Optimal deterministic group testing algorithms to estimate the number of defectives

In this work, we study the problem of estimating the number of defectives d in a set of n items up to a multiplicative factor of Δ>1 using deterministic group testing algorithms. Particularly, given Δ>1 and D>d, we give upper and lower bounds on the number of tests required for the task of generating an estimation dˆ such that d/Δ≤dˆ≤dΔ. In the adaptive settings, we prove that any adaptive deterministic algorithm for estimating d must perform at least Ω((D/Δ²)log⁡(n/D)) tests. This extends the same lower bound achieved in [1] for non-adaptive algorithms. In addition, we show that our bound is tight up to a small additive term by constructing an adaptive algorithm for this task. Furthermore, we give a non-constructive proof of an upper bound of O((D/Δ²) (log⁡(n/D)+log⁡Δ)) for the non-adaptive settings. This bound is an improvement over the upper bound O((log⁡D)/(log⁡Δ))Dlog⁡n) from [1], and matches the lower bound up to a small additive term. Moreover, we use existing techniques for building expander regular bipartite graphs, extractors and condensers from [2,3] to construct two polynomial-time non-adaptive algorithms for estimating d. Using the construction from [2], a polynomial non-adaptive algorithm that makes O((D^1+o(1)/Δ²)⋅log⁡n) tests are defined. The second algorithm exploits the construction from [3] to build another non-adaptive algorithm that performs (D/Δ²)⋅Quazipoly (log⁡n) tests. This is the first explicit construction with an almost optimal test complexity.