Strongly connected orientation with minimum lexicographic order of indegrees

TL;DR

This paper proves that the SC-Path-Reversal algorithm not only minimizes the maximum indegree in a strongly connected orientation but also optimally minimizes the lexicographic order of indegrees, confirming a conjecture.

Contribution

The paper confirms that the SC-Path-Reversal algorithm is optimal for minimizing the lexicographic order of indegrees in strongly connected orientations.

Findings

SC-Path-Reversal finds a strongly connected orientation minimizing lexicographic indegree order

The algorithm is polynomial-time and optimal for the lexicographic objective

Confirms the conjecture by Borradaile et al. about the algorithm's optimality

Abstract

Given a simple undirected graph $G$, an orientation of $G$ is to assign every edge of $G$ a direction. Borradaile et al gave a greedy algorithm SC-Path-Reversal (in polynomial time) which finds a strongly connected orientation that minimizes the maximum indegree, and conjectured that SC-Path-Reversal is indeed optimal for the "minimizing the lexicographic order" objective as well. In this note, we give a positive answer to the conjecture, that is we show that the algorithm SC-PATH-REVERSAL finds a strongly connected orientation that minimizes the lexicographic order of indegrees.