A Multifacility Location Problem on Median Spaces

This paper is concerned with the problem of locating II new facilities in the median space when there are k facilities already located. The objective is to minimize the weighted sum of distances. Necessary and sufficient conditions are established. Based on these results a polynomial algorithm is presented. The algorithm requires the solution of a sequence of minimum-cut problems. The complexity of this algorithm for median graphs and networks and for finite median spaces with 1 V 1 points is 0( 1 V I3 + 1 V I+(n)), where $(n) is the complexity of the applied maximum-flow algorithm. For a simple rectilinear polygon P with N edges and equipped with the rectilinear distance the analogical algorithm requires O(N + k(log N + log k + I,!+))) time and O(N + k+(n)) time in the case of the vertex-restricted multifacility location problem.