The Dichotomy of Evaluating Homomorphism-Closed Queries on Probabilistic Graphs

TL;DR

This paper establishes a complete complexity classification for evaluating homomorphism-closed queries on probabilistic graphs, showing a dichotomy between polynomial-time solvability and #P-hardness, extending previous results to infinite unions of conjunctive queries.

Contribution

It proves that all unbounded homomorphism-closed queries on probabilistic graphs are #P-hard, completing the complexity classification for this class of queries.

Findings

Probabilistic query evaluation is #P-hard for all unbounded homomorphism-closed queries.

Bounded homomorphism-closed queries are classified by existing dichotomy results.

The results apply to various query fragments, including Datalog, regular path queries, and ontology-mediated queries.

Abstract

We study the problem of query evaluation on probabilistic graphs, namely, tuple-independent probabilistic databases over signatures of arity two. We focus on the class of queries closed under homomorphisms, or, equivalently, the infinite unions of conjunctive queries. Our main result states that the probabilistic query evaluation problem is #P-hard for all unbounded queries from this class. As bounded queries from this class are equivalent to a union of conjunctive queries, they are already classified by the dichotomy of Dalvi and Suciu (2012). Hence, our result and theirs imply a complete data complexity dichotomy, between polynomial time and #P-hardness, on evaluating homomorphism-closed queries over probabilistic graphs. This dichotomy covers in particular all fragments of infinite unions of conjunctive queries over arity-two signatures, such as negation-free (disjunctive) Datalog, regular path queries, and a large class of ontology-mediated queries. The dichotomy also applies to a restricted case of probabilistic query evaluation called generalized model counting, where fact probabilities must be 0, 0.5, or 1. We show the main result by reducing from the problem of counting the valuations of positive partitioned 2-DNF formulae, or from the source-to-target reliability problem in an undirected graph, depending on properties of minimal models for the query.