Shortest Longest-Path Graph Orientations for Trees

Graph orientation transforms an undirected graph into a directed graph by assigning a direction to each edge. Among the many different optimization problems related to graph orientations, we focus here on the Shortest Longest-Path Orientation problem (SLPO) which is a generalization of the well-known Minimum Graph Coloring problem. The input to SLPO is an edge-bi-weighted undirected graph in which every edge has two (possibly different and not necessarily positive) lengths associated with its two directions. The goal is to find an orientation of the input graph that minimizes the length of the longest simple directed path. Recently, polynomial-time algorithms for simple graph structures such as paths, cycles, stars, and trees were proposed, and a new polynomial-time inapproximability result was also established. This paper presents (i) an O(n²logn)-time algorithm for trees, which is a significant improvement over the previously fastest algorithm whose time complexity was Ω(n¹⁴) and (ii) polynomial-time algorithms for trees and spiders that run even faster than (i) as long as every edge weight is an integer and the total weight of the edges is sub-exponential.