A Switching Framework for Online Interval Scheduling with Predictions

TL;DR

This work tackles irrevocable online interval scheduling with predictions by introducing Trust-and-Switch and its refinement SemiTrust-and-Switch to balance prediction reliance with robust decision-making. The authors provide strong theoretical guarantees, including 1-consistency and additive/multiplicative robustness bounds, and prove tightness for two-value instances. They also design SmoothMerge, a randomized merging algorithm that blends predictive and robust strategies to achieve graceful degradation when predictions are imperfect, supported by experimental validation on real data. The framework offers a modular approach to leveraging predictions in scheduling tasks while preserving performance under adversarial inputs, with potential applicability to related online optimization problems. Overall, the paper advances the understanding of consistency-robustness trade-offs in learning-augmented online interval scheduling and delivers practical algorithms that adapt to prediction quality.

Abstract

We study online interval scheduling in the irrevocable setting, where each interval must be immediately accepted or rejected upon arrival. The objective is to maximize the total length of accepted intervals while ensuring that no two accepted intervals overlap. We consider this problem in a learning-augmented setting, where the algorithm has access to (machine-learned) predictions. The goal is to design algorithms that leverage these predictions to improve performance while maintaining robust guarantees in the presence of prediction errors. Our main contribution is the SemiTrust-and-Switch framework, which provides a unified approach for combining prediction-based and classical interval scheduling algorithms. This framework applies to both deterministic and randomized algorithms and captures the trade-off between consistency (performance under accurate predictions) and robustness (performance under adversarial inputs). Moreover, we provide lower bounds, proving the tightness of this framework in particular settings. We further design a randomized algorithm that smoothly interpolates between prediction-based and robust algorithms. This algorithm achieves both robustness and smoothness--its performance degrades gracefully with the quality of the prediction.

Figures (7)

• Figure 1: Classification of intervals when, at timepoint $r_j$, the framework decides to switch based on the evaluation up to $t_j = d_i$, where $I_i \in \hat{\mathbf{O}}$. The three classes $\mathcal{I}_1$, $\mathcal{I}_2$, and $\mathcal{I}_3$ are illustrated with distinct visual styles in the figure.
• Figure 2: Illustration of prediction $\hat{\mathbf{I}}$. The red intervals represent the optimal solution for this set of intervals.
• Figure 3: The two cases of the adversarial input. In both figures, the red intervals represent the intervals in the optimal solution. In case \ref{['fig:adversary-instance-case1']}, the blue interval appears only in the Alg, whereas in case \ref{['fig:adversary-instance-case2']}, it is included in both the optimal solution and the Alg.
• Figure 5: The last connected component of the conflict graph $G(\textsc{Opt}, \textsc{Alg})$ before switching. The styles of intervals represent their mapped counterparts. Note that $I_1$ is the only interval taken by Opt that is not mapped to any interval. All intervals $I_4$ and $I_5$, which are mapped to $I_i$, are contained within the range $[r_i, d_i + k]$.
• Figure 6: The two cases of the adversarial input. In both figures, the red intervals represent the intervals in the optimal solution. In case \ref{['fig:adversary-instance-case1']}, the blue interval appears only in the Alg, whereas in case \ref{['fig:adversary-instance-case2']}, it is included in both the optimal solution and the Alg.

Theorems & Definitions (22)

• Lemma 1
• proof
• Theorem 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Proposition 1
• Lemma 4