The Complexity of Two Graph Orientation Problems

We consider two orientation problems in a graph, namely the minimization of the sum of all the shortest path lengths and the minimization of the diameter. We show that it is NP-complete to decide whether a graph has an orientation such that the sum of all the shortest paths lengths is at most an integer specified in the input. The proof is a short reduction from a result of Chv\'atal and Thomassen showing that it is NP-complete to decide whether a graph can be oriented so that its diameter is at most 2. In contrast, for each positive integer k, we describe a linear-time algorithm that decides for a planar graph G whether there is an orientation for which the diameter is at most k. We also extend this result from planar graphs to any minor-closed family not containing all apex graphs.