For $n\geq 3$, let $(H_n, E)$ denote the $n$-th Henson graph, i.e., the unique countable homogeneous graph with exactly those finite graphs as induced subgraphs that do not embed the complete graph on $n$ vertices. We show that for all structures $\Gamma$ with domain $H_n$ whose relations are first-order definable in $(H_n,E)$ the constraint satisfaction problem for $\Gamma$ is either in P or is NP-complete. We moreover show a similar complexity dichotomy for all structures whose relations are first-order definable in a homogeneous graph whose reflexive closure is an equivalence relation. Together with earlier results, in particular for the random graph, this completes the complexity classification of constraint satisfaction problems of structures first-order definable in countably infinite homogeneous graphs: all such problems are either in P or NP-complete.
翻译:对于 $\ geq 3 美元, 请用$( h_ n, E) 表示 $( gamma ) 的制约满意度问题, 要么在 P 中, 要么在 NP 中完成 。 此外, 我们展示了相似的复杂度二分法, 所有结构, 其关系在一阶中可以确定, 在一阶中, 直径关闭是等同关系 。 与先前的结果, 特别是随机图一样, 这完成了结构约束满意度问题的复杂性分类, 其第一阶是可量化的无限的单一图: 所有这些问题都是在 P 或 NP 中完成 。