Online 3-Taxi on General Metrics

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.