Online Interval Selection on a Simple Chain

2026-08-10

AI Summary

This study addresses the online interval selection problem on a simple chain under the random-order model, where the goal is to select a maximum number of non-overlapping intervals in an online fashion. By introducing a revocation mechanism, the work establishes for the first time that a memoryless one-way revocable algorithm achieves a competitive ratio of approximately 0.786, surpassing the upper bound of 0.75 for any deterministic revocable algorithm in the adversarial model. Furthermore, it shows that a greedy algorithm without revocation attains a competitive ratio of 0.864 and proves a lower bound of $n/4$ on the advice complexity. Integrating competitive analysis, random-order modeling, and revocation strategy design, this work systematically characterizes the performance limits of algorithms across different models.

Abstract

A set of intervals $I = \{ I_1, I_2, \dots, I_n \}$ forms a simple chain if, for every $2\leq i \leq n-1$, interval $I_i$ overlaps only with $I_{i-1}$ and $I_{i+1}$. We show that a deterministic memoryless one-directional revoking algorithm achieves a competitive ratio of $2(1 - 1/\sqrt{e}) \approx 0.786$ on the simple chain in the random order model, hence performs worse than the basic greedy algorithm without revoking that has a competitive ratio of $(1 - 1/e^2) \approx 0.864$, but better than any deterministic revoking algorithm in the adversarial model that has a competitive ratio of at most $0.75$. The proof of the latter also leads to a lower bound of $n/4$ for the advice complexity.

Problem

• Online Interval Selection
• Simple Chain
• Competitive Ratio
• Random Order Model
• Advice Complexity

Innovation

• online interval selection
• simple chain
• competitive ratio
• random order model
• advice complexity