On the minimum load coloring problem

Abstract

Given a graph G = ( V , E ) with n vertices, m edges and maximum vertex degree Δ, the load distribution of a coloring φ : V → { red, blue } is a pair d φ = ( r φ , b φ ) , where r φ is the number of edges with at least one end-vertex colored red and b φ 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, l φ : = 1 m ⋅ max { r φ , b φ } , 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 ) log 2 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 Δ = o ( m 2 n g ) and an embedding is given. In the general situation we show that a coloring with load at most 3 4 + O ( Δ / 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 .