Improved algorithms to network p-center location problems

In this paper we show that a p(⩾2)p(⩾2)-center location problem in general networks can be transformed to the well-known Kleeʼs measure problem (Overmars and Yap, 1991) [15]. This results in a significantly improved algorithm for the continuous case with running time O(mpnp/22log⁎nlogn) for p⩾3p⩾3, where n is the number of vertices, m   is the number of edges, and log⁎n denotes the iterated logarithm of n (Cormen et al., 2001) [10]. For p=2p=2, the running time of the improved algorithm is O(m2nlog2n). The previous best result for the problem is O(mpnpα(n)logn) where α(n)α(n) is the inverse Ackermann function (Tamir, 1988) [17]. When the underlying network is a partial k-tree (k fixed), we exploit the geometry inherent in the problem and propose a two-level tree decomposition data structure which can be used to efficiently solve discrete p-center problems for small values of p.