  We investigate conditions for the unique recoverability of sparse
integer-valued signals from a small number of linear measurements. Both the
objective of minimizing the number of nonzero components, the so-called
$\ell_0$-norm, as well as its popular substitute, the $\ell_1$-norm, are
covered. Furthermore, integrality constraints and possible bounds on the
variables are investigated. Our results show that the additional prior
knowledge of signal integrality allows for recovering more signals than what
can be guaranteed by the established recovery conditions from (continuous)
compressed sensing. Moreover, even though the considered problems are \NP-hard
in general (even with an $\ell_1$-objective), we investigate testing the
$\ell_0$-recovery conditions via some numerical experiments. It turns out that
the corresponding problems are quite hard to solve in practice using black-box
software. However, medium-sized instances of $\ell_0$- and
$\ell_1$-minimization with binary variables can be solved exactly within
reasonable time.
