The Secretary Problem with Predictions and a Chosen Order
This work addresses online decision-making problems like interview scheduling by enhancing algorithms with predictions, offering incremental improvements in competitive ratios for both random and chosen order scenarios.
The paper tackles the secretary problem with machine-learned predictions, proposing a randomized algorithm that balances consistency and robustness by switching from trusting predictions to a threshold rule upon detecting large errors. For the Random Order Secretary Problem, it achieves a competitive ratio of max{0.221, (1-ε)/(1+ε)}, improving on prior work, and for the Chosen Order Secretary Problem, it achieves max{0.262, (1-ε)/(1+ε)}, surpassing previous bounds and approaching the classical benchmark of 1/e ≈ 0.368.
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 $ε\in [0,1]$ denote the maximum multiplicative prediction error. For ROSP, our algorithm achieves a competitive ratio of $\max\{0.221, (1-ε)/(1+ε)\}$, improving upon the prior bound of $\max\{0.215, (1-ε)/(1+ε)\}$. For COSP, we achieve $\max\{0.262, (1-ε)/(1+ε)\}$, 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.