Approximation algorithms and hardness results for shortest path based graph orientations

The graph orientation problem calls for orienting the edges of an undirected graph so as to maximize the number of pre-specified source-target vertex pairs that admit a directed path from the source to the target. Most algorithmic approaches to this problem share a common preprocessing step, in which the input graph is reduced to a tree by repeatedly contracting its cycles. While this reduction is valid from an algorithmic perspective, the assignment of directions to the edges of the contracted cycles becomes arbitrary, and the connecting source-target paths may be arbitrarily long. In the context of biological networks, the connection of vertex pairs via shortest paths is highly motivated, leading to the following variant: Given an undirected graph and a collection of source-target vertex pairs, assign directions to the edges so as to maximize the number of pairs that are connected by a shortest (in the original graph) directed path. Here we study this variant, provide strong inapproximability results for it and propose an approximation algorithm for the problem, as well as for relaxations of it where the connecting paths need only be approximately shortest.