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.

翻译：对于固定的有限锦标赛集合 ${\mathcal F}$，${\mathcal F}$-自由定向问题询问给定的有限无向图 $G$ 是否具有 ${\mathcal F}$-自由定向，即能否对 $G$ 的边进行定向，使得所得有向图不包含 ${\mathcal F}$ 中的任何锦标赛。我们证明，对于每个 ${\mathcal F}$，该问题要么属于 P，要么是 NP 完全的。我们的证明将分类任务归结为 ${\mathcal F}$ 的定向完成问题的完全复杂度分类，后者是上述问题的一个变体（由 Bang-Jensen、Huang 和 Zhu 于 2017 年提出），其输入为有向图而非无向图。我们的证明使用了约束满足理论的结果，以及 Agarwal 和 Kompatscher（2018）关于无限置换群与变换幺半群的一个结论。