  In this work, we consider the sparsity-constrained community-based group
testing problem, where the population follows a community structure. In
particular, the community consists of $F$ families, each with $M$ members. A
number $k_f$ out of the $F$ families are infected, and a family is said to be
infected if $k_m$ out of its $M$ members are infected. Furthermore, the
sparsity constraint allows at most $\rho_T$ individuals to be grouped in each
test. For this sparsity-constrained community model, we propose a probabilistic
group testing algorithm that can identify the infected population with a
vanishing probability of error and we provide an upper-bound on the number of
tests. When $k_m = \Theta(M)$ and $M \gg \log(FM)$, our bound outperforms the
existing sparsity-constrained group testing results trivially applied to the
community model. If the sparsity constraint is relaxed, our achievable bound
reduces to existing bounds for community-based group testing. Moreover, our
scheme can also be applied to the classical dilution model, where it
outperforms existing noise-level-independent schemes in the literature.
