Fast Edge Colorings with Fixed Number of Colors to Minimize Imbalance

We study the following optimization problem: the input is a multigraph G = (V,E) and an integer parameter g. A feasible solution consists of a (not necessarily proper) coloring of E with colors 1, 2, ..., g. Denote by d(v,i) the number of edges colored i incident to v. The objective is to minimize $\sum_{v \in{V}} \mbox{max}_{i}d(v,i)$, which roughly corresponds to the “imbalance” of the edge coloring. This problem was proposed by Berry and Modiano (INFOCOM 2004), with the goal of optimizing the deployment of tunable ports in optical networks. Following them we call the optimization problem MTPS – Minimum Tunable Port with Symmetric Assignments. Among other results, they give a reduction from Edge Coloring showing that MTPS is NP-Hard and then give a 2-approximation algorithm. We give a (3/2)-approximation algorithm. Key to this problem is the following question: given a multigraph G = (V,E) of maximum degree g, what fraction of the vertices can be properly edge-colored in a coloring with g colors, where a vertex is properly edge-colored if the edges incident to it have different colors? Our main lemma states that there is such a coloring with half of the vertices properly edge-colored. For g ≤4, two thirds of vertices can be made properly edge-colored. Our algorithm is based on g Maximum Matching computations (total running time $O(g m \sqrt{n + m/g})$) and a local optimization procedure, which by itself gives a 2-approximation. An interesting analysis gives an expected O((gn + m) log(gn +m)) running time for the local optimization procedure.