How Hard is it to Decide if a Fact is Relevant to a Query?

We consider the following fundamental problem: given a database D, Boolean conjunctive query (CQ) q, and fact f in D, decide whether f is relevant to q wrt. D, i.e., does f belong to a minimal subset S of D such that S |= q. Despite being of central importance to query answer explanation, the combined complexity of deciding query relevance has not been studied in detail, leaving open what makes this problem hard, and which restrictions can yield lower complexity. Relevance has already been shown to be harder than query evaluation: namely, $Σ^p_2$-complete for CQs, even over a binary signature. We further observe that NP-hardness applies already to (acyclic) chain CQs. Our work identifies self-joins (multiple atoms with the same relation) as the culprit. Indeed, we prove that if we forbid or bound the occurrence of self-joins, then relevance has the same complexity as query evaluation, namely, NP (without structural restrictions) and LogCFL (for bounded hypertreewidth classes). In the ontology setting, we establish an analogous result for ontology-mediated queries consisting of a CQ and DL-Lite_R ontology, namely that relevance is no harder than query answering provided that we bound the interaction width (which generalizes both self-join width and a recently introduced 'interaction-free' condition). Our results thus pinpoint what makes relevance harder than query evaluation and identify natural classes of queries which admit efficient relevance computation.

Shapley Revisited: Tractable Responsibility Measures for Query Answers

The Shapley value, originating from cooperative game theory, has been employed to define responsibility measures that quantify the contributions of database facts to obtaining a given query answer. For non-numeric queries, this is done by considering a cooperative game whose players are the facts and whose wealth function assigns 1 or 0 to each subset of the database, depending on whether the query answer holds in the given subset. While conceptually simple, this approach suffers from a notable drawback: the problem of computing such Shapley values is #P-hard in data complexity, even for simple conjunctive queries. This motivates us to revisit the question of what constitutes a reasonable responsibility measure and to introduce a new family of responsibility measures -- weighted sums of minimal supports (WSMS) -- which satisfy intuitive properties. Interestingly, while the definition of WSMSs is simple and bears no obvious resemblance to the Shapley value formula, we prove that every WSMS measure can be equivalently seen as the Shapley value of a suitably defined cooperative game. Moreover, WSMS measures enjoy tractable data complexity for a large class of queries, including all unions of conjunctive queries. We further explore the combined complexity of WSMS computation and establish (in)tractability results for various subclasses of conjunctive queries.

Shapley Value Computation in Ontology-Mediated Query Answering

The Shapley value, originally introduced in cooperative game theory for wealth distribution, has found use in KR and databases for the purpose of assigning scores to formulas and database tuples based upon their contribution to obtaining a query result or inconsistency. In the present paper, we explore the use of Shapley values in ontology-mediated query answering (OMQA) and present a detailed complexity analysis of Shapley value computation (SVC) in the OMQA setting. In particular, we establish a PF/#P-hard dichotomy for SVC for ontology-mediated queries (T,q) composed of an ontology T formulated in the description logic ELHI_\bot and a connected constant-free homomorphism-closed query q. We further show that the #P-hardness side of the dichotomy can be strengthened to cover possibly disconnected queries with constants. Our results exploit recently discovered connections between SVC and probabilistic query evaluation and allow us to generalize existing results on probabilistic OMQA.