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Learning-Augmented Algorithms for the Bahncard Problem

Hailiang Zhao, Xueyan Tang, Peng Chen, Shuiguang Deng

TL;DR

A new learning-augmented algorithm, named PFSUM, is developed that incorporates both history and short-term future to improve online decision making and outperforms the primal-dual-based algorithm.

Abstract

In this paper, we study learning-augmented algorithms for the Bahncard problem. The Bahncard problem is a generalization of the ski-rental problem, where a traveler needs to irrevocably and repeatedly decide between a cheap short-term solution and an expensive long-term one with an unknown future. Even though the problem is canonical, only a primal-dual-based learning-augmented algorithm was explicitly designed for it. We develop a new learning-augmented algorithm, named PFSUM, that incorporates both history and short-term future to improve online decision making. We derive the competitive ratio of PFSUM as a function of the prediction error and conduct extensive experiments to show that PFSUM outperforms the primal-dual-based algorithm.

Learning-Augmented Algorithms for the Bahncard Problem

TL;DR

A new learning-augmented algorithm, named PFSUM, is developed that incorporates both history and short-term future to improve online decision making and outperforms the primal-dual-based algorithm.

Abstract

In this paper, we study learning-augmented algorithms for the Bahncard problem. The Bahncard problem is a generalization of the ski-rental problem, where a traveler needs to irrevocably and repeatedly decide between a cheap short-term solution and an expensive long-term one with an unknown future. Even though the problem is canonical, only a primal-dual-based learning-augmented algorithm was explicitly designed for it. We develop a new learning-augmented algorithm, named PFSUM, that incorporates both history and short-term future to improve online decision making. We derive the competitive ratio of PFSUM as a function of the prediction error and conduct extensive experiments to show that PFSUM outperforms the primal-dual-based algorithm.

Paper Structure

This paper contains 25 sections, 13 theorems, 112 equations, 30 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Learning-Augmented Algorithms for the Bahncard Problem
  5. Initial Attempt
  6. Second Attempt
  7. PFSUM Algorithm
  8. Experiments
  9. Discussions and Conclusions
  10. Appendix
  11. A Detailed Comparison between Our Work and im2023online
  12. Consistency of SUM$_w$ is at least $(3 - \beta) / (1 + \beta)$
  13. Proof of Theorem \ref{['theorem-fsum']}
  14. Proof of Proposition \ref{['prop-pfsum-merge-iii-to-v']}
  15. Proof for Pattern III
  16. ...and 10 more sections

Key Result

Lemma 3.1

fleischer2001bahncard For any time $t$, if $c (\sigma; [t, t+T) ) \geq \gamma := C/(1- \beta)$, OPT has at least one reduced request in $[t, t+T)$. The same holds for the time interval $(t, t+T]$.

Figures (30)

  • Figure 1: Illustration for Lemma \ref{['prop-off-cost2']}. The shaded rectangle is the valid time of a Bahncard purchased by OPT.
  • Figure 2: All the 6 patterns of concerned time intervals in which either PFSUM or OPT has a Bahncard. In Patterns III to VI, $x$ is the number of Bahncards purchased by OPT in an on phase and expiring in the next on phase. $x$ can be any non-negative integer.
  • Figure 3: Pattern graph.
  • Figure 4: The cost ratios for commuters ($\beta = 0.8$, $T = 10$, $C = 100$). "U", "N" and "P" represent Uniform, Normal and Pareto ticket price distributions respectively.
  • Figure 5: The cost ratios for occasional travelers ($\beta = 0.8$, $T = 10$, $C = 100$). "U", "N" and "P" represent Uniform, Normal and Pareto ticket price distributions respectively.
  • ...and 25 more figures

Theorems & Definitions (17)

  • Lemma 3.1
  • Corollary 3.2
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • Proposition 4.5
  • ...and 7 more