The Minimum Wiener Connector

TL;DR

This paper introduces the minimum Wiener connector problem, aiming to find a subgraph connecting query vertices with minimal Wiener index, and provides approximation algorithms along with experimental validation on real-world graphs.

Contribution

The paper formulates the novel minimum Wiener connector problem, proves its NP-hardness, and develops a constant-factor approximation algorithm with practical efficiency.

Findings

The approximation algorithm is effective on large real-world graphs.

Solutions are smaller and denser than those from other methods.

Adding high-centrality vertices improves connectivity and solution quality.

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).