The Secretary Problem with Predictions and a Chosen Order

2026-01-12

Information Technology Convergence and Services

AI Summary

This work studies the secretary problem with predictions, aiming to achieve near-optimal performance when predictions are accurate while maintaining a bounded competitive ratio when they are not. The authors propose a unified randomized algorithm for both the random-order secretary problem (ROSP) and the controllable-order secretary problem (COSP), dynamically balancing trust in predictions and threshold-based strategies to harmonize consistency and robustness. Notably, this is the first approach to integrate predictions with order control in COSP, thereby improving worst-case performance. Theoretical analysis shows that the algorithm attains competitive ratios of $\max\{0.221, (1-\varepsilon)/(1+\varepsilon)\}$ for ROSP and $\max\{0.262, (1-\varepsilon)/(1+\varepsilon)\}$ for COSP, outperforming existing methods and approaching the classical $1/e$ benchmark.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study a learning-augmented variant of the secretary problem, recently introduced by Fujii and Yoshida (2023), in which the decision-maker has access to machine-learned predictions of candidate values. The central challenge is to balance consistency and robustness: when predictions are accurate, the algorithm should select a near-optimal secretary, while under inaccurate predictions it should still guarantee a bounded competitive ratio. We consider both the classical Random Order Secretary Problem (ROSP), where candidates arrive in a uniformly random order, and a more natural learning-augmented model in which the decision-maker may choose the arrival order based on predicted values. We call this model the Chosen Order Secretary Problem (COSP), capturing scenarios such as interview schedules set in advance. We propose a new randomized algorithm applicable to both ROSP and COSP. Our method switches from fully trusting predictions to a threshold-based rule once a large prediction deviation is detected. Let $\epsilon \in [0,1]$ denote the maximum multiplicative prediction error. For ROSP, our algorithm achieves a competitive ratio of $\max\{0.221, (1-\epsilon)/(1+\epsilon)\}$, improving upon the prior bound of $\max\{0.215, (1-\epsilon)/(1+\epsilon)\}$. For COSP, we achieve $\max\{0.262, (1-\epsilon)/(1+\epsilon)\}$, surpassing the $0.25$ worst-case bound for prior approaches and moving closer to the classical secretary benchmark of $1/e \approx 0.368$. These results highlight the benefit of combining predictions with arrival-order control in online decision-making.

Problem

• Secretary Problem
• Learning-Augmented Algorithms
• Competitive Ratio
• Prediction Error
• Online Decision-Making

Innovation

• learning-augmented algorithms
• secretary problem
• chosen order
• competitive ratio
• prediction error