Using the algebraic approach to promise constraint satisfaction problems, we establish complexity classifications of three natural variants of hypergraph colourings: standard nonmonochromatic colourings, conflict-free colourings, and linearly-ordered colourings. Firstly, we show that finding an $\ell$-colouring of a $k$-colourable $r$-uniform hypergraph is NP-hard for all constant $2\leq k\leq \ell$ and $r\geq 3$. This provides a shorter proof of a celebrated result by Dinur et al. [FOCS'02/Combinatorica'05]. Secondly, we show that finding an $\ell$-conflict-free colouring of an $r$-uniform hypergraph that admits a $k$-conflict-free colouring is NP-hard for all constant $3\leq k\leq\ell$ and $r\geq 4$, except for $r=4$ and $k=2$ (and any $\ell$); this case is solvable in polynomial time. The case of $r=3$ is the standard nonmonochromatic colouring, and the case of $r=2$ is the notoriously difficult open problem of approximate graph colouring. Thirdly, we show that finding an $\ell$-linearly-ordered colouring of an $r$-uniform hypergraph that admits a $k$-linearly-ordered colouring is NP-hard for all constant $3\leq k\leq\ell$ and $r\geq 4$, thus improving on the results of Nakajima and Živný [ICALP'22/ACM TocT'23].

翻译：利用承诺约束满足问题的代数方法，我们建立了超图着色的三种自然变体的复杂性分类：标准非单色着色、无冲突着色和线性有序着色。首先，我们证明对于所有常数 $2\leq k\leq \ell$ 和 $r\geq 3$，在 $k$-可着色的 $r$-一致超图中寻找 $\ell$-着色是 NP 难的。这为 Dinur 等人 [FOCS'02/Combinatorica'05] 的著名结果提供了一个更短的证明。其次，我们证明对于所有常数 $3\leq k\leq\ell$ 和 $r\geq 4$，在允许 $k$-无冲突着色的 $r$-一致超图中寻找 $\ell$-无冲突着色是 NP 难的，除了 $r=4$ 且 $k=2$（以及任意 $\ell$）的情况；该情况可在多项式时间内求解。$r=3$ 的情况对应标准非单色着色，而 $r=2$ 的情况则是臭名昭著的困难开放问题——近似图着色。第三，我们证明对于所有常数 $3\leq k\leq\ell$ 和 $r\geq 4$，在允许 $k$-线性有序着色的 $r$-一致超图中寻找 $\ell$-线性有序着色是 NP 难的，从而改进了 Nakajima 和 Živný [ICALP'22/ACM TocT'23] 的结果。