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

TL;DR

This paper introduces an optimal deterministic protocol for a parallel generalization of the secretary problem involving multiple queues and selection criteria, improving theoretical bounds and solving open problems.

Contribution

It provides the first optimal deterministic protocol for the shared Q-queue J-choice K-best secretary problem with multiple queues, extending classical solutions.

Findings

Proposed a simple, efficient protocol with multiple phases and adaptive criteria.

Established a lower bound of 1 - O(ln^2 K / K^2) for the 1-queue K-best case.

Achieved an optimal competitive ratio of approximately 0.372 for the 2-queue 2-choice 2-best problem.

Abstract

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 $1\over e$-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 $1-O(\frac{\ln^2K}{K^2})$ of the competitive ratio. For the latter, we show the optimal competitive ratio is $\approx0.372$ while previously the best known result is 0.356 [14].