  This work investigates multiple testing by considering minimax separation
rates in the sparse sequence model, when the testing risk is measured as the
sum FDR+FNR (False Discovery Rate plus False Negative Rate). First using the
popular beta-min separation condition, with all nonzero signals separated from
$0$ by at least some amount, we determine the sharp minimax testing risk
asymptotically and thereby explicitly describe the transition from "achievable
multiple testing with vanishing risk" to "impossible multiple testing".
Adaptive multiple testing procedures achieving the corresponding optimal
boundary are provided: the Benjamini--Hochberg procedure with a properly tuned
level, and an empirical Bayes $\ell$-value (`local FDR') procedure. We prove
that the FDR and FNR make non-symmetric contributions to the testing risk for
most optimal procedures, the FNR part being dominant at the boundary. The
multiple testing hardness is then investigated for classes of arbitrary sparse
signals. A number of extensions, including results for classification losses
and convergence rates in the case of large signals, are also investigated.
