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Diversity of Answers to Conjunctive Queries

Timo Camillo Merkl, Reinhard Pichler, Sebastian Skritek

TL;DR

The paper studies Diverse-$\mathcal{Q}$: selecting a size-$k$ diversity set of answers to a CQ (and extensions) such that a diversity measure $\delta$ (built from pairwise Hamming distances via a polynomial-time aggregator $f$) reaches at least $d$. It provides a detailed parameterized complexity landscape across query classes: for acyclic CQs (ACQ) with diverse-set size $k$, Diverse-$\mathcal{ACQ}$ is in XP for combined complexity and becomes FPT in data complexity; a W[1]-hardness result holds for ws-monotone measures. Extensions to unions of CQs (UCQ/UACQ) and CQs with negation (CQ$^\neg$) show a mix of tractability and hardness, with Diverse-wsm-UACQ being NP-hard even for $k=2$, while Diverse-sum-ACQ remains FPT in query complexity and efficient in data settings. The work also outlines algorithmic strategies based on Yannakakis-style join-tree DP, and discusses structural width measures (treewidth, hypertree width) and their impact on tractability, as well as future directions like smw-bounded queries, beta-acyclicity, and approximation approaches. Overall, the results map the computational boundaries of producing diverse CQ-answer subsets, informing both exact and heuristic diversification approaches in practice.

Abstract

Enumeration problems aim at outputting, without repetition, the set of solutions to a given problem instance. However, outputting the entire solution set may be prohibitively expensive if it is too big. In this case, outputting a small, sufficiently diverse subset of the solutions would be preferable. This leads to the Diverse-version of the original enumeration problem, where the goal is to achieve a certain level d of diversity by selecting k solutions. In this paper, we look at the Diverse-version of the query answering problem for Conjunctive Queries and extensions thereof. That is, we study the problem if it is possible to achieve a certain level d of diversity by selecting k answers to the given query and, in the positive case, to actually compute such k answers.

Diversity of Answers to Conjunctive Queries

TL;DR

The paper studies Diverse-: selecting a size- diversity set of answers to a CQ (and extensions) such that a diversity measure (built from pairwise Hamming distances via a polynomial-time aggregator ) reaches at least . It provides a detailed parameterized complexity landscape across query classes: for acyclic CQs (ACQ) with diverse-set size , Diverse- is in XP for combined complexity and becomes FPT in data complexity; a W[1]-hardness result holds for ws-monotone measures. Extensions to unions of CQs (UCQ/UACQ) and CQs with negation (CQ) show a mix of tractability and hardness, with Diverse-wsm-UACQ being NP-hard even for , while Diverse-sum-ACQ remains FPT in query complexity and efficient in data settings. The work also outlines algorithmic strategies based on Yannakakis-style join-tree DP, and discusses structural width measures (treewidth, hypertree width) and their impact on tractability, as well as future directions like smw-bounded queries, beta-acyclicity, and approximation approaches. Overall, the results map the computational boundaries of producing diverse CQ-answer subsets, informing both exact and heuristic diversification approaches in practice.

Abstract

Enumeration problems aim at outputting, without repetition, the set of solutions to a given problem instance. However, outputting the entire solution set may be prohibitively expensive if it is too big. In this case, outputting a small, sufficiently diverse subset of the solutions would be preferable. This leads to the Diverse-version of the original enumeration problem, where the goal is to achieve a certain level d of diversity by selecting k solutions. In this paper, we look at the Diverse-version of the query answering problem for Conjunctive Queries and extensions thereof. That is, we study the problem if it is possible to achieve a certain level d of diversity by selecting k answers to the given query and, in the positive case, to actually compute such k answers.
Paper Structure (11 sections, 19 theorems, 49 equations, 1 figure)

This paper contains 11 sections, 19 theorems, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Diversity of Conjunctive Queries
  4. Combined and Query Complexity
  5. Basic Algorithm
  6. W[1]-Hardness
  7. Speeding up the Basic Algorithm
  8. Data Complexity
  9. Diversity of Unions of Conjunctive Queries
  10. Diversity of Conjunctive Queries with Negation
  11. Conclusion and Future Work

Key Result

Theorem 3.2

The Diverse-ACQ problem is in $\mathsf{XP}$in combined complexity when parameterized by the size $k$ of the diversity set. More specifically, for an ACQ $Q(X)$, a database $I$, and integers $k$ and $d$, algo:one decides the Diverse-ACQ problem in time $\mathcal{O}(|R^I|^{2k} \cdot (|X| + 1)^{k(k-1)}

Figures (1)

  • Figure 1: Example Execution of the Basic Algorithm.

Theorems & Definitions (39)

  • Theorem 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • proof : Proof (of \ref{['theorem:acqXP']})
  • ...and 29 more