We present a structural classification of constraint satisfaction problems (CSP) described by reflexive complete $2$-edge-coloured graphs. In particular, this classification extends the structural dichotomy for graph homomorphism problems known as the Hell--Nešetřil theorem (1990). Our classification is also efficient: we can check in polynomial time whether the CSP of a reflexive complete $2$-edge-coloured graph is in P or NP-complete, whereas for arbitrary $2$-edge-coloured graphs, this task is NP-complete. We then apply our main result in the context of matrix partition problems and sandwich problems. Firstly, we obtain one of the few algorithmic solutions to general classes of matrix partition problems. And secondly, we present a P vs. NP-complete classification of sandwich problems for matrix partitions.

翻译：本文对由自反完全$2$边着色图描述的约束满足问题（CSP）提出了结构分类。特别地，该分类扩展了图同态问题的结构二分定理（即Hell–Nešetřil定理，1990年）。我们的分类同时具有高效性：对于自反完全$2$边着色图，可在多项式时间内判定其CSP属于P类还是NP完全问题；而对于任意$2$边着色图，该判定任务本身是NP完全的。随后，我们将主要结果应用于矩阵划分问题与三明治问题的研究。首先，我们获得了针对广义矩阵划分问题的少数算法解之一；其次，我们提出了矩阵划分三明治问题的P类与NP完全分类。