Approximation algorithms for querying incomplete databases

ACM Transactions on Database Systems, 2014

The term naïve evaluation refers to evaluating queries over incomplete databases as if nulls were usual data values, i.e., to using the standard database query evaluation engine. Since the semantics of query answering over incomplete databases is that of certain answers, we would like to know when naïve evaluation computes them: i.e., when certain answers can be found without inventing new specialized algorithms. For relational databases it is well known that unions of conjunctive queries possess this desirable property, and results on preservation of formulae under homomorphisms tell us that within relational calculus, this class cannot be extended under the open-world assumption. Our goal here is twofold. First, we develop a general framework that allows us to determine, for a given semantics of incompleteness, classes of queries for which naïve evaluation computes certain answers. Second, we apply this approach to a variety of semantics, showing that for many classes of queries beyond unions of conjunctive queries, naïve evaluation makes perfect sense under assumptions different from open-world. Our key observations are: (1) naïve evaluation is equivalent to monotonicity of queries with respect to a semantics-induced ordering, and (2) for most reasonable semantics of incompleteness, such monotonicity is captured by preservation under various types of homomorphisms. Using these results we find classes of queries for which naïve evaluation works, e.g., positive first-order formulae for the closed-world semantics. Even more, we introduce a general relation-based framework for defining semantics of incompleteness, show how it can be used to capture many known semantics and to introduce new ones, and describe classes of first-order queries for which naïve evaluation works under such semantics.

SIAM Journal on Computing, 2014

When finding exact answers to a query over a large database is infeasible, it is natural to approximate the query by a more efficient one that comes from a class with good bounds on the complexity of query evaluation. In this paper we study such approximations for conjunctive queries. These queries are of special importance in databases, and we have a very good understanding of the classes that admit fast query evaluation, such as acyclic, or bounded (hyper)treewidth queries. We define approximations of a given query Q as queries from one of those classes that disagree with Q as little as possible. We concentrate on approximations that are guaranteed to return correct answers. We prove that for the above classes of tractable conjunctive queries, approximations always exist, and are at most polynomial in the size of the original query. This follows from general results we establish that relate closure properties of classes of conjunctive queries to the existence of approximations. We also show that in many cases, the size of approximations is bounded by the size of the query they approximate. We establish a number of results showing how combinatorial properties of queries affect properties of their approximations, study bounds on the number of approximations, as well as the complexity of finding and identifying approximations. The technical toolkit of the paper comes from the theory of graph homomorphisms, as we mainly work with tableaux of queries and characterize approximations via preorders based on the existence of homomorphisms. In particular, most of our results can be also interpreted as approximation or complexity results for directed graphs.

1999

Semistructured data occur in situations where information lacks a homogeneous structure and is incomplete. Yet, up to now the incompleteness of information has not been re ected by special features of query languages for semistructured data. Our goal is to investigate the principles of queries that allow for incomplete answers. We do not present, however, a concrete query language.

2018

Consistent query answering is a principled approach for querying inconsistent knowledge bases. It relies on the notion of a repair, that is, a maximal consistent subset of the facts in the knowledge base. One drawback of this approach is that entire facts are deleted to resolve inconsistency, even if they may still contain useful “reliable” information. To overcome this limitation, we propose a new notion of repair allowing values within facts to be updated for restoring consistency. This more finegrained repair primitive allows us to preserve more information in the knowledge base. We also introduce the notion of a universal repair, which is a compact representation of all repairs. Then, we show that consistent query answering in our framework is intractable (coNP-complete). In light of this result, we develop a polynomial time approximation algorithm for computing a sound (but possibly incomplete) set of consistent query answers.

Proceedings of the 2015 ACM SIGMOD International Conference on Management of Data, 2015

In many applications including loosely coupled cloud databases, collaborative editing and network monitoring, data from multiple sources is regularly used for query answering. For reasons such as system failures, insufficient author knowledge or network issues, data may be temporarily unavailable or generally nonexistent. Hence, not all data needed for query answering may be available. In this paper, we propose a natural class of completeness patterns, expressed by selections on database tables, to specify complete parts of database tables. We then show how to adapt the operators of relational algebra so that they manipulate these completeness patterns to compute completeness patterns pertaining to query answers. Our proposed algebra is computationally sound and complete with respect to the information that the patterns provide. We show that stronger completeness patterns can be obtained by considering not only the schema but also the database instance and we extend the algebra to take into account this additional information. We develop novel techniques to efficiently implement the computation of completeness patterns on query answers and demonstrate their scalability on real data.

Semistructured data occur in situations where information lacks a homogeneous structure and is incomplete. Yet, up to now the incompleteness of information has not been re ected by special features of query languages. Our goal is to investigate the principles of queries that allow for incomplete answers. We do not present, however, a concrete query language. Queries over classical structured data models contain a numb e r o f v ariables and constraints on these variables. An answer is a binding of the variables by elements of the database such that the constraints are satis ed. In the present paper, we loosen this concept in so far as we allow also answers that are partial, that is, not all variables in the query are bound by such an answer. Partial answers make it necessary to re ne the model of query evaluation. The rst modi cation relates to the satisfaction of constraints: under some circumstances we consider constraints involving unbound variables as satis ed. Second, in order to prevent a proliferation of answers, we only accept answers that are maximal in the sense that there are no assignments that bind more variables and satisfy the constraints of the query. Our model of query evaluation consists of two phases, a search phase and a lter phase. Semistructured databases are essentially labeled directed graphs. In the search phase, we use a query graph containing variables to match a maximal portion of the database graph. We i n v estigate three di erent semantics for query graphs, which give rise to three variants of matching. For each v ariant, we provide algorithms and complexity results. In the lter phase, the maximal matchings resulting from the search phase are subjected to constraints, which may b e weak or strong. Strong constraints require all their variables to be bound, while weak constraints do not. We describe a polynomial algorithm for evaluating a special type of queries with lter constraints, and assess the complexity o f e v aluating other queries for several kinds of constraints. In the nal part, we i n v estigate the containment problem for queries consisting only of search constraints under the di erent semantics.

Standard databases convey Reiter's closed-world as- sumption that an atom not in the database is false. This assumption is relaxed in locally closed databases that are sound but only partially complete about their do- main. One of the consequences of the weakening of the closed-world assumption is that query answering in locally closed databases is undecidable. In this paper, we develop efficient approximate methods for query an- swering, based on fixpoint computations, and investi- gate conditions that assure the optimality of these meth- ods. Our approach of approximative reasoning may be incorporated in different contexts where incompleteness plays a major role and efficient reasoning is imperative.

Standard databases convey Reiter's closed-world assumption that an atom not in the database is false. This assumption is relaxed in locally complete databases that are sound but only partially complete about their domain. One of the consequences of the weakening of the closed-world assumption is that query answering in locally closed databases is not tractable. In this paper we develop efficient approximate methods for query answering, based on fixpoint computations. We present preliminary results for a broad class of locally closed databases in which this method produces complete answers to queries.

2018

Consistent query answering is a principled approach for querying inconsistent knowledge bases. It relies on the central notion of repair, that is, a maximal consistent subset of the facts in the knowledge base. One drawback of this approach is that entire facts are deleted to resolve inconsistency, even if they may still contain useful "reliable" information. To overcome this limitation, we propose an inconsistency-tolerant semantics for query answering based on a new notion of repair, allowing values within facts to be updated for restoring consistency. This more fine-grained repair primitive allows us to preserve more information in the knowledge base. We also introduce the notion of a universal repair, which is a compact representation of all repairs and can be computed in polynomial time. Then, we show that consistent query answering in our framework is intractable (coNP-complete). In light of this result, we develop a polynomial time approximation algorithm for computing a sound (but possibly incomplete) set of consistent query answers.

Proceedings of the VLDB Endowment, 2014

A database is called uncertain if two or more tuples of the same relation are allowed to agree on their primary key. Intuitively, such tuples act as alternatives for each other. A repair (or possible world) of such uncertain database is obtained by selecting a maximal number of tuples without ever selecting two tuples of the same relation that agree on their primary key. For a Boolean query q , the problem CERTAINTY( q ) takes as input an uncertain database db and asks whether q evaluates to true on every repair of db. In recent years, the complexity of CERTAINTY( q ) has been studied under different restrictions on q . These complexity studies have assumed no restrictions on the uncertain databases that are input to CERTAINTY( q ). In practice, however, it may be known that these input databases are partially consistent, in the sense that they satisfy some dependencies (e.g., functional dependencies). In this article, we introduce the problem CERTAINTY( q ) in the presence of a set...