  We propose necessary and sufficient conditions for a sensing matrix to be
"s-semigood" -- to allow for exact $\ell_1$-recovery of sparse signals with at
most $s$ nonzero entries under sign restrictions on part of the entries. We
express the error bounds for imperfect $\ell_1$-recovery in terms of the
characteristics underlying these conditions. Furthermore, we demonstrate that
these characteristics, although difficult to evaluate, lead to verifiable
sufficient conditions for exact sparse $\ell_1$-recovery and to efficiently
computable upper bounds on those $s$ for which a given sensing matrix is
$s$-semigood. We concentrate on the properties of proposed verifiable
sufficient conditions of $s$-semigoodness and describe their limits of
performance.
