Online Coloring of Short Intervals

TL;DR

This work investigates online coloring of intersection graphs formed by intervals with lengths in $[1,\sigma]$, bridging unit-interval and general interval graphs. It introduces a block-based online coloring algorithm that achieves an asymptotic ratio of $1+\sigma$, matching the best-known bounds for $\sigma=1$ and surpassing prior results for $1<\sigma<2$, by organizing the line into overlapping blocks and assigning colors via private class counters. On the negativity side, the authors develop a recursive Presenter–Algorithm framework that yields tight lower bounds: no online algorithm can achieve asymptotic ratios below $5/3$ for $\sigma>1$, below $7/4$ for $\sigma>2$, and no algorithm can beat $5/2$ asymptotically for all large $\sigma$, using a series of structured lower-bound schemas built through multiple rounds of separation and composition. The core technical contribution is the recursive composition of strategies, enabling lower bounds above 2 and illustrating that the restricted $\sigma$-interval coloring can be strictly easier than general interval coloring, while leaving open gaps and guiding future work toward tightness (and a potential $5/2$-competitive algorithm for all $\sigma$).

Abstract

We study the online graph coloring problem restricted to the intersection graphs of intervals with lengths in $[1,σ]$. For $σ=1$ it is the class of unit interval graphs, and for $σ=\infty$ the class of all interval graphs. Our focus is on intermediary classes. We present a $(1+σ)$-competitive algorithm, which beats the state of the art for $1 < σ< 2$, 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 $σ>1$, nor better than $7/4$-competitive for any $σ>2$, and that no algorithm beats the $5/2$ asymptotic competitive ratio for all, arbitrarily large, values of $σ$. 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 $2$.

Figures (11)

• Figure 1: Small blocks (above), and large blocks (below), for $b=3$
• Figure 2: Separation strategy
• Figure 3: Strategy construction in Lemma \ref{['lem:lower_32']}
• Figure 4: Strategy construction in Lemma \ref{['lem:lower_53']}
• Figure 5: Lemma \ref{['lem:lower_74']}, Case 1: ${\left|\mathcal{C}_{2} \setminus \mathcal{C}_{1}\right|} \geqslant \frac{\omega}{2\alpha+2}$

Theorems & Definitions (39)

• Definition 1
• Definition 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Corollary 7
• Theorem 7
• proof
• Corollary 8