On the minimum load coloring problem

Given a graph G=(V,E) with n vertices, m edges and maximum vertex degree @D, the load distribution of a coloring @f:V->{red, blue} is a pair d"@f=(r"@f,b"@f), where r"@f is the number of edges with at least one end-vertex colored red and b"@f is the number of edges with at least one end-vertex colored blue. Our aim is to find a coloring @f such that the (maximum) load, l"@f:[email protected]?max{r"@f,b"@f}, 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+(@D/m)log"2n. 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 @D=o(m^2ng) and an embedding is given. In the general situation we show that a coloring with load at most 34+O(@D/m) 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 34-n/m. Finally, we state some conjectures on how our results generalize to k-colorings for k>2.