Forbidden Tournaments and the Orientation Completion Problem
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917763049586688 |
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| author | Bodirsky, Manuel Guzmán-Pro, Santiago |
| author_facet | Bodirsky, Manuel Guzmán-Pro, Santiago |
| contents | For a fixed finite set of finite tournaments ${\mathcal F}$, the ${\mathcal F}$-free orientation problem asks whether a given finite undirected graph $G$ has an $\mathcal F$-free orientation, i.e., whether the edges of $G$ can be oriented so that the resulting digraph does not embed any of the tournaments from ${\mathcal F}$. We prove that for every ${\mathcal F}$, this problem is in P or NP-complete. Our proof reduces the classification task to a complete complexity classification of the orientation completion problem for ${\mathcal F}$, which is the variant of the problem above where the input is a directed graph instead of an undirected graph, introduced by Bang-Jensen, Huang, and Zhu (2017). Our proof uses results from the theory of constraint satisfaction, and a result of Agarwal and Kompatscher (2018) about infinite permutation groups and transformation monoids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_08327 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Forbidden Tournaments and the Orientation Completion Problem Bodirsky, Manuel Guzmán-Pro, Santiago Combinatorics Computational Complexity Logic 05C60 (Primary) 03C98 (Secondary) For a fixed finite set of finite tournaments ${\mathcal F}$, the ${\mathcal F}$-free orientation problem asks whether a given finite undirected graph $G$ has an $\mathcal F$-free orientation, i.e., whether the edges of $G$ can be oriented so that the resulting digraph does not embed any of the tournaments from ${\mathcal F}$. We prove that for every ${\mathcal F}$, this problem is in P or NP-complete. Our proof reduces the classification task to a complete complexity classification of the orientation completion problem for ${\mathcal F}$, which is the variant of the problem above where the input is a directed graph instead of an undirected graph, introduced by Bang-Jensen, Huang, and Zhu (2017). Our proof uses results from the theory of constraint satisfaction, and a result of Agarwal and Kompatscher (2018) about infinite permutation groups and transformation monoids. |
| title | Forbidden Tournaments and the Orientation Completion Problem |
| topic | Combinatorics Computational Complexity Logic 05C60 (Primary) 03C98 (Secondary) |
| url | https://arxiv.org/abs/2309.08327 |