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 $Q\subseteq V$ 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 $\mathbf{P}=\mathbf{NP}$. Our main contribution is a constant-factor approximation algorithm running in time $\tilde{O}(|Q||E|)$. 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).

Published in Proceedings of the 2015 ACM SIGMOD International Conference on Management of Data

References in corpus (2)

• Cooperative Game Theory Approaches for Network Partitioning
• Identifying high betweenness centrality nodes in large social networks

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• Span-core Decomposition for Temporal Networks: Algorithms and Applications