Near-Optimal Sparsity-Constrained Group Testing: Improved Bounds and Algorithms

Recent advances in noiseless non-adaptive group testing have led to a precise asymptotic characterization of the number tests required for high-probability recovery sublinear regime $k = n^{\theta }$ (with notation="LaTeX">$\theta \in (0,1)$ ), with notation="LaTeX">$n$ individuals among which notation="LaTeX">$k$ are infected. However, may increase substantially under real-world practical constraints, notably including bounds on maximum notation="LaTeX">$\Delta $ an individual can be placed in, or notation="LaTeX">$\Gamma given test. While previous works guarantees these settings, significant gaps remain between achievability and converse bounds. In this paper, we completely close several most prominent gaps. case -divisible items, show that definite defectives (DD) algorithm coupled random regular design is asymptotically optimal dense scaling regimes, within factor e more generally; establish by strengthening both best known -sized tests, provide comprehensive analysis \Theta (1)$ , again threshold proving optimality SCOMP (a slight refinement DD) equipped tailored pooling scheme. Finally, each two near-optimal adaptive algorithms based sequential splitting, provably demonstrate performance algorithms.