Optimal Algorithms for Free Order Multiple-Choice Secretary

TL;DR

We address the free-order $k$-secretary problem under adversarial, unknown distributions and design randomness-efficient strategies. The authors introduce a framework that combines atomic/positive-event analysis, Chernoff-based derandomization, and dimension reduction (via Reed-Solomon constructions) to produce low-entropy distributions over arrival orders and simple deterministic multi-threshold algorithms. They prove near-optimal competitiveness $1 - O\left(\sqrt{\frac{\log k}{k}}\right)$ for $k < \frac{\log n}{\log \log n}$ with entropy $\Theta(\log\log n)$, and provide entropy lower bounds that match these upper bounds, thereby establishing near-tight results. The work also extends to the classic secretary problem, delivering refined, entropy-efficient results, and introduces a robust toolkit (atomic/positive-event decomposition, conditional-probability derandomization, and dimension reduction) with potential applicability to other online selection problems. Overall, the paper advances randomness-efficient strategies for free-order secretary problems and sharpens the understanding of how entropy influences achievable competitiveness.

Abstract

Suppose we are given integer $k \leq n$ and $n$ boxes labeled $1,\ldots, n$ by an adversary, each containing a number chosen from an unknown distribution. We have to choose an order to sequentially open these boxes, and each time we open the next box in this order, we learn its number. If we reject a number in a box, the box cannot be recalled. Our goal is to accept the $k$ largest of these numbers, without necessarily opening all boxes. This is the free order multiple-choice secretary problem. Free order variants were studied extensively for the secretary and prophet problems. Kesselheim, Kleinberg, and Niazadeh KKN (STOC'15) initiated a study of randomness-efficient algorithms (with the cheapest order in terms of used random bits) for the free order secretary problems. We present an algorithm for free order multiple-choice secretary, which is simultaneously optimal for the competitive ratio and used amount of randomness. I.e., we construct a distribution on orders with optimal entropy $Θ(\log\log n)$ such that a deterministic multiple-threshold algorithm is $1-O(\sqrt{\log k/k})$-competitive. This improves in three ways the previous best construction by KKN, whose competitive ratio is $1 - O(1/k^{1/3}) - o(1)$. Our competitive ratio is (near)optimal for the multiple-choice secretary problem; it works for exponentially larger parameter $k$; and our algorithm is a simple deterministic multiple-threshold algorithm, while that in KKN is randomized. We also prove a corresponding lower bound on the entropy of optimal solutions for the multiple-choice secretary problem, matching entropy of our algorithm, where no such previous lower bound was known. We obtain our algorithmic results with a host of new techniques, and with these techniques we also improve significantly the previous results of KKN about constructing entropy-optimal distributions for the classic free order secretary.

Theorems & Definitions (32)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 1: Atomic events with respect to a bucketing
• Definition 2: Positive events based on atomic family $\mathcal{A}_{k, \mathcal{B}}$
• Theorem 5
• Theorem 6
• Theorem 7
• Lemma 1