Egalitarian Graph Orientations

TL;DR

This paper studies how to assign directions to edges in an undirected graph to achieve fairness in indegree distribution, providing optimal algorithms for certain cases and showing NP-hardness for others.

Contribution

It introduces polynomial-time algorithms for egalitarian orientations minimizing lexicographic indegree order and maximum indegree, and proves NP-hardness for acyclic cases.

Findings

Optimal algorithms for lexicographic indegree minimization

Optimal algorithms for strongly-connected orientations with minimal maximum indegree

NP-hardness of acyclic lexicographic indegree minimization

Abstract

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.