Approximation Algorithms for k-Source Bottleneck Routing Cost Spanning Tree Problems

In this paper, 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 bottleneck vertex routing cost spanning tree (k-BVRT) 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, i.e., we want to minimize maxν ∈ v{Σ s ∈ S d T (s,υ)}, in which d T (s,v) is the length of the path between s and v on T. The other problem is the k-source bottleneck source routing cost spanning tree (k-BSRT) problem, in which the objective function is the maximum total distance from any source to all vertices, i.e., maxs ∈ S{Σ ν ∈ V d T (s,υ)}. In this paper, we present a polynomial time approximation scheme (PTAS) for the 2-BVRT problem. For the 2-BSRT problem, we first give (2+e)-approximation algorithm for any e> 0, and then present a PTAS for the case that the input graphs are restricted to metric graphs. Finally we show that there is a simple 3-approximation algorithm for both the two problems with arbitrary k.

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