Efficient enumeration of solutions produced by closure operations

In this paper we address the problem of generating all elements obtained by the saturation of an initial set by some operations. More precisely, we prove that we can generate the closure of a boolean relation (a set of boolean vectors) by polymorphisms with a polynomial delay. Therefore we can compute with polynomial delay the closure of a family of sets by any set of "set operations": union, intersection, symmetric difference, subsets, supersets $\dots$). To do so, we study the $Membership_{\mathcal{F}}$ problem: for a set of operations $\mathcal{F}$, decide whether an element belongs to the closure by $\mathcal{F}$ of a family of elements. In the boolean case, we prove that $Membership_{\mathcal{F}}$ is in P for any set of boolean operations $\mathcal{F}$. When the input vectors are over a domain larger than two elements, we prove that the generic enumeration method fails, since $Membership_{\mathcal{F}}$ is NP-hard for some $\mathcal{F}$. We also study the problem of generating minimal or maximal elements of closures and prove that some of them are related to well known enumeration problems such as the enumeration of the circuits of a matroid or the enumeration of maximal independent sets of a hypergraph. This article improves on previous works of the same authors.Comment: 30 pages, 1 figure. Long version of the article arXiv:1509.05623 of the same name which appeared in STACS 2016. Final version for DMTCS journa

Certainty Closure: Reliable Constraint Reasoning with Incomplete or Erroneous Data

Constraint Programming (CP) has proved an effective paradigm to model and solve difficult combinatorial satisfaction and optimisation problems from disparate domains. Many such problems arising from the commercial world are permeated by data uncertainty. Existing CP approaches that accommodate uncertainty are less suited to uncertainty arising due to incomplete and erroneous data, because they do not build reliable models and solutions guaranteed to address the user's genuine problem as she perceives it. Other fields such as reliable computation offer combinations of models and associated methods to handle these types of uncertain data, but lack an expressive framework characterising the resolution methodology independently of the model. We present a unifying framework that extends the CP formalism in both model and solutions, to tackle ill-defined combinatorial problems with incomplete or erroneous data. The certainty closure framework brings together modelling and solving methodologies from different fields into the CP paradigm to provide reliable and efficient approches for uncertain constraint problems. We demonstrate the applicability of the framework on a case study in network diagnosis. We define resolution forms that give generic templates, and their associated operational semantics, to derive practical solution methods for reliable solutions.Comment: Revised versio

Optimization of Regular Path Queries in Graph Databases

Regular path queries offer a powerful navigational mechanism in graph databases. Recently, there has been renewed interest in such queries in the context of the Semantic Web. The extension of SPARQL in version 1.1 with property paths offers a type of regular path query for RDF graph databases. While eminently useful, such queries are difficult to optimize and evaluate efficiently, however. We design and implement a cost-based optimizer we call Waveguide for SPARQL queries with property paths. Waveguide builds a query planwhich we call a waveplan (WP)which guides the query evaluation. There are numerous choices in the con- struction of a plan, and a number of optimization methods, so the space of plans for a query can be quite large. Execution costs of plans for the same query can vary by orders of magnitude with the best plan often offering excellent performance. A WPs costs can be estimated, which opens the way to cost-based optimization. We demonstrate that Waveguide properly subsumes existing techniques and that the new plans it adds are relevant. We analyze the effective plan space which is enabled by Waveguide and design an efficient enumerator for it. We implement a pro- totype of a Waveguide cost-based optimizer on top of an open-source relational RDF store. Finally, we perform a comprehensive performance study of the state of the art for evaluation of SPARQL property paths and demonstrate the significant performance gains that Waveguide offers