Minimum restricted diameter spanning trees

Let G = (V,E) be a requirement graph. Let d = (d_ij)_{i,j = 1}ⁿ be a length metric. For a tree T denote by d _T(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, D_T, 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 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 exists 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 312. We use this last result to present an O(log(n))-approximation algorithm.