On the Complexity of Bounded View Propagation for Conjunctive Queries

The view propagation problem is a class of view update problem in relational databases  <xref ref-type="bibr" rid="ref7">[7]</xref> , involving deletion and insertion propagations. Given source database <inline-formula><tex-math notation="LaTeX">$D$ </tex-math><alternatives><inline-graphic xlink:href="cai-ieq1-2758361.gif"/></alternatives></inline-formula>, conjunctive query <inline-formula><tex-math notation="LaTeX">$Q$</tex-math><alternatives> <inline-graphic xlink:href="cai-ieq2-2758361.gif"/></alternatives></inline-formula>, view <inline-formula> <tex-math notation="LaTeX">$V$</tex-math><alternatives><inline-graphic xlink:href="cai-ieq3-2758361.gif"/> </alternatives></inline-formula> generated by query <inline-formula><tex-math notation="LaTeX">$Q(D)$</tex-math> <alternatives><inline-graphic xlink:href="cai-ieq4-2758361.gif"/></alternatives></inline-formula> and a deletion (insertion) on view <inline-formula><tex-math notation="LaTeX">$\Delta V$</tex-math><alternatives> <inline-graphic xlink:href="cai-ieq5-2758361.gif"/></alternatives></inline-formula>, deletion (insertion) propagation is to find a side effect free update <inline-formula><tex-math notation="LaTeX">$\Delta D$</tex-math><alternatives> <inline-graphic xlink:href="cai-ieq6-2758361.gif"/></alternatives></inline-formula> on <inline-formula> <tex-math notation="LaTeX">$D$</tex-math><alternatives><inline-graphic xlink:href="cai-ieq7-2758361.gif"/> </alternatives></inline-formula> such that the deletion (insertion) of <inline-formula><tex-math notation="LaTeX"> $\Delta D$</tex-math><alternatives><inline-graphic xlink:href="cai-ieq8-2758361.gif"/></alternatives></inline-formula> from (into) <inline-formula><tex-math notation="LaTeX">$D$</tex-math><alternatives> <inline-graphic xlink:href="cai-ieq9-2758361.gif"/></alternatives></inline-formula> will delete (insert) the intentional ones <inline-formula><tex-math notation="LaTeX">$\Delta V$</tex-math><alternatives> <inline-graphic xlink:href="cai-ieq10-2758361.gif"/></alternatives></inline-formula> without resulting in the deletion (insertion) of additional tuples from (into) the view. Generally, such a deletion (insertion) is side effect free. The related data management applications include query result explanation, data debugging, and anonymizing datasets, which rely on understanding how interventions in a database affect the output of a query. View propagation is a natural and typical way to define such interventions, which seems to be well-studied. However, in general, the candidate update on a source database is picked up aimlessly in advance, making the updated database to be very distant from the original one no matter whether it is the maximum one. In this paper, we formally define the bounded view propagation problem, where candidate update <inline-formula><tex-math notation="LaTeX">$\Delta D$</tex-math><alternatives> <inline-graphic xlink:href="cai-ieq11-2758361.gif"/></alternatives></inline-formula> is bounded as a subset of <italic> potential</italic> <inline-formula><tex-math notation="LaTeX">$C$</tex-math><alternatives> <inline-graphic xlink:href="cai-ieq12-2758361.gif"/></alternatives></inline-formula> which is a fixed small tuple set of <inline-formula><tex-math notation="LaTeX">$D$</tex-math><alternatives> <inline-graphic xlink:href="cai-ieq13-2758361.gif"/></alternatives></inline-formula>. We study the complexity of this problem for conjunctive queries, and make contributions to the previous results of the problems of side-effect free deletion propagation. Specifically, our bounded view propagation problem decreases computational complexity regardless of conjunctive query structure. We show the fixed <italic>potential</italic> is actually a dichotomy for both deletion and insertion propagations, and figure out the results on combined complexity which is neglected previously. Based on our results, for view propagation, we map out a complete picture of the computational complexity hierarchy for conjunctive queries on both data and combined complexities. Moreover, this bounded version is an update forbidden case of view propagation, and our results can be applied to it.

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