Approximation shortest paths algorithms for some k-source spanning tree problems
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 subset of 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 max(v is an element of V){Sigma(s is an element of S) d(T) (S, V)} 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 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(s is an element of S){Sigma(v is an element of V) d(T)(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 + epsilon)-approximation algorithm for any epsilon > 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. (c) 2006 Wiley Periodicals, Inc.
spanning tree polynomial time approximation scheme combinatorial optimization problem approximation algorithm