Graph orientation is a fundamental problem in graph theory that has recently arisen in the study of signaling-regulatory pathways in protein networks. Given a graph and a list of source-target vertex pairs, one wishes to assign directions to the edges so as to maximize the number of pairs that admit a directed source-to-target path. When the input graph is undirected, a sub-logarithmic approximation is known for this problem. However, the approximability of the biologically-relevant variant, in which the input graph has both directed and undirected edges, was left open. Here we give the first approximation algorithms to this problem. Our algorithms provide a sub-linear guarantee in the general case, and logarithmic guarantees for structured instances. © 2012 Elsevier B.V. All rights reserved.
Elberfeld, M., Segev, D., Davidson, C. R., Silverbush, D., & Sharan, R. (2013). Approximation algorithms for orienting mixed graphs. In Theoretical Computer Science (Vol. 483, pp. 96–103). https://doi.org/10.1016/j.tcs.2012.03.044