Solving Multi-choice Secretary Problem in Parallel: An Optimal Observation-Selection Protocol

The classical secretary problem investigates the question of how to hire the best secretary from n candidates who come in a uniformly random order. In this work we investigate a parallel generalizations of this problem introduced by Feldman and Tennenholtz [14]. We call it shared Q-queue J-choice K-best secretary problem. In this problem, n candidates are evenly distributed into Q queues, and instead of hiring the best one, the employer wants to hire J candidates among the best K persons. The J quotas are shared by all queues. This problem is a generalized version of J-choice K-best problem which has been extensively studied and it has more practical value as it characterizes the parallel situation. Although a few of works have been done about this generalization, to the best of our knowledge, no optimal deterministic protocol was known with general Q queues. In this paper, we provide an optimal deterministic protocol for this problem. The protocol is in the same style of the 1 e -solution for the classical secretary problem, but with multiple phases and adaptive criteria. Our protocol is very simple and efficient, and we show that several generalizations, such as the fractional J-choice K-best secretary problem and exclusive Q-queue J-choice K-best secretary problem, can be solved optimally by this protocol with slight modification and the latter one solves an open problem of Feldman and Tennenholtz [14]. In addition, we provide theoretical analysis for two typical cases, including the 1-queue 1-choice K-best problem and the shared 2-queue 2-choice 2-best problem. For the former, we prove a lower bound 1−O( ln 2 K K2 ) of the competitive ratio. For the latter, we show the optimal competitive ratio is ≈ 0.372 while previously the best known result is 0.356 [14].