On the approximability of reachability-preserving network orientations

We introduce a graph-orientation problem arising in the study of biological networks. Given an undirected graph and a list of ordered source–target vertex pairs, the goal is to orient the graph such that a maximum number of pairs admit a directed source-to-target path. We study the complexity and approximability of this problem. We show that the problem is NP-hard even on star graphs and hard to approximate to within some constant factor. On the positive side, we provide an Ω(log log n/ log n) factor approximation algorithm for the problem on n-vertex graphs. We further show that for any instance of the problem there exists an orientation of the input graph that satisfies a logarithmic fraction of all pairs and that this bound is tight up to a constant factor. Our techniques also lead to constant-factor approximation algorithms for some restricted variants of the problem.