The Online $k$-Taxi Problem

We consider the online $k$-taxi problem, a generalization of the $k$-server problem, in which $k$ taxis serve a sequence of requests in a metric space. A request consists of two points $s$ and $t$, representing a passenger that wants to be carried by a taxi from $s$ to $t$. The goal is to serve all requests while minimizing the total distance traveled by all taxis. The problem comes in two flavors, called the easy and the hard $k$-taxi problem: In the easy $k$-taxi problem, the cost is defined as the total distance traveled by the taxis; in the hard $k$-taxi problem, the cost is only the distance of empty runs. The hard $k$-taxi problem is substantially more difficult than the easy version with at least an exponential deterministic competitive ratio, $\Omega(2^k)$, admitting a reduction from the layered graph traversal problem. In contrast, the easy $k$-taxi problem has exactly the same competitive ratio as the $k$-server problem. We focus mainly on the hard version. For hierarchically separated trees (HSTs), we present a memoryless randomized algorithm with competitive ratio $2^k-1$ against adaptive online adversaries and provide two matching lower bounds: for arbitrary algorithms against adaptive adversaries and for memoryless algorithms against oblivious adversaries. Due to well-known HST embedding techniques, the algorithm implies a randomized $O(2^k\log n)$-competitive algorithm for arbitrary $n$-point metrics. This is the first competitive algorithm for the hard $k$-taxi problem for general finite metric spaces and general $k$. For the special case of $k=2$, we obtain a precise answer of $9$ for the competitive ratio in general metrics. With an algorithm based on growing, shrinking and shifting regions, we show that one can achieve a constant competitive ratio also for the hard $3$-taxi problem on the line (abstracting the scheduling of three elevators).