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 spanning tree T such that the maximum total distance from any vertex to all sources is minimized, that is, we want to minimize maxvεV{&Sgr;sεSdT(s,v)}, in which dT(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, max sεS{&Sgr;vεVdT(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.
