The approximate graph colouring problem concerns colouring a $k$-colourable graph with $c$ colours, where $c\geq k$. This problem naturally generalises to promise graph homomorphism and further to promise constraint satisfaction problems. Complexity analysis of all these problems is notoriously difficult. In this paper, we introduce two new techniques to analyse the complexity of promise CSPs: one is based on topology and the other on adjunction. We apply these techniques, together with the previously introduced algebraic approach, to obtain new NP-hardness results for a significant class of approximate graph colouring and promise graph homomorphism problems.

中文翻译：

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承诺约束满足中的拓扑和附加

近似图着色问题涉及用 $c$ 颜色为 $k$ 可着色图着色，其中 $c\geq k$。这个问题自然地推广到承诺图同态，并进一步推广到承诺约束满足问题。众所周知，对所有这些问题的复杂性分析非常困难。在本文中，我们介绍了两种新技术来分析 promise CSP 的复杂性：一种基于拓扑，另一种基于附加。我们将这些技术与之前介绍的代数方法一起应用，为一类重要的近似图着色和承诺图同态问题获得新的 NP 硬度结果。