On the Minimum Load Coloring Problem

Given a graph G = (V,E) with n vertices, m edges and maximum vertex degree ∆, the load distribution of a coloring φ : V → {red, blue} 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, lφ := 1 m · max{rφ, bφ}, 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+(∆/m) log2 n. For graphs with genus g > 0, we show that a coloring with load OPT(1 + o(1)) can be computed in O(n + g log n)-time, if the maximum degree satisfies ∆ = o( 2 ng ) and an embedding is given. In the general situation we show that a coloring with load at most 3 4+O( √ ∆/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 3 4 − √ n/m. Finally, we state some conjectures on how our results generalize to k–colorings for k > 2.