Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning-augmented Online Algorithm for Two-level Ski-rental Problem

Keyuan Zhang, Zhongdong Liu, Nakjung Choi, Bo Ji

TL;DR

This work formulates the two-level ski-rental problem with rent, single, and combo payment options, and proposes a robust deterministic online algorithm (RDTSR) with a worst-case competitive ratio of $3 - \frac{1}{C_s} - \frac{1}{C_c}(2 - \frac{1}{C_s})$. To leverage data-driven advantages while preserving guarantees, it introduces LADTSR, a learning-augmented algorithm that adjusts purchase thresholds based on ML predictions via a trust parameter $\\theta$, achieving both consistency and robustness. Theoretical results bound LADTSR’s performance by $CR \\le \\min \{1+\\theta+\\theta^2 + \frac{1+2\\theta}{1-\\theta} \\cdot \frac{\\eta}{OPT(\\mathbf{D})}, 1+\\theta^{-1} + \\theta^{-3}\\}$, with $\\theta$ controlling the trade-off, and demonstrate lower risk under imperfect predictions. Empirical evaluations on synthetic and real-world traces validate robustness of RDTSR and show LADTSR can exceed RDTSR when predictions are accurate, while maintaining bounded performance under prediction errors, thereby offering a practical, prediction-informed approach for multi-resource rental and purchase decisions.

Abstract

In this paper, we study the two-level ski-rental problem,where a user needs to fulfill a sequence of demands for multiple items by choosing one of the three payment options: paying for the on-demand usage (i.e., rent), buying individual items (i.e., single purchase), and buying all the items (i.e., combo purchase). Without knowing future demands, the user aims to minimize the total cost (i.e., the sum of the rental, single purchase, and combo purchase costs) by balancing the trade-off between the expensive upfront costs (for purchase) and the potential future expenses (for rent). We first design a robust online algorithm (RDTSR) that offers a worst-case performance guarantee. While online algorithms are robust against the worst-case scenarios, they are often overly cautious and thus suffer a poor average performance in typical scenarios. On the other hand, Machine Learning (ML) algorithms typically show promising average performance in various applications but lack worst-case performance guarantees. To harness the benefits of both methods, we develop a learning-augmented algorithm (LADTSR) by integrating ML predictions into the robust online algorithm, which outperforms the robust online algorithm under accurate predictions while ensuring worst-case performance guarantees even when predictions are inaccurate. Finally, we conduct numerical experiments on both synthetic and real-world trace data to corroborate the effectiveness of our approach.

Learning-augmented Online Algorithm for Two-level Ski-rental Problem

TL;DR

This work formulates the two-level ski-rental problem with rent, single, and combo payment options, and proposes a robust deterministic online algorithm (RDTSR) with a worst-case competitive ratio of . To leverage data-driven advantages while preserving guarantees, it introduces LADTSR, a learning-augmented algorithm that adjusts purchase thresholds based on ML predictions via a trust parameter , achieving both consistency and robustness. Theoretical results bound LADTSR’s performance by , with controlling the trade-off, and demonstrate lower risk under imperfect predictions. Empirical evaluations on synthetic and real-world traces validate robustness of RDTSR and show LADTSR can exceed RDTSR when predictions are accurate, while maintaining bounded performance under prediction errors, thereby offering a practical, prediction-informed approach for multi-resource rental and purchase decisions.

Abstract

In this paper, we study the two-level ski-rental problem,where a user needs to fulfill a sequence of demands for multiple items by choosing one of the three payment options: paying for the on-demand usage (i.e., rent), buying individual items (i.e., single purchase), and buying all the items (i.e., combo purchase). Without knowing future demands, the user aims to minimize the total cost (i.e., the sum of the rental, single purchase, and combo purchase costs) by balancing the trade-off between the expensive upfront costs (for purchase) and the potential future expenses (for rent). We first design a robust online algorithm (RDTSR) that offers a worst-case performance guarantee. While online algorithms are robust against the worst-case scenarios, they are often overly cautious and thus suffer a poor average performance in typical scenarios. On the other hand, Machine Learning (ML) algorithms typically show promising average performance in various applications but lack worst-case performance guarantees. To harness the benefits of both methods, we develop a learning-augmented algorithm (LADTSR) by integrating ML predictions into the robust online algorithm, which outperforms the robust online algorithm under accurate predictions while ensuring worst-case performance guarantees even when predictions are inaccurate. Finally, we conduct numerical experiments on both synthetic and real-world trace data to corroborate the effectiveness of our approach.
Paper Structure (38 sections, 14 theorems, 162 equations, 5 figures, 4 algorithms)

This paper contains 38 sections, 14 theorems, 162 equations, 5 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work.
  3. System Model and Problem Formulation
  4. System Model.
  5. Problem Formulation.
  6. Competitive Ratio.
  7. Robust Online Algorithm
  8. Indicative Cost and Purchase Threshold.
  9. Algorithm Description.
  10. Competitive Analysis.
  11. Learning-augmented Online Algorithm
  12. Machine Learning Predictions
  13. A Simple Algorithm: Follow the Prediction
  14. Design of Learning-augmented Algorithm
  15. Analysis of Learning-augmented Algorithm
  16. ...and 23 more sections

Key Result

Theorem 3.1

The CR of RDTSR is upper bounded by In particular, this upper bound is achieved by choosing thresholds $\lambda_s = C_s$ and $\lambda_c = C_c$.

Figures (5)

  • Figure 1: Empirical CR on the synthetic dataset.
  • Figure 2: Average cost ratio under varying prediction errors.
  • Figure 3: Average performance on the synthetic dataset (unit demands). Figs. \ref{['fig:u20']}, \ref{['fig:u25']}, and \ref{['fig:u31']} are under the uniform sequence setting; Figs. \ref{['fig:m20']}, \ref{['fig:m25']}, and \ref{['fig:m33']} are under the mixed sequence setting.
  • Figure 4: Average performance on the synthetic dataset (multi-unit demands). Figs. \ref{['fig:mu20']}, \ref{['fig:mu25']}, and \ref{['fig:mu31']} are under the uniform sequence setting; Figs. \ref{['fig:mm20']}, \ref{['fig:mm25']}, and \ref{['fig:mm33']} are under the mixed sequence setting.
  • Figure 5: Average performance on the real-world datasets. Figs. \ref{['fig:azure20']}, \ref{['fig:azure25']}, and \ref{['fig:azure30']} are under the Azure dataset; Figs. \ref{['fig:app20']}, \ref{['fig:app25']}, and \ref{['fig:app30']} are under the AppUsage dataset.

Theorems & Definitions (39)

  • Theorem 3.1
  • proof : Proof sketch
  • Remark 3.1
  • Lemma 4.1
  • proof : Proof sketch
  • Remark 4.1
  • Theorem 4.1
  • proof : Proof sketch
  • Remark 4.2
  • Remark 4.3
  • ...and 29 more