  In group testing, the goal is to identify a subset of defective items within
a larger set of items based on tests whose outcomes indicate whether any
defective item is present. This problem is relevant in areas such as medical
testing, data science, communications, and many more. Motivated by physical
considerations, we consider a sparsity-based constrained setting (Gandikota et
al., 2019) in which the testing procedure is subject to one of the following
two constraints: items are finitely divisible and thus may participate in at
most $\gamma$ tests; or tests are size-constrained to pool no more than $\rho$
items per test. While information-theoretic limits and algorithms are known for
the non-adaptive setting, relatively little is known in the adaptive setting.
We address this gap by providing an information-theoretic converse that holds
even in the adaptive setting, as well as a near-optimal noiseless adaptive
algorithm for $\gamma$-divisible items. In broad scaling regimes, our upper and
lower bounds asymptotically match up to a factor of $e$. We also present a
simple asymptotically optimal adaptive algorithm for $\rho$-sized tests. In
addition, in the non-adaptive setting with $\gamma$-divisible items, we use the
Definite Defectives (DD) decoding algorithm and study bounds on the required
number of tests for vanishing error probability under the random near-constant
test-per-item design. We show that the number of tests required can be
significantly less than the Combinatorial Orthogonal Matching Pursuit (COMP)
decoding algorithm, and is asymptotically optimal in broad scaling regimes.
