Online coloring of short interval graphs and two-count interval graphs
This work provides tight lower bounds for online coloring of σ-interval graphs and new bounds for 2-count interval graphs, advancing the understanding of online graph coloring for restricted interval families.
The paper studies online coloring of σ-interval graphs and 2-count interval graphs. It shows that for σ-interval graphs, the competitive ratio of any online algorithm can be arbitrarily close to 3, and for 2-count interval graphs, First-Fit has competitive ratio at most 4, with lower bounds of 2.5 (unknown representation) and 2 (known representation).
We study the online coloring of $σ$-interval graphs, which are interval graphs with interval lengths in $[1,σ]$ and 2-count interval graphs, which are interval graphs that require at most two distinct interval lengths. For $σ$-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 $ε>0$, there is a $σ>1$ such that there is no online algorithm for $σ$-interval coloring with competitive ratio less than $3-ε$. 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