Efficient Graph Algorithms for Network Analysis

We provide efficient algorithms for two major problems in network analysis. One is the Graph Center (GC) problem and the other is the Graph Bottleneck (GB) problem. The GC problem is relevant for facility location, while the GB problem is relevant for transportation and logistics. The GC problem is to identify a pre-determined number of center vertices such that the distances or costs from (or to) the centers to (or from) other vertices is minimized. Let us take an example of dispatching fire engines from fire stations. The distance here is defined by the shortest distance from the closest station to a house. The problem is then to determine the center vertices (i.e. fire stations) so that the maximum distance to each house is minimized. In the case of hospitals the distance is defined by that from a house to the hospital. In the case of renewable resource centers, we may need to consider both distances, “to” and “from”. The bottleneck of a path is the minimum capacity of edges on the path. The Bottleneck Paths (BP) problem is to compute the paths that give us the maximum bottleneck values between pairs of vertices. The Graph Bottleneck (GB) problem is to find the minimum bottleneck value out of bottleneck paths for all possible pairs of vertices. We give two similar algorithms that are based on binary search to solve the 1-center GC problem and the GB problem on directed graphs with unit edge costs. We achieve Õ(n2.373)1 worst case time complexity for both the 1-center GC problem and the GB problem, where n is the number of vertices in the graph. This is better than the straightforward methods of solving