Uniform Reliability of Self-Join-Free Conjunctive Queries

TL;DR

This paper explores the computational complexity of counting fact sets satisfying self-join-free conjunctive queries in probabilistic databases, establishing a dichotomy that links query hierarchy to tractability.

Contribution

It generalizes the known reliability complexity dichotomy to the uniform case, proving hierarchical structure is necessary for polynomial-time solvability.

Findings

Uniform reliability is polynomial-time computable for hierarchical queries.

Non-hierarchical queries are #P-hard for uniform reliability.

The hierarchical condition is necessary for tractability under certain assumptions.

Abstract

The reliability of a Boolean Conjunctive Query (CQ) over a tuple-independent probabilistic database is the probability that the CQ is satisfied when the tuples of the database are sampled one by one, independently, with their associated probability. For queries without self-joins (repeated relation symbols), the data complexity of this problem is fully characterized by a known dichotomy: reliability can be computed in polynomial time for hierarchical queries, and is #P-hard for non-hierarchical queries. Inspired by this dichotomy, we investigate a fundamental counting problem for CQs without self-joins: how many sets of facts from the input database satisfy the query? This is equivalent to the uniform case of the query reliability problem, where the probability of every tuple is required to be 1/2. Of course, for hierarchical queries, uniform reliability is solvable in polynomial time, like the reliability problem. We show that being hierarchical is also necessary for this tractability (under conventional complexity assumptions). In fact, we establish a generalization of the dichotomy that covers every restricted case of reliability in which the probabilities of tuples are determined by their relation.