Charting the Tractability Frontier of Certain Conjunctive Query Answering

TL;DR

This paper explores the computational complexity of determining whether a Boolean conjunctive query holds across all repairs of an uncertain database, classifying attack graph cycles to delineate tractable and intractable cases.

Contribution

It introduces a cycle classification in attack graphs and characterizes the complexity boundary of CERTAINTY(q) for acyclic Boolean conjunctive queries without self-join.

Findings

CERTAINTY(q) is coNP-complete with strong cycles in attack graphs.

CERTAINTY(q) is in P when no strong cycle exists and all weak cycles are terminal.

Partial results are provided for cases with nonterminal cycles and no strong cycles.

Abstract

An uncertain database is defined as a relational database in which primary keys need not be satisfied. A repair (or possible world) of such database is obtained by selecting a maximal number of tuples without ever selecting two distinct tuples with the same primary key value. For a Boolean query q, the decision problem CERTAINTY(q) takes as input an uncertain database db and asks whether q is satisfied by every repair of db. Our main focus is on acyclic Boolean conjunctive queries without self-join. Previous work has introduced the notion of (directed) attack graph of such queries, and has proved that CERTAINTY(q) is first-order expressible if and only if the attack graph of q is acyclic. The current paper investigates the boundary between tractability and intractability of CERTAINTY(q). We first classify cycles in attack graphs as either weak or strong, and then prove among others the following. If the attack graph of a query q contains a strong cycle, then CERTAINTY(q) is coNP-complete. If the attack graph of q contains no strong cycle and every weak cycle of it is terminal (i.e., no edge leads from a vertex in the cycle to a vertex outside the cycle), then CERTAINTY(q) is in P. We then partially address the only remaining open case, i.e., when the attack graph contains some nonterminal cycle and no strong cycle. Finally, we establish a relationship between the complexities of CERTAINTY(q) and evaluating q on probabilistic databases.