Approximation and hardness results for the maximum edge q-coloring problem

Abstract

We consider the problem of coloring edges of a graph subject to the following constraints: for every vertex v, all the edges incident with v have to be colored with at most q colors. The goal is to find a coloring satisfying the above constraints and using the maximum number of colors. Notice that the notion of coloring is different than in the classical edge coloring problem, as neighboring edges can have the same color.This problem has been studied in the past from the combinatorial and algorithmic point of view. The optimal coloring is known for some special classes of graphs. There is also an approximation algorithm for general graphs, which in the case q=2 gives a 2-approximation. However, the complexity of finding the optimal coloring was not known.In this paper, we prove that the problem is NP-Hard for any q≥2. Moreover, we prove that it cannot be approximated within a factor of 1+1/q−ϵ for any ϵ>0 and any q≥2 assuming the unique games conjecture (UGC), or 1+q−2(q−1)2−ϵ for any ϵ>0 and any q≥3 (≈1.19 for q=2) assuming P≠NP. These results hold even when the considered graphs are bipartite.On the algorithmic side, we restrict to the case q=2, since this is the most important in practice and we show a 5/3-approximation algorithm for graphs which have a perfect matching.