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arXiv · 1504.00513

The Minimum Wiener Connector

Natali Ruchansky·Francesco Bonchi·David Garcia-Soriano·Francesco Gullo·Nicolas Kourtellis
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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 $\widetilde{O}(|Q||E|)$. 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).

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Natali Ruchansky, Francesco Bonchi, David Garcia-Soriano, Francesco Gullo, Nicolas Kourtellis. 2015-04-02. The Minimum Wiener Connector. https://doi.org/10.1145/2723372.2749449

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