Fagin defined the class $NP$ by the means of Existential Second-Order logic. Feder and Vardi expressed it (up to polynomial equivalence) by special fragments of Existential Second-Order logic (SNP), while the authors used forbidden expanded substructures (cf. lifts and shadows). Consequently, for such problems there is no dichotomy, unlike for CSPs. We prove that ordering problems for graphs defined by finitely many forbidden ordered subgraphs capture the full power of the class $NP$, that is, any language in the class $NP$ is polynomially equivalent to an ordering problem. In particular, we refute a conjecture of Hell, Mohar and Rafiey that dichotomy holds for this class. On the positive side, we confirm the conjecture of Duffus, Ginn and Rödl that ordering problems defined by a single obstruction which is a biconnected ordered graph are $NP$-complete if the graph is not complete. We initiate the study of meta-theorems for classes which have the full power of the class $NP$. For example, homomorphism problems (or CSPs) do not have full power (similarly to coloring problems). On the other hand, we show that problems defined by the existence of an ordering, which avoids certain ordered patterns, have full power. We find it surprising that such simple structures can express the full power of $NP$. A principal tool for obtaining these results is the Sparse Incomparability Lemma in many of its variants, which are classical results in the theory of homomorphisms of graphs and structures. We prove it here in the setting of ordered stuctures as a Temporal Sparse Incomparability Lemma. This is a non-trivial result, even in the random setting, and a deterministic algorithm requires more effort. Interestingly, our proof involves the Lovász Local Lemma.

翻译：Fagin 通过存在二阶逻辑定义了 $NP$ 类。Feder 和 Vardi 使用存在二阶逻辑的特殊片段（SNP）表达了它（直至多项式等价），而作者们则使用了禁止的扩展子结构（参见提升与阴影）。因此，与约束满足问题（CSPs）不同，此类问题不存在二分性。我们证明了，由有限多个禁止有序子图定义的图的有序化问题能够捕获 $NP$ 类的全部能力，即 $NP$ 类中的任何语言都与某个有序化问题多项式等价。特别地，我们反驳了 Hell、Mohar 和 Rafiey 关于此类问题存在二分性的猜想。从积极的一面看，我们证实了 Duffus、Ginn 和 Rödl 的猜想：由单个双连通有序图作为障碍物定义的有序化问题，若该图不是完全图，则是 $NP$ 完全的。我们开始研究具有 $NP$ 类全部能力的各类问题的元定理。例如，同态问题（或 CSPs）不具备全部能力（类似于着色问题）。另一方面，我们证明了通过存在一个避免特定有序模式的排序来定义的问题具有全部能力。我们发现，如此简单的结构能够表达 $NP$ 的全部能力是令人惊讶的。获得这些结果的一个主要工具是稀疏不可比性引理及其多种变体，这些是图和结构同态理论中的经典结果。我们在此在有序结构的背景下证明了它，称之为时序稀疏不可比性引理。这是一个非平凡的结果，即使在随机设置下也是如此，而确定性算法则需要更多努力。有趣的是，我们的证明涉及 Lovász 局部引理。