Packing returning secretaries. Issue 3 (26th November 2020) - Hoefer, Martin; Wilhelmi, Lisa
Abstract: 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 2 n 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 Θ( 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.