# Online Coloring of Short Intervals

## TL;DR

This paper investigates online coloring of interval graphs with lengths in [1,σ], presenting a new algorithm with competitive ratio (1+σ) and establishing bounds that show the problem's complexity varies with σ.

## Contribution

It introduces a $(1+\sigma)$-competitive algorithm for intermediate interval lengths and provides lower bounds, revealing the problem's complexity spectrum.

## Key findings

- The algorithm outperforms previous methods for 1<σ<2.
- Lower bounds are established at 5/3, 7/4, and 5/2 competitive ratios.
- The problem's difficulty varies with σ, being easier than general interval graphs but harder than unit interval graphs.

## Abstract

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 $1 < \sigma < 2$, and proves that the problem we study can be strictly easier than online coloring of general 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 $\sigma>2$, and that no algorithm beats the $5/2$ asymptotic competitive ratio for all, arbitrarily large, values of $\sigma$. That last result shows that the problem we study can be strictly harder than unit interval coloring. Our main technical contribution is a recursive composition of strategies, which seems essential to prove any lower bound higher than $2$.