Online Coloring of Short Intervals

Grzegorz Gutowski, Konstanty Junosza-Szaniawski, Patryk Mikos, Adam Polak, Joanna Sokół

Published 2018-02-26Version 1

We study the online graph coloring problem restricted to the intersection graphs of intervals with lengths in $[1,\sigma]$. For $\sigma=1$ it is the class of unit interval graphs, and for $\sigma=\infty$ the class of all interval graphs. Our focus is on intermediary classes. We present a $(1+\sigma)$-competitive algorithm, which beats the state of the art for $1 < \sigma < 2$. For $\sigma = 1$ our algorithm matches the performance of FirstFit, which is $2$-competitive for unit interval graphs. For $\sigma=2$ it matches the Kierstead-Trotter algorithm, which is $3$-competitive for all interval graphs. On the lower bound side, we prove that no algorithm is better than $5/3$-competitive for any $\sigma>1$, nor better than $7/4$-competitive for any $\sigma>2$, nor better than $5/2$-competitive for arbitrarily large values of $\sigma$.