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 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.

中文翻译：

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包装回国秘书

我们研究了在线秘书问题，该问题在组合包装域中具有n个候选人，并随时间顺序随机到达。目的是确定可行的最大总价值候选人。在第一个变体中，每个候选人都准确到达两次。所有2 n次到达均以随机顺序发生。我们提出了一个简单的0.5竞争算法。用于在线二分匹配问题，我们得到具有比至少一个算法0.5721 -  Ö（1） ，并用比率至少0.5459对所有的算法Ñ  ≥1 。我们扩展所有的算法和比率以ķ  ≥2每个候选人的到来人数。在第二个变体中，有许多未定的候选人。在每个回合中，都有来自池中的随机候选人到达。到达后，可以决定（接受/拒绝）候选人，也可以推迟候选人。在计算最佳解决方案时，我们专注于最大程度地减少延迟。预期数量的Θ（n  log  n）总是足够的。对于二分匹配，我们可以显示O（r  log  n）的紧密边界，其中r是最佳匹配的大小。对于拟阵，我们可以改善这进一步结合的紧密Ø（[R ^' 日志（ñ /r ^'）），其中r ^'是拟阵和对偶拟阵的最小秩。