Minimum Restricted Diameter Spanning Trees

Let G = (V,E) be a requirements graph. Let d = (dij)i,j=1 be a length metric. For a tree T denote by dT (i, j) the distance between i and j in T (the length according to d of the unique i − j path in T ). The restricted diameter of T , DT , is the maximum distance in T between pair of vertices with requirement between them. The minimum restricted diameter spanning tree problem is to find a spanning tree T such that the minimum restricted diameter is minimized. We prove that the minimum restricted diameter spanning tree problem is NP hard and that unless P = NP there is no polynomial time algorithm with performance guarantee of less than 2. In the case that G contains isolated vertices and the length matrix is defined by distances over a tree we prove that there exist a tree over the non-isolated vertices such that its restricted diameter is at most 4 times the minimum restricted diameter and that this constant is at least 3 2 . We use this last result to present an O(log(n))-approximation algorithm.