We study the complexity of evaluating well-designed pattern trees, a query language extending conjunctive queries with the possibility to define parts of the query to be optional. This possibility of optional parts is important for obtaining meaningful results over incomplete data sources as it is common in semantic web settings. Recently, a structural characterization of the classes of well-designed pattern trees that can be evaluated in polynomial time was shown. However, projection—a central feature of many query languages—was not considered in this study. We work towards closing this gap by giving a characterization of all tractable classes of simple well-designed pattern trees with projection (under some common complexity theoretic assumptions). Since well-designed pattern trees correspond to the fragment of well-designed {AND, OPTIONAL}-SPARQL queries this gives a complete description of the tractable classes of queries with projections in this fragment that can be characterized by the underlying graph structures of the queries. For non-simple pattern trees the tractability criteria for simple pattern trees do not capture all tractable classes. We thus extend the characterization for the non-simple case in order to capture some additional tractable cases.

我们研究了评估精心设计的模式树的复杂性，这种模式树是扩展联合查询的查询语言，并有可能将查询的各个部分定义为可选。可选部分的这种可能性对于在不完整的数据源上获得有意义的结果非常重要，这在语义Web设置中很常见。最近，显示了可以在多项式时间内评估的精心设计的模式树的类别的结构表征。但是，本研究未考虑投影（许多查询语言的主要特征）。我们通过给出所有易于处理的简单精心设计的带有投影的模式树（在某些常见的复杂性理论假设下）的类的表征，来努力弥合这一差距。由于精心设计的模式树对应于精心设计的{AND，可选的} -SPARQL查询可完整描述该片段中具有投影的易处理查询类别，这些类别可以由查询的基础图结构来表征。对于非简单模式树，简单模式树的易处理性准则未涵盖所有易处理的类。因此，我们扩展了非简单案例的表征，以便捕获一些其他易于处理的案例。