Egalitarian Graph Orientations

Given an undirected graph, one can assign directions to each of the edges of the graph, thus orienting the graph. To be as egalitarian as possible, one may wish to find an orientation such that no vertex is unfairly hit with too many arcs directed into it. We discuss how this objective arises in problems resulting from telecommunications. We give optimal, polynomial-time algorithms for: finding an orientation that minimizes the lexicographic order of the indegrees and finding a strongly-connected orientation that minimizes the maximum indegree. We show that minimizing the lexicographic order of the indegrees is NP-hard when the resulting orientation is required to be acyclic. Submitted: July 2016 Reviewed: October 2016 Revised: December 2016 Reviewed: March 2017 Revised: May 2017 Accepted: May 2017 Final: May 2017 Published: June 2017 Article type: Regular paper Communicated by: A. Lubiw Research supported by NSF CCF-0963921 and NSF 0852030. E-mail addresses: [email protected] (Glencora Borradaile) [email protected] (Jennifer Iglesias) [email protected] (Theresa Migler) [email protected] (Antonio Ochoa) [email protected] (Gordon Wilfong) [email protected] (Lisa Zhang) 688 Borradaile et al. Egalitarian Graph Orientations