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.

[1]  H. Whitney.  Non-Separable and Planar Graphs. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[2]  H. Robbins.  A Theorem on Graphs, with an Application to a Problem of Traffic Control , 1939 .

[3]  D. R. Fulkerson,et al.  Maximal Flow Through a Network , 1956 .

[4]  Robert E. Tarjan,et al.  Computing an st -Numbering , 1976, Theor. Comput. Sci..

[5]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[6]  Ronald V. Book,et al.  Review: Michael R. Garey and David S. Johnson, Computers and intractability: A guide to the theory of $NP$-completeness , 1980 .

[7]  Wolfgang Wimmer.  Ein Verfahren zur Verhinderung von Verklemmungen in Vermittlernetzen , 1982 .

[8]  Nicola Santoro,et al.  Labelling and Implicit Routing in Networks , 1985, Computer/law journal.

[9]  Jan van Leeuwen,et al.  Interval Routing , 1987, Computer/law journal.

[10]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[11]  Pierre Fraigniaud,et al.  A characterization of networks supporting linear interval routing , 1994, PODC '94.

[12]  Patrice Ossona de Mendez,et al.  Regular Orientations, Arboricity, and Augmentation , 1994, Graph Drawing.

[13]  William Klostermeyer,et al.  Pushing Vertices and Orienting Edges , 1999, Ars Comb..

[14]  Konstantin Salikhov.  On the orientation of graphs , 2000, math/0012252.

[15]  Cyril Gavoille,et al.  A survey on interval routing , 2000, Theor. Comput. Sci..

[16]  Jonathan L. Gross,et al.  Handbook of graph theory , 2007, Discrete mathematics and its applications.

[17]  Venkat Venkateswaran,et al.  Minimizing maximum indegree , 2004, Discret. Appl. Math..

[18]  Timothy M. Chan,et al.  Balanced vertex-orderings of graphs , 2005, Discret. Appl. Math..

[19]  Yuichi Asahiro,et al.  Graph Orientation Algorithms to minimize the Maximum Outdegree , 2006, Int. J. Found. Comput. Sci..

[20]  Yuichi Asahiro,et al.  Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree , 2007, AAIM.

[21]  A. Frank.  Connections in Combinatorial Optimization , 2011 .

[22]  Yuichi Asahiro,et al.  Upper and Lower Degree Bounded Graph Orientation with Minimum Penalty , 2011, CATS.

[23]  Yuichi Asahiro,et al.  Degree-Constrained Graph Orientation: Maximum Satisfaction and Minimum Violation , 2014, Theory of Computing Systems.