Two major milestones on the road to the full complexity dichotomy for finite-domain constraint satisfaction problems were Bulatov's proof of the dichotomy for conservative templates, and the structural dichotomy for smooth digraphs of algebraic length 1 due to Barto, Kozik, and Niven. We lift the combined scenario to the infinite, and prove that any smooth digraph of algebraic length 1 pp-constructs, together with pairs of orbits of an oligomorphic subgroup of its automorphism group, every finite structure -- and hence its conservative graph-colouring problem is NP-hard -- unless the digraph has a pseudo-loop, i.e. an edge within an orbit. We thereby overcome, for the first time, previous obstacles to lifting structural results for digraphs in this context from finite to $\omega$-categorical structures; the strongest lifting results hitherto not going beyond a generalisation of the Hell-Ne\v{s}et\v{r}il theorem for undirected graphs. As a consequence, we obtain a new algebraic invariant of arbitrary $\omega$-categorical structures enriched by pairs of orbits which fail to pp-construct some finite structure.

翻译：在有限域约束满足问题的完全复杂性二分法研究道路上，两个重要里程碑分别是Bulatov关于保守模板二分法的证明，以及Barto、Kozik和Niven针对代数长度为1的光滑有向图提出的结构二分法。我们将这一综合框架提升至无限域情形，证明任何代数长度为1的光滑有向图，若与其自同构群的寡态子群轨道对共同进行pp构造，可生成所有有限结构——从而导致其保守图着色问题具有NP难度——除非该有向图存在伪环（即轨道内部的边）。由此我们首次克服了将此类有向图结构结果从有限域提升至ω-范畴结构时面临的障碍；此前最强的提升结果仅止于无向图的Hell-Nešetřil定理的推广形式。作为推论，我们获得了一个新的代数不变量，适用于任意通过轨道对增强的ω-范畴结构，这些结构无法通过pp构造生成某些有限结构。