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