Packing Returning Secretaries

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 determine 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. For the online bipartite matching problem, we obtain an algorithm with ratio at least 0.5721 - o(1), and an algorithm with ratio at least 0.5459 for all n >= 1. We extend all algorithms and ratios to k >= 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. We 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 bipartite matching, we can show a tight bound of O(r log n), where r is the size of the optimum matching. For matroids, we can improve this further to a tight bound of O(r(') log(n/r('))), where r(') is the minimum rank of the matroid and the dual matroid.