Consistent query answering is the problem of computing the answers from a database that are consistent with respect to certain integrity constraints that the database as a whole may fail to satisfy. Those answers are characterized as those that are invariant under minimal forms of restoring the consistency of the database. In this context, we study the problem of repairing databases by fixing integer numerical values at the attribute level with respect to denial and aggregation constraints. We introduce a quantitative definition of database fix, and investigate the complexity of several decision and optimization problems, including DFP, i.e. the existence of fixes within a given distance from the original instance, and CQA, i.e. deciding consistency of answers to aggregate conjunctive queries under different semantics. We provide sharp complexity bounds, identify relevant tractable cases; and introduce approximation algorithms for some of those that are intractable. More specifically, we obtain results like undecidability of existence of fixes for aggregation constraints; MAXSNP-hardness of DFP, but a good approximation algorithm for a relevant special case; and intractability but good approximation for CQA for aggregate queries for one database atom denials (plus built-ins).

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Bertossi, L, Bravo, L. (Loreto), Franconi, E. (Enrico), & Lopatenko, A. (Andrei). (2005). Complexity and approximation of fixing numerical attributes in databases under integrity constraints.