Fast Approximation Algorithm for the 1-Median Problem

We present a fast approximation algorithm for the 1-median problem. Our algorithm can be applied to metric undirected graphs with node weight. Given a node v, our algorithm repeatedly executes a process of finding a node with higher centrality,in which an approximate centrality of each node v’ is calculated for the subgraph called the best kNSPGS of (v,v’). The best kNSPGS of (v,v’) is a subgraph that contains the shortest path tree of v, and approximate centralities of all the nodes v’ for the subtrees can be calculated more efficiently than their exact centralities for the original graph. We empirically show that our algorithm runs much faster and has better approximation ratio than a sophisticated existing method called DTZ. We demonstrate the effectiveness of our algorithm through experiments. We can use graphs to describe many kinds of relationships in daily life. For example, consider a graph with node weight to describe the transportation network. The node weight means the customer’s demand and the edges mean transport routes with the cost of its length. In this situation, the facility location problem seeks an optimal location of facilities to minimize the total cost of the transport and the opening costs of the facilities. This problem has been studied since 1960s because it can be applied to many other situations. The k-median problem [1] is a special case of this problem. In the k-median problem, the number of facilities is fixed to k, but there is no cost to open facilities. Even now, this is one of the important subjects of research. Approximation algorithms for the k-median problem have been presented so far ([2], [3]), since k-median problem is NP-hard. Jain et al. proposed an approximation algorithm for k-median problem by using LP-relaxation [2]. Indyk proposed an approximation algorithm based on the random sampling [3]. Freeman defined some centrality measures [4]. The closeness centrality is one of them. This centrality of node v is defined as the reciprocal number of the sum of the distances from v to other nodes. The solution of the 1-median problem, also known as the Fermat-Weber problem, is the highest closeness centrality node. In the field of data mining, a high closeness centrality node has important meaning. For example, the closeness centrality is utilized to seek authorities from social networks and citation networks of papers. Recently, also in the field of biomedicine, this