摘要
In this article, we investigate two spanning tree problems of graphs with k given sources. Let G = (V, E, w) be an undirected graph with nonnegative edge lengths and S ⊂ V a set of k specified sources. The first problem is the k-source maximum vertex shortest paths spanning tree (k-MVST) problem, in which we want to find a spanany tree T such that the maximum total distance from any vertex to all sources is minimized, that is, we want to minimize max <sub>v∈V</sub> {σ <sub>s∈S</sub> d <sub>T</sub> (s, v)}, in which d <sub>T</sub> (s, v) is the length of the path between s and v on T. The other problem is the k-source maximum source shortest paths spanning tree (k-MSST) problem, in which the objective function is the maximum total distance from any source to all vertices, that is, maxs <sub>∈</sub> S{σ <sub>v∈V</sub> d <sub>T</sub> (s, v)}. In this article, we present a polynomial time approximation scheme (PTAS) for the 2-MVST problem. For the 2-MSST problem, we first give (2 + ε)-approximation algorithm for any ε > 0, and then present a PTAS for the case that the input graphs are restricted to metric graphs. Finally, we show that there are simple 3-approximation algorithms for both problems with arbitrary k. © 2006 Wiley Periodicals, Inc.