Buffer minimization using max-coloring

Given a graph <i>G</i> = (<i>V,E</i>) and positive integral vertex weights <i>w</i> : <i>V</i> → N, the <i>max-coloring problem</i> seeks to find a proper vertex coloring of <i>G</i> whose color classes <i>C</i><inf>1,</inf> <i>C</i><inf>2,</inf>...,<i>C</i><inf><i>k</i></inf>, minimize Σ^<i>k</i><inf><i>i</i> = 1</inf> <i>max</i><inf>ν∈<i>C</i><inf>i</inf></inf><i>w</i>(ν). 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. Specifically, companies have tried to solve this problem in the design of memory managers for wireless protocol stacks such as GPRS or 3G.Though this problem seems similar to the wellknown dynamic storage allocation problem, we point out 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 a 10-approximation algorithm. We show this result by proving that the first-fit algorithm for on-line coloring an interval graph <i>G</i> uses no more than 10.<i>x</i>(<i>G</i>) colors, significantly improving the bound of 26.<i>x</i>(<i>G</i>) by Kierstead and Qin (<i>Discrete Math.</i>, 144, 1995). We also show that the max-coloring problem is NP-hard.