最小Wiener连接器
The Minimum Wiener Connector
摘要：
图的Wiener指数是其顶点之间所有成对最短路距离的和。本文研究了寻找最小Wiener连通子的新问题：给定一个连通图$G=(V,E)$和一个查询顶点集$Q\subseteq V$，寻找一个连通所有查询顶点且具有最小Wiener指数的子图$G$.我们表明，最小维纳连接器承认一个多项式时间（虽然不切实际）的特殊情况下，查询顶点的数量是有界的精确算法。我们表明，在一般情况下，该问题是NP-难的，并没有PTAS，除非$\mathbf{P} = \mathbf{NP}$。我们的主要贡献是一个常数因子近似算法运行在时间$\widetilde{O}(|Q||E|)$。对大量真实世界图的彻底实验证实，我们的方法比其他方法返回更小和更密集的解决方案，并且通过向查询集$Q$添加少量重要顶点（即，具有高中心性的顶点）。
英文摘要：
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).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
