Coloring tournaments with few colors: Algorithms and complexity
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866915030140715008 |
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| author | Klingelhoefer, Felix Newman, Alantha |
| author_facet | Klingelhoefer, Felix Newman, Alantha |
| contents | A $k$-coloring of a tournament is a partition of its vertices into $k$ acyclic sets. Deciding if a tournament is 2-colorable is NP-hard. A natural problem, akin to that of coloring a 3-colorable graph with few colors, is to color a 2-colorable tournament with few colors. This problem does not seem to have been addressed before, although it is a special case of coloring a 2-colorable 3-uniform hypergraph with few colors, which is a well-studied problem with super-constant lower bounds.
We present a new efficient decomposition lemma for tournaments, which we use to design polynomial-time algorithms to color various classes of tournaments with few colors, notably, to color a 2-colorable tournament with ten colors. We also use this lemma to prove equivalence between the problems of coloring 3-colorable tournaments and coloring 3-colorable graphs with constantly many colors. For the classes of tournaments considered, we complement our upper bounds with strengthened lower bounds, painting a comprehensive picture of the algorithmic and complexity aspects of coloring tournaments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_02922 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Coloring tournaments with few colors: Algorithms and complexity Klingelhoefer, Felix Newman, Alantha Data Structures and Algorithms A $k$-coloring of a tournament is a partition of its vertices into $k$ acyclic sets. Deciding if a tournament is 2-colorable is NP-hard. A natural problem, akin to that of coloring a 3-colorable graph with few colors, is to color a 2-colorable tournament with few colors. This problem does not seem to have been addressed before, although it is a special case of coloring a 2-colorable 3-uniform hypergraph with few colors, which is a well-studied problem with super-constant lower bounds. We present a new efficient decomposition lemma for tournaments, which we use to design polynomial-time algorithms to color various classes of tournaments with few colors, notably, to color a 2-colorable tournament with ten colors. We also use this lemma to prove equivalence between the problems of coloring 3-colorable tournaments and coloring 3-colorable graphs with constantly many colors. For the classes of tournaments considered, we complement our upper bounds with strengthened lower bounds, painting a comprehensive picture of the algorithmic and complexity aspects of coloring tournaments. |
| title | Coloring tournaments with few colors: Algorithms and complexity |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2305.02922 |