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Parameterised Complexity of Consistent Query Answering via Graph Representations

Teemu Hankala, Miika Hannula, Yasir Mahmood, Arne Meier

TL;DR

This work investigates the parameterised complexity of consistent query answering (CQA) through two graph-based representations: the solution-conflict hypergraph and Gaifman graphs derived from MSO descriptions. The first framework yields a direct fixed-parameter tractable algorithm for counting repairs that satisfy a monotonic query by performing dynamic programming on a tree decomposition of the solution-conflict hypergraph, enabling data- and sometimes combined- complexity insights for various query and constraint classes. The second framework leverages Courcelle’s theorem on Gaifman graphs to obtain fixed-parameter tractability results in combined complexity for expressive queries under FD and DC constraints, and extends to BU CQ^{\neq} via an MSO encoding whose size is bounded by key parameters. The paper also shows that the two graph representations induce incomparable treewidth measures, highlighting the complementary nature of the approaches. Together, these results advance a parameterised understanding of CQA, offer a potentially practical repair-counting method, and map out future directions for extending to broader constraints and queries as well as practical experimentation with the proposed algorithms.

Abstract

We study consistent query answering via different graph representations. First, we introduce solution-conflict hypergraphs in which nodes represent facts and edges represent either conflicts or query solutions. Considering a monotonic query and a set of antimonotonic constraints, we present an explicit algorithm for counting the number of repairs satisfying the query based on a tree decomposition of the solution-conflict hypergraph. The algorithm not only provides fixed-parameter tractability results for data complexity over expressive query and constraint classes, but also introduces a novel and potentially implementable approach to repair counting. Second, we consider the Gaifman graphs arising from MSO descriptions of consistent query answering. Using a generalization of Courcelle's theorem, we then present fixed-parameter tractability results for combined complexity over expressive query and constraint classes.

Parameterised Complexity of Consistent Query Answering via Graph Representations

TL;DR

This work investigates the parameterised complexity of consistent query answering (CQA) through two graph-based representations: the solution-conflict hypergraph and Gaifman graphs derived from MSO descriptions. The first framework yields a direct fixed-parameter tractable algorithm for counting repairs that satisfy a monotonic query by performing dynamic programming on a tree decomposition of the solution-conflict hypergraph, enabling data- and sometimes combined- complexity insights for various query and constraint classes. The second framework leverages Courcelle’s theorem on Gaifman graphs to obtain fixed-parameter tractability results in combined complexity for expressive queries under FD and DC constraints, and extends to BU CQ^{\neq} via an MSO encoding whose size is bounded by key parameters. The paper also shows that the two graph representations induce incomparable treewidth measures, highlighting the complementary nature of the approaches. Together, these results advance a parameterised understanding of CQA, offer a potentially practical repair-counting method, and map out future directions for extending to broader constraints and queries as well as practical experimentation with the proposed algorithms.

Abstract

We study consistent query answering via different graph representations. First, we introduce solution-conflict hypergraphs in which nodes represent facts and edges represent either conflicts or query solutions. Considering a monotonic query and a set of antimonotonic constraints, we present an explicit algorithm for counting the number of repairs satisfying the query based on a tree decomposition of the solution-conflict hypergraph. The algorithm not only provides fixed-parameter tractability results for data complexity over expressive query and constraint classes, but also introduces a novel and potentially implementable approach to repair counting. Second, we consider the Gaifman graphs arising from MSO descriptions of consistent query answering. Using a generalization of Courcelle's theorem, we then present fixed-parameter tractability results for combined complexity over expressive query and constraint classes.

Paper Structure

This paper contains 20 sections, 26 theorems, 28 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Databases and queries
  4. Parameterised Complexity
  5. Hypergraphs and treewidth
  6. Warm-up: Conflict Graphs
  7. Solution-conflict graphs
  8. Gaifman Graphs
  9. The Case of Functional Dependencies
  10. Generalisation to Denial Constraints
  11. Conclusion
  12. Solution-Conflict Hypergraphs
  13. Base steps
  14. Induction step (\ref{['it:three']})
  15. Induction step (\ref{['it:one']})
  16. ...and 5 more sections

Key Result

proposition 1

Consider a set of PKs $\Sigma= \{R_1\colon \{1\} , R_2 \colon \{1\} \}$ and a $\mathrm{sjfBCQ}$$q=\exists x y z (R_1(x,y)\land R_2(z,y))$. Then $\mathsf{CERTAINTY}{(\Sigma, q)}$ is ${\normalfont\textbf{co}}{\normalfont\textbf{NP}}\xspace$-hard.

Figures (1)

  • Figure 1: Given a subset of bags $S\subseteq E$ and sets $r \subseteq s \subseteq d \subseteq \mathrm{usub}(S)$, $\mathrm{eset}(r,s,d,S)$ consists of all $r'\subseteq \mathrm{usub}(S)\setminus d$ such that $r \cup r'$ form a max-repair of the hatched area $s\cup (\mathrm{usub}(S)\setminus d)$, and $e \not\subseteq(r\cup r')\cap b$ for any $b\in S,e\in E_s$.

Theorems & Definitions (50)

  • definition 1
  • definition 2
  • definition 3: Treewidth
  • definition 4: Conflict hypergraph
  • proposition 1: FuxmanM07
  • corollary 1
  • definition 5: Solution-conflict hypergraph
  • theorem 1
  • proof
  • lemma 1
  • ...and 40 more