  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 $\eul$ 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.
