Abstract
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 edgeweighted bipartite multigraph G=(V,E) with weights
w:E\rightarrow [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 2m1 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 \lceil 2.2223m \rceil colors improving on the previous result of \frac{9m}{4} by Feige and Singh.
Our algorithm is purely combinatorial and combines the König's theorem for edge coloring bipartite graphs and firstfit decreasing heuristic for bin packing. However, our analysis uses configuration linear program for the bin packing problem to give the improved result.
BibTeX  Entry
@InProceedings{khan_et_al:LIPIcs:2015:5643,
author = {Arindam Khan and Mohit Singh},
title = {{On Weighted Bipartite Edge Coloring}},
booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)},
pages = {136150},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897972},
ISSN = {18688969},
year = {2015},
volume = {45},
editor = {Prahladh Harsha and G. Ramalingam},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5643},
URN = {urn:nbn:de:0030drops56437},
doi = {10.4230/LIPIcs.FSTTCS.2015.136},
annote = {Keywords: Edge coloring, Bin packing, Clos networks, Approximation algorithms, Graph algorithms}
}
Keywords: 

Edge coloring, Bin packing, Clos networks, Approximation algorithms, Graph algorithms 
Collection: 

35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015) 
Issue Date: 

2015 
Date of publication: 

14.12.2015 