Online coloring of short interval graphs and two-count interval graphs

TL;DR

This paper investigates the limits of online coloring algorithms for special classes of interval graphs, establishing bounds on their competitive ratios and analyzing the performance of greedy algorithms.

Contribution

It proves new lower bounds on the competitive ratios for online coloring of $\sigma$-interval and 2-count interval graphs, and analyzes the effectiveness of the First-Fit algorithm.

Findings

No online algorithm for $\sigma$-interval coloring can have a competitive ratio less than 3 for some $\sigma$.

First-Fit has a competitive ratio at most 4 for 2-count interval graphs.

Lower bounds of 2.5 and 2 are established for online algorithms when the interval representation is unknown or known, respectively.

Abstract

We study the online coloring of $\sigma$-interval graphs, which are interval graphs with interval lengths in $[1,\sigma]$ and 2-count interval graphs, which are interval graphs that require at most two distinct interval lengths. For $\sigma$-interval graphs, the Kierstead-Trotter algorithm has competitive ratio 3 and no online algorithm has competitive ratio better than 2. In this paper, we show that for every $\varepsilon>0$, there is a $\sigma>1$ such that there is no online algorithm for $\sigma$-interval coloring with competitive ratio less than $3-\varepsilon$. For 2-count interval graphs, we show that the greedy algorithm First-Fit has competitive ratio at most $4$, that there is no online algorithm with competitive ratio less than $2.5$ when the interval representation is unknown, and that there is no online algorithm with competitive ratio less than $2$ when the interval representation is known.