On Weighted Bipartite Edge Coloring

We study weighted bipartite edge coloring problem, which is a generalization of two classical problems: bin packing and edge coloring. This problem has been inspired from the study of Clos networks in multirate switching environment in communication networks. In weighted bipartite edge coloring problem, we are given an edge-weighted bipartite multigraph G = (V,E) with weights w : E → [0, 1]. The goal is to find a proper weighted coloring of the edges with as few colors as possible. An edge coloring of the weighted graph is called a proper weighted coloring if the sum of the weights of the edges incident to a vertex of any color is at most one. Chung and Ross conjectured 2m − 1 colors are sufficient for a proper weighted coloring, where m denotes the minimum number of unit sized bins needed to pack the weights of all edges incident at any vertex. We give an algorithm that returns a coloring with at most d2.2223me colors improving on the previous result of 9m 4 by Feige and Singh. Our algorithm is purely combinatorial and combines the König’s theorem for edge coloring bipartite graphs and first-fit decreasing heuristic for bin packing. However, our analysis uses configuration linear program for the bin packing problem to give the improved result. 1998 ACM Subject Classification G.2.2 Graph Theory