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

TL;DR

The paper addresses online coloring for restricted interval-graph classes, focusing on $\sigma$-interval graphs and 2-count interval graphs. It proves an upper bound of $3$ on the competitive ratio for online $\sigma$-interval coloring via the Kierstead–Trotter framework, while showing that no online algorithm can beat $3-\varepsilon$ for some $\sigma>1$ and many $\sigma$ in the unknown-representation setting. For 2-count interval graphs, the authors analyze First-Fit and several information regimes, establishing non-trivial lower bounds across four cases and demonstrating how knowledge of interval representations and lengths affects the achievable competitive ratios. The work also develops and employs systematic adversarial strategies (EL, CS, B) to construct inputs that enforce higher color usage, thereby refining our understanding of online coloring limits in these interval-based classes. Overall, the results sharpen the landscape of online coloring for structured interval graphs and provide concrete lower bounds that differentiate the impact of information availability on online algorithms.

Abstract

We study the online coloring of $σ$-interval graphs which are interval graphs where the interval lengths are between 1 and $σ$ and 2-count interval graphs which are interval graphs that require at most $2$ distinct interval lengths. For online $σ$-interval graph coloring, we focus on online algorithms that do not have knowledge of the interval representation. The Kierstead-Trotter algorithm has competitive ratio 3 for all $σ$ and no online algorithm has competitive ratio better than 2, even for $σ=1$. In this paper, we show that for every $ε>0$, there is a $σ>1$ such that there is no online algorithm for $σ$-interval coloring with competitive ratio less than $3-ε$. Our strategy also improves the best known lower bounds for the greedy algorithm First-Fit for many values of $σ$. For online 2-count interval graph coloring, we analyze the performance of First-Fit and algorithms under various scenarios. We consider algorithms that receive the interval representation as input and algorithms that do not. We also consider algorithms that have prior knowledge of the interval lengths and algorithms that do not. We provide non-trivial lower bounds for each of the four cases. In particular, we show that there is no online algorithm with competitive ratio less than $2.5$ when the interval lengths are known, there is no online algorithm with competitive ratio less than $2$ when the interval representation is known, and there is no online algorithm with competitive ratio less than $1.75$ when both the interval lengths and interval representation are known.

Figures (4)

• Figure 1: An example of a 2-count interval graph that is not a unit-interval graph and an example of an interval graph that is not a 2-count interval graph.
• Figure 2: The strategy $\mathop{\mathrm{KT}}\nolimits\langle{\omega\rangle}$.
• Figure 3: Strategy $S\langle{\omega\rangle}$.
• Figure 4: Strategy for Theorem \ref{['thm:final']}

Theorems & Definitions (29)

• Theorem 2.1: Kierstead and Trotter kie-tro-81
• Theorem 2.2: Chrobak and Ślusarek chr-slu-88
• Theorem 2.3: Epstein and Levy eps-lev-05
• Theorem 2.4: Bosek et al. survey
• Corollary 2.5: Bosek et al. survey
• Corollary 4.1: Kierstead and Trotter kie-tro-81
• Proposition 4.2
• proof
• Lemma 4.3: Epstein and Levy eps-lev-05
• proof