Abstract
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.
Highlights
Well-designed pattern trees are a query formalism well-suited to deal with the ever increasing amount of incomplete data
Well-designed pattern trees over SPARQL triple patterns are equivalent to the class of well-designed {AND, OPTIONAL}-SPARQL queries, see Perez et al [21], and were originally introduced as a formalism to more study SPARQL queries
By replacing triple patterns with relational atoms, Well-designed pattern trees (wdPTs) can be seen as an extension of Conjunctive Queries (CQs): a wdPT is a rooted tree where each node represents a conjunction of atoms, and the tree structure represents a nesting of optional matching
Summary
Well-designed pattern trees (wdPTs) are a query formalism well-suited to deal with the ever increasing amount of incomplete data. In addition to the increased complexity of the evaluation problem, this has the effect that the classes of wdPTs with projection for which query evaluation is in PTIME and in FPT no longer coincide, as observed by Kroll et al [16] In this setting, the choice of the tractability notion makes a difference when describing all tractable classes. To extend the applicability of our techniques in the case of non-simple queries, we generalize our criteria by incorporating the notion of cores into well-designed pattern trees as it was done in the projection free case by Romero [24] and for conjunctive queries by Dalmau et al [7] While this allows us to show tractability for more classes of wdPTs, we do not achieve a full dichotomy in this setting. Several minor improvements have been made throughout the article, including the extension of existing and addition of new examples
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