Minimizing movement in mobile facility location problems

In the mobile facility location problem, which is a variant of the classical facility location, each facility and client is assigned to a start location in a metric graph and our goal is to find a destination node for each client and facility such that every client is sent to a node which is the destination of some facility. The quality of a solution can be measured either by the total distance clients and facilities travel or by the maximum distance traveled by any client or facility. As we show in this article (by an approximation-preserving reduction), the problem of minimizing the total movement of facilities and clients generalizes the classical k -median problem. The class of movement problems was introduced by Demaine et al. [2007] where a simple 2-approximation was proposed for the minimum maximum movement mobile facility location problem while an approximation for the minimum total movement variant and hardness results for both were left as open problems. Our main result here is an 8-approximation algorithm for the minimum total movement mobile facility location problem. Our algorithm is obtained by rounding an LP relaxation in five phases. For the minimum maximum movement mobile facility location problem, we show that we cannot have a better than a 2-approximation for the problem, unless P = NP so the simple algorithm proposed by Demaine et al. [2007] is essentially best possible.