  We study online secretary problems with returns in combinatorial packing
domains with $n$ candidates that arrive sequentially over time in random order.
The goal is to accept a feasible packing of candidates of maximum total value.
In the first variant, each candidate arrives exactly twice. All $2n$ arrivals
occur in random order. We propose a simple 0.5-competitive algorithm that can
be combined with arbitrary approximation algorithms for the packing domain,
even when the total value of candidates is a subadditive function. For
bipartite matching, we obtain an algorithm with competitive ratio at least
$0.5721 - o(1)$ for growing $n$, and an algorithm with ratio at least $0.5459$
for all $n \ge 1$. We extend all algorithms and ratios to $k \ge 2$ arrivals
per candidate.
  In the second variant, there is a pool of undecided candidates. In each
round, a random candidate from the pool arrives. Upon arrival a candidate can
be either decided (accept/reject) or postponed (returned into the pool). We
mainly focus on minimizing the expected number of postponements when computing
an optimal solution. An expected number of $\Theta(n \log n)$ is always
sufficient. For matroids, we show that the expected number can be reduced to
$O(r \log (n/r))$, where $r \le n/2$ is the minimum of the ranks of matroid and
dual matroid. For bipartite matching, we show a bound of $O(r \log n)$, where
$r$ is the size of the optimum matching. For general packing, we show a lower
bound of $\Omega(n \log \log n)$, even when the size of the optimum is $r =
\Theta(\log n)$.
