The Minimum Wiener Connector
Abstract
The Wiener index of a graph is the sum of all pairwise shortest-path distances between its vertices. In this paper we study the novel problem of finding a minimum Wiener connector: given a connected graph G=(V,E) and a set Q⊂eq V of query vertices, find a subgraph of G that connects all query vertices and has minimum Wiener index. We show that The Minimum Wiener Connector admits a polynomial-time (albeit impractical) exact algorithm for the special case where the number of query vertices is bounded. We show that in general the problem is NP-hard, and has no PTAS unless P = NP. Our main contribution is a constant-factor approximation algorithm running in time O(|Q||E|). A thorough experimentation on a large variety of real-world graphs confirms that our method returns smaller and denser solutions than other methods, and does so by adding to the query set Q a small number of important vertices (i.e., vertices with high centrality).