In this article, we show that the completion problem, i.e. the decision problem whether a partial structure can be completed to a full structure, is NP-complete for many combinatorial structures. While the gadgets for most reductions in literature are found by hand, we present an algorithm to construct gadgets in a fully automated way. Using our framework which is based on SAT, we present the first thorough study of the completion problem on sign mappings with forbidden substructures by classifying thousands of structures for which the completion problem is NP-complete. Our list in particular includes interior triple systems, which were introduced by Knuth towards an axiomatization of planar point configurations. Last but not least, we give an infinite family of structures generalizing interior triple system to higher dimensions for which the completion problem is NP-complete.

翻译：本文证明，对于许多组合结构，完备化问题（即判断部分结构能否扩展为完整结构的决策问题）是NP完全的。虽然文献中大多数归约的构造由人工设计完成，但我们提出一种算法，能以完全自动化的方式构造归约中的构件。基于SAT框架，我们首次对带有禁止子结构的符号映射上的完备化问题进行了系统研究，通过分类数千种结构证明其完备化问题具有NP完全性。特别地，我们的分类清单包含由Knuth为公理化平面点配置而引入的内部三元系统。最后但同样重要的是，我们给出了将内部三元系统推广至更高维度的无限结构族，并证明其完备化问题同样具有NP完全性。