The aim of quantitative group testing problem is to recover $k$ defective items from a set of $n$ items using minimal number of quantitative/additive group tests, where each test reveals the number of defective items present in the group. Fully adaptive strategies have been proposed and shown to, in general, require significantly fewer measurements. In the sublinear regime, where $k=n^{\alpha}$ for $0 < \alpha < 1$, for instance, this means a gain proportional to $\alpha \log n$. However, this gain is obtained at the cost of significant complexity associated with adapting tests to previous outcomes. This paper introduces a family of low-complexity strategies with efficient construction and decoding. These strategies adapt tests only in stages, allowing for a flexible trade-off between the number of stages, i.e., the complexity cost of adaptation, and the overall number of tests i.e., the adaptivity gain. Specifically, our algorithm matches the state-of-the-art adaptive algorithm in terms of the number of measurements while reducing the required number of stages by a multiplicative factor of $1-\alpha$. Our results demonstrates that a small number of stages can lead to substantial improvements over non-adaptive strategies. Furthermore, in comparison to existing non-adaptive algorithms, our algorithm achieves a 47% improvement in the overall number of tests, with the addition of just one stage.