Max-coloring and online coloring with bandwidths on interval graphs
ACM Transactions on Algorithms
Given a graph G = (V, E) and positive integral vertex weights w : V → N, the max-coloring problem seeks to find a proper vertex coloring of G whose color classes C 1 , C 2 , . . . , C k , minimize k i=1 max v∈C i w(v). This problem, restricted to interval graphs, arises whenever there is a need to design dedicated memory managers that provide better performance than the general purpose memory management of the operating system. Though this problem seems similar to the dynamic storage allocation
... problem, there are fundamental differences. We make a connection between max-coloring and on-line graph coloring and use this to devise a simple 2-approximation algorithm for max-coloring on interval graphs. We also show that a simple first-fit strategy, that is a natural choice for this problem, yields an 8-approximation algorithm. We show this result by proving that the first-fit algorithm for on-line coloring an interval graph G uses no more than 8 · χ(G) colors, significantly improving the bound of 26·χ(G) by Kierstead and Qin (Discrete Math., 144, 1995). We also show that the max-coloring problem is NP-hard. The problem of online coloring of intervals with bandwidths is a simultaneous generalization of online interval coloring and online bin packing. The input is a set I of intervals, each interval i ∈ I having an associated bandwidth b(i) ∈ (0, 1]. We seek an online algorithm that produces a coloring of the intervals such that for any color c and any real r, the sum of the bandwidths of intervals containing r and colored c is at most 1. Motivated by resource allocation problems, Adamy and Erlebach (Proceedings of the First International Workshop on Online and Approximation Algorithms, 2003, LNCS 2909, pp 1-12 ) consider this problem and present an algorithm that uses at most 195 times the number of colors used by an optimal off-line algorithm. Using the new analysis of first-fit coloring of interval graphs, we show that the Adamy-Erlebach algorithm is 35-competitive. Finally, we generalize the Adamy-Erlebach algorithm to a class of algorithms and show that a different instance from this class is 30-competitive.