Finite dominating sets for the multi-facility ordered median problem in networks and algorithmic applications

We identify a finite dominating set (FDS) for a special case of the multi-facility ordered median problem in networks, in which the k-weights can take at least two different values. This FDS result not only includes the FDS research for the p-center problem, but also extends the case of a < b in the k-weights provided by Kalcsics et al. (Kalcsics, J., Nickel, S., Puerto, J. (2003). Multi-facility ordered median problems: A further analysis. Networks, 41(1), 1–12. Furthermore, based on the FDS result, we devise an exact algorithm for the problem. In addition, we present a polynomial time algorithm for the problem with at most three different values in the k-weights in tree networks. Finally, we pose an open problem on identifying the existence of a polynomial size FDS for the multi-facility convex ordered median problem in networks.