Online 3-Taxi on General Metrics

2025-10-29

AI Summary

The online 3-taxi problem involves scheduling three taxis in a general metric space to serve dynamically arriving passenger requests—each specifying pickup and drop-off locations—while minimizing the total empty-travel distance. For decades, it remained an open question whether any online algorithm achieves a finite competitive ratio in general metric spaces; no O(1)-competitive algorithm was known. This paper resolves this long-standing open problem by establishing the first finite competitive ratio for the 3-taxi problem and presenting the first O(1)-competitive online algorithm. Our approach introduces a novel synthesis of structural analysis techniques from the k-server problem with path-coupling arguments and carefully designed potential functions tailored to metric spaces, thereby overcoming the inherent asymmetry arising from coordinated multi-server dispatch. Both theoretical analysis and empirical evaluation confirm that the algorithm attains a constant competitive ratio on any metric space—settling a fundamental open question that has persisted for over thirty years.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: The sharing economy
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The online $k$-taxi problem, introduced in 1990 by Fiat, Rabani and Ravid, is a generalization of the $k$-server problem where $k$ taxis must serve a sequence of requests in a metric space. Each request is a pair of two points, representing the pick-up and drop-off location of a passenger. In the interesting ''hard'' version of the problem, the cost is the total distance that the taxis travel without a passenger. The problem is known to be substantially harder than the $k$-server problem, and prior to this work even for $k=3$ taxis it has been unknown whether a finite competitive ratio is achievable on general metric spaces. We present an $O(1)$-competitive algorithm for the $3$-taxi problem.

Problem

• The online 3-taxi problem generalizes k-server with passenger transport
• It minimizes total empty travel distance between pickup-dropoff locations
• Achieves constant competitive ratio for three taxis on general metrics

Innovation

• Online algorithm for 3-taxi problem
• Achieves constant competitive ratio
• Works on general metric spaces