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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. [ABSTRACT FROM AUTHOR]

We investigate the complexity of three optimization problems in Boolean propositional logic related to information theory: Given a conjunctive formula over a set of relations, find a satisfying assignment with minimal Hamming distance to a given assignment that satisfies the formula (NearestOtherSolution, NOSol) or that does not need to satisfy it (NearestSolution, NSol). The third problem asks for two satisfying assignments with a minimal Hamming distance among all such assignments (MinSolutionDistance, MSD). For all three problems we give complete classifications with respect to the relations admitted in the formula. We give polynomial time algorithms for several classes of constraint languages. For all other cases we prove hardness or completeness regarding APX, poly-APX, or equivalence to well-known hard optimization problems. [ABSTRACT FROM AUTHOR]