Confidence-Aware Learning-Augmented Algorithms for the Bahncard Problem

Learning-augmented algorithms integrate predictions into online decision-making while maintaining worst-case guarantees. In this work, we revisit the learning-augmented algorithm design for the classical Bahncard problem, and propose a confidence-aware algorithm that can dynamically adjust its trust on predictions. We first identify subtle issues in existing analysis, including restrictive assumptions on interval dominance and missing edge cases in the interaction between the online algorithm and the optimal solution. To address these issues, we develop a refined pattern-based analysis and derive improved competitive-ratio bounds that explicitly capture the trade-off between prediction accuracy and robustness. Our results demonstrate that incorporating confidence into prediction-driven decisions leads to strictly better performance in favorable regimes, while preserving strong worst-case guarantees.