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 . $\varphi: V \longrightarrow$ 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 := max{r _ϕ , b _ϕ }, is minimized. The problem has applications in broadcast WDM communication networks (Ageev et al., 2004). 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 m/2+${\it \Delta}$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)-time, if the maximum degree satisfies ${\it \Delta} = o(\frac{m^{2}}{ng})$ and an embedding is given. In the general situation we show that a coloring with load at most ${\frac{3} {4}}m+O({\sqrt{{\it \Delta}m}})$ can be found in deterministic polynomial time using a derandomized version of Azuma’s martingale inequality. This bound describes the “typical” situation: in the random multi-graph model we prove that for almost all graphs, the optimal load is at least $\frac{3}{4}m - {\sqrt{3mn}}$ . Finally, we generalize our results to k–colorings for k > 2.