[Paper Review] Answer Counting under Guarded TGDs

Cristina Feier, Carsten Lutz+1 more|arXiv (Cornell University)|Jan 8, 2021

Semantic Web and Ontologies40 references4 citations

TL;DR

This paper provides a comprehensive parameterized complexity classification for counting answers to conjunctive queries and unions of conjunctive queries (UCQs) under guarded tuple-generating dependencies (TGDs), in both ontology-mediated querying and querying under constraints. The key contribution is a precise characterization of when answer counting is fixed-parameter tractable (FPT), W[1]-equivalent, #W[1]-equivalent, #W[2]-hard, or #A[2]-equivalent, based on structural measures like treewidth, contract treewidth, starsize, and linked matching number, under bounded arity and full data schema assumptions.

ABSTRACT

We study the complexity of answer counting for ontology-mediated queries and for querying under constraints, considering conjunctive queries and unions thereof (UCQs) as the query language and guarded TGDs as the ontology and constraint language, respectively. Our main result is a classification according to whether answer counting is fixed-parameter tractable (FPT), W[1]-equivalent, #W[1]-equivalent, #W[2]-hard, or #A[2]-equivalent, lifting a recent classification for UCQs without ontologies and constraints due to Dell et al. The classification pertains to various structural measures, namely treewidth, contract treewidth, starsize, and linked matching number. Our results rest on the assumption that the arity of relation symbols is bounded by a constant and, in the case of ontology-mediated querying, that all symbols from the ontology and query can occur in the data (so-called full data schema). We also study the meta-problems for the mentioned structural measures, that is, to decide whether a given ontology-mediated query or constraint-query specification is equivalent to one for which the structural measure is bounded.

Motivation & Objective

• To classify the parameterized complexity of answer counting for UCQs under guarded TGDs in both ontology-mediated querying and querying under constraints.
• To determine for which classes of queries answer counting is fixed-parameter tractable (FPT), and for which it is harder, based on structural properties of the query.
• To extend prior work by Dell et al. (2019) on UCQs without ontologies to the more complex setting with guarded TGDs.
• To investigate meta-problems: deciding whether a given OMQ or CQS is equivalent to one with bounded structural measures like treewidth or starsize.
• To analyze the impact of bounded arity and full data schema assumptions on the complexity classification.

Proposed method

• The authors use structural measures—treewidth, contract treewidth, starsize, and linked matching number—as parameters to classify the complexity of answer counting.
• They establish reductions and hardness results via homomorphism and chase-based constructions, particularly analyzing the behavior of the chase process on guarded TGDs.
• The proof technique involves constructing homomorphisms from query rewritings to the chase of a database, using properties of guardedness to control variable sharing and path structures.
• They show that if a query has high starsize, then a database instance with high starsize can be constructed that preserves homomorphisms, leading to a contradiction under bounded starsize assumptions.
• The analysis relies on the fact that guarded TGDs ensure that paths in the chase can be traced back to the original database using only original constants, under bounded arity.
• The classification is lifted from the case without ontologies to the guarded TGD setting by exploiting the closure properties of the chase and the guardedness condition.

Experimental results

Research questions

• RQ1 Under what conditions is answer counting for UCQs under guarded TGDs fixed-parameter tractable?
• RQ2 How do structural measures like treewidth and starsize affect the parameterized complexity of answer counting in the presence of guarded TGDs?
• RQ3 Is there a precise classification of answer counting complexity into FPT, W[1]-equivalent, #W[1]-equivalent, #W[2]-hard, or #A[2]-equivalent classes for guarded TGDs?
• RQ4 Can the meta-problem of deciding whether an OMQ or CQS is equivalent to one with bounded treewidth or starsize be solved efficiently?
• RQ5 How do assumptions such as bounded arity and full data schema affect the complexity classification?

Key findings

• Answer counting for UCQs under guarded TGDs is FPT if and only if the query has bounded treewidth, contract treewidth, or starsize, under bounded arity and full data schema.
• If the linked matching number is unbounded, answer counting is #W[2]-hard, indicating a sharp threshold in complexity.
• The problem is #W[1]-equivalent when the linked matching number is bounded but treewidth is unbounded, showing a hierarchy of complexity.
• The classification lifts from the case without ontologies to the guarded TGD setting, preserving the FPT/W[1]/#W[1] boundary.
• The meta-problem of deciding whether an OMQ or CQS is equivalent to one with bounded treewidth is decidable, though complexity depends on the ontology structure.
• The results extend to the OMQ language (ELIH, UCQ) when ontologies are in normal form, and to frontier-guarded TGDs with bounded treewidth bodies.