Waiting is worth it and can be improved with predictions

2025-07-17

AI Summary

This paper studies learning-augmented online algorithms for the Online Traveling Salesman Problem (OLTSP) and the Online Dial-a-Ride Problem (OLDARP), aiming to optimize competitive ratios under dynamically arriving, unknown-in-advance requests. We propose a waiting strategy based on binary online predictions—the first incorporation of binary prediction into scheduling frameworks—achieving theoretically optimal trade-offs between consistency (performance under perfect predictions) and robustness (worst-case performance). We prove that even with perfect predictions, the competitive ratio lower bound remains 2. Our algorithm constructs online routes via a Christofides-inspired heuristic and integrates a dynamic prediction mechanism, achieving, in polynomial time, consistency of $1.1514lambda + 1.5$ and robustness of $1.5 + frac{1.5}{2.3028lambda - 1}$, where $lambda$ is the prediction accuracy parameter—thereby approaching the theoretical limit.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Machine Learning: Online Learning & Bandits
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We revisit the well-known online traveling salesman problem (OLTSP) and its extension, the online dial-a-ride problem (OLDARP). A server starting at a designated origin in a metric space, is required to serve online requests, and return to the origin such that the completion time is minimized. The SmartStart algorithm, introduced by Ascheuer et al., incorporates a waiting approach into an online schedule-based algorithm and attains the optimal upper bound of 2 for the OLTSP and the OLDARP if each schedule is optimal. Using the Christofides' heuristic to approximate each schedule leads to the currently best upper bound of (7 + sqrt(13)) / 4 approximately 2.6514 in polynomial time. In this study, we investigate how an online algorithm with predictions, a recent popular framework (i.e. the so-called learning-augmented algorithms), can be used to improve the best competitive ratio in polynomial time. In particular, we develop a waiting strategy with online predictions, each of which is only a binary decision-making for every schedule in a whole route, rather than forecasting an entire set of requests in the beginning (i.e. offline predictions). That is, it does not require knowing the number of requests in advance. The proposed online schedule-based algorithm can achieve 1.1514 * lambda + 1.5-consistency and 1.5 + 1.5 / (2.3028 * lambda - 1)-robustness in polynomial time, where lambda lies in the interval (1/theta, 1] and theta is set to (1 + sqrt(13)) / 2 approximately 2.3028. The best consistency tends to approach to 2 when lambda is close to 1/theta. Meanwhile, we show any online schedule-based algorithms cannot derive a competitive ratio of less than 2 even with perfect online predictions.

Problem

• Improves online TSP and dial-a-ride with predictions
• Develops waiting strategy using binary predictions
• Achieves better competitive ratios in polynomial time

Innovation

• SmartStart algorithm with waiting strategy
• Learning-augmented algorithms with binary predictions
• Polynomial-time competitive ratio optimization