Online coloring of short interval graphs and two-count interval graphs
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arXiv
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| Format: | Preprint |
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2024
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| author | Curbelo, Israel R. |
| author_facet | Curbelo, Israel R. |
| contents | 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 $\varepsilon>0$, there is a $σ>1$ such that there is no online algorithm for $σ$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_17193 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Online coloring of short interval graphs and two-count interval graphs Curbelo, Israel R. Data Structures and Algorithms Combinatorics 05D99, 68W27, 05C15 F.2.2 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 $\varepsilon>0$, there is a $σ>1$ such that there is no online algorithm for $σ$-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. |
| title | Online coloring of short interval graphs and two-count interval graphs |
| topic | Data Structures and Algorithms Combinatorics 05D99, 68W27, 05C15 F.2.2 |
| url | https://arxiv.org/abs/2412.17193 |