Online 3-Taxi on General Metrics

TL;DR

This work resolves the long-standing open question of whether a finite competitive ratio exists for the hard online $3$-taxi problem on general metric spaces by introducing TripodTracker, a deterministic $O(1)$-competitive algorithm. The method combines a novel interval-based memory on top of a tripod decomposition of the metric, with a carefully crafted potential function that tracks a distorted online–offline matching and a secondary Σ component to handle edge cases. The algorithm moves the active taxi slowly while the two passive taxis move along a tripod, using reorganization and a discounted matching to ensure that online progression can be charged to offline optimal progress, yielding a constant competitive ratio. The paper also outlines pathways to extend the approach to larger $k$, including using hierarchical embeddings (HSTs) and generalized interval structures, potentially advancing the broader understanding of the online $k$-taxi problem on general metrics and related $k$-server generalizations.

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.

Figures (4)

• Figure 1: The tripod $B(x,y,z)$ with center $e$.
• Figure 2: The matching contribution of the $(x_2,y_2)$-pair is determined by the undiscounted factor of $1$ and the discounted factor of $1-\psi$ over the different portions of the path $P_2$ from $x_2$ to $y_2$.
• Figure 3: When the two passive online taxis $x_2$ and $x_3$ are matched to two offline taxis $y$ and $\hat{y}$, the relative distance of the branching points $g$ and $\hat{g}$ to each passive taxi determines the minimum matching. In the minimum matching of the above scenario, $x_2$ is matched to $\hat{y}$ and $x_3$ is matched to $y$.
• Figure 4: Example of TripodTracker serving a request. Since the bridges and tripods change over the course of serving a request, each diagram displays the current state of each bridge or tripod at that point in the algorithm.

Theorems & Definitions (25)

• Theorem 1.1
• Lemma 2.1
• Claim 3.0
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Claim 3.3
• Lemma 3.4