Max-Coloring Paths: Tight Bounds and Extensions

Abstract

The max-coloring problem is to compute a legal coloring of the vertices of a graph G = (V,E) with a non-negative weight function w on V such that $\sum_{i=1}^k \max_{v\in C_i} w(v_i)$ is minimized, where C ₁,...,C _k are the various color classes. Max-coloring general graphs is as hard as the classical vertex coloring problem, a special case where vertices have unit weight. In fact, in some cases it can even be harder: for example, no polynomial time algorithm is known for max-coloring trees. In this paper we consider the problem of max-coloring paths and its generalization, max-coloring a broad class of trees and show it can be solved in time $O(|V| + \text{time for sorting the vertex weights})$. When vertex weights belong to ℝ, we show a matching lower bound of Ω(|V|log|V|) in the algebraic computation tree model.