A Dichotomy in the Complexity of Consistent Query Answering for Two Atom Queries With Self-Join

TL;DR

The paper establishes a complete complexity dichotomy for consistent query answering under primary key constraints for fixed two-atom Boolean conjunctive queries with self-joins: the problem is either PTIME or coNP-complete. The authors introduce 2way-determined queries and tripath-based structures, showing coNP-hardness via fork-tripath gadgets and PTIME results via a greedy fixpoint Cert_k(q) or a matching-based approach for triangle-tripath cases. A hybrid algorithm that combines Cert_k(q) with bipartite matching achieves PTIME for the broad 2way-determined class without fork-tripaths, while fork-tripath cases yield coNP-hardness; triangle-tripath cases require the matching-based method, with clique-databases providing clean PTIME scenarios. Overall, the work provides decidable, implementable criteria to classify two-atom queries and delineates precise algorithmic strategies, advancing the understanding of consistent query answering under primary keys.

Abstract

We consider the dichotomy conjecture for consistent query answering under primary key constraints. It states that, for every fixed Boolean conjunctive query q, testing whether q is certain (i.e. whether it evaluates to true over all repairs of a given inconsistent database) is either polynomial time or coNP-complete. This conjecture has been verified for self-join-free and path queries. We show that it also holds for queries with two atoms.

Figures (2)

• Figure 1: Tripath illustrations
• Figure 2: Consider the SAT formula $(\neg s \lor t \lor u) \land (\neg s \lor \neg t \lor u) \land (s \lor \neg t \lor \neg u)$. Each clause has a corresponding block denoted by $C_1, C_2$ and $C_3$ respectively. Each such block has three facts corresponding to the literals of the clause. The figure illustrates the gadget for the variable $s$. Similar construction is also done for the variables $t$ and $u$. Note that because the tripath is solution-nice, if a repair $r$ makes the query false and picks $\neg s$ from $C_1$ and/or $C_2$ then it cannot have $s$ from $C_3$. Conversely if $r$ contains $s$ from $C_3$ then it cannot have $\neg s$ from both $C_1$ and $C_2$. The variable-niceness of the tripath provides the necessary variables to encode the clause and the literals.

Theorems & Definitions (19)

• theorem 1
• proposition 1
• theorem 2
• theorem 3
• lemma 1
• lemma 2
• lemma 3
• proposition 2
• theorem 4
• proposition 3