On the Minimum Load Coloring Problem

Ahuja, N., Baltz, Andreas, Doerr, B., Privetivy, A. and Srivastav, Amand (2007) On the Minimum Load Coloring Problem Journal of Discrete Algorithms, 5 (3). pp. 533-545.

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Abstract

Given a graph G=(V,E) with n vertices, m edges and maximum vertex degree Δ, the load distribution of a coloring View the MathML source is a pair dφ=(rφ,bφ), where rφ is the number of edges with at least one end-vertex colored red and bφ is the number of edges with at least one end-vertex colored blue. Our aim is to find a coloring φ such that the (maximum) load, View the MathML source, is minimized. This problems arises in Wavelength Division Multiplexing (WDM), the technology currently in use for building optical communication networks. After proving that the general problem is NP-hard we give a polynomial time algorithm for optimal colorings of trees and show that the optimal load is at most 1/2+(Δ/m)log2n. For graphs with genus g>0, we show that a coloring with load OPT(1+o(1)) can be computed in O(n+glogn)-time, if the maximum degree satisfies View the MathML source and an embedding is given. In the general situation we show that a coloring with load at most View the MathML source can be found by analyzing a random coloring with Chebychev's inequality. This bound describes the “typical” situation: in the random graph model G(n,m) we prove that for almost all graphs, the optimal load is at least View the MathML source. Finally, we state some conjectures on how our results generalize to k-colorings for k>2.

Document Type:	Article
Keywords:	Graph coloring; Graph partitioning
Research affiliation:	OceanRep > The Future Ocean - Cluster of Excellence Kiel University
Refereed:	Yes
ISSN:	1570-8667
Projects:	Future Ocean
Date Deposited:	23 Mar 2011 12:51
Last Modified:	27 Jan 2012 05:54
URI:	http://eprints.uni-kiel.de/id/eprint/11025

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