On the Minimum Load Coloring Problem

Abstract

AbstractGiven 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.KeywordsPlanar GraphLoad DistributionPolynomial Time AlgorithmOptimal LoadOuterplanar GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.