  In this paper, we study the problem of non-adaptive group testing, in which
one seeks to identify which items are defective given a set of
suitably-designed tests whose outcomes indicate whether or not at least one
defective item was included in the test. The most widespread recovery criterion
seeks to exactly recover the entire defective set, and relaxed criteria such as
approximate recovery and list decoding have also been considered. In this
paper, we study the fundamental limits of group testing under the significantly
relaxed {\em weak recovery} criterion, which only seeks to identify a small
fraction (e.g., $0.01$) of the defective items. Given the near-optimality of
i.i.d.~Bernoulli testing for exact recovery in sufficiently sparse scaling
regimes, it is natural to ask whether this design additionally succeeds with
much fewer tests under weak recovery. Our main negative result shows that this
is not the case, and in fact, under i.i.d.~Bernoulli random testing in the
sufficiently sparse regime, an {\em all-or-nothing} phenomenon occurs: When the
number of tests is slightly below a threshold, weak recovery is impossible,
whereas when the number of tests is slightly above the same threshold,
high-probability exact recovery is possible. In establishing this result, we
additionally prove similar negative results under Bernoulli designs for the
weak detection problem (distinguishing between the group testing model
vs.~completely random outcomes) and the problem of identifying a single item
that is definitely defective. On the positive side, we show that all three
relaxed recovery criteria can be attained using considerably fewer tests under
suitably-chosen non-Bernoulli designs.
