One-way and round-trip center location problems

Abstract

In the classical p-center problem there is a set V of points (customers) in some metric space, and the objective is to locate p centers (servers), minimizing the maximum distance between a customer and his respective nearest server. In this paper we consider an extension, where each customer is associated with a set of existing depots or distribution stations he can use. The service of a customer consists of the travel of a server to some permissible depot, loading of some package (e.g., a spare part) at the depot, and the delivery of the package to the customer. This model is called the customer one-way problem. In the round-trip version of the problem, the service also includes the travel from the customer to the home base of the server. In both problems the customer cost of the service is a linear function of the distance travelled by the server. The objective is to locate p servers, minimizing the maximum customer cost (weighted distance travelled by the respective server).Since the classical p-center problem is NP-hard, so are the one-way and the round-trip models we study. We present efficient constant factor approximation algorithms for these problems on general networks. Turning to special networks, we prove that the one-way problem is strongly NP-hard even on path networks. We then present polynomial time algorithms for the round-trip problem on general tree networks. We also discuss the single center case, and provide polynomial time algorithms for general networks, tree networks and planar Euclidean and rectilinear metric spaces.

Collection depots facility location problems in trees

“…Tamir and Halman [12] studied the k-center problem in the plane, in graphs, in trees, and in paths. They introduced three different versions for the CDFL problem.…”

“…Let c be the centroid of T (i). The centroid of any tree T is a vertex of T , not necessarily unique, whose removal from the tree disconnects it in components with size no more than |T | [12]) and determining the client x max with the largest weighted trip distance. We denote by S 1 (c) the component rooted at c that contains x max and by S 2 (c) the component, also rooted at c, that does not contain x max (see Fig.…”

“…The restricted version of the collection depots problem proposed by Tamir and Halman [12] specifies that every client v i has a set X(v i ) ⊂ D of depots that are allowed for that client. If X(v i ) = D for all clients v i , then we have an instance of the unrestricted CDFL problem.…”

“…The unrestricted model is solved in [12] in O(n log n) time. We generalize the iterative approach of Megiddo [9] used to solve the classic weighted 1-center problem.…”

“…We generalize the iterative approach of Megiddo [9] used to solve the classic weighted 1-center problem. The main idea, as in [12], is to compute trip distances from a source vertex to all the "active" clients and decide which of the subtrees adjacent to the source must contain the optimal center. To achieve linear time performance, each iteration must take time linear in the size of the tree and a constant fraction of the clients must be eliminated from the tree.…”