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Soft and Constrained Hypertree Width

Matthias Lanzinger, Cem Okulmus, Reinhard Pichler, Alexander Selzer, Georg Gottlob

TL;DR

This work introduces soft hypertree width (shw), a tractable relaxation of hypertree width that preserves fixed-$k$ decidability while potentially yielding smaller decompositions. By grounding shw in candidate tree decompositions (CTDs) and dropping the HD-specific

Abstract

Hypertree decompositions provide a way to evaluate Conjunctive Queries (CQs) in polynomial time, where the exponent of this polynomial is determined by the width of the decomposition. In theory, the goal of efficient CQ evaluation therefore has to be a minimisation of the width. However, in practical settings, it turns out that there are also other properties of a decomposition that influence the performance of query evaluation. It is therefore of interest to restrict the computation of decompositions by constraints and to guide this computation by preferences. To this end, we propose a novel framework based on candidate tree decompositions, which allows us to introduce soft hypertree width (shw). This width measure is a relaxation of hypertree width (hw); it is never greater than hw and, in some cases, shw may actually be lower than hw. Most importantly, shw preserves the tractability of deciding if a given CQ is below some fixed bound, while offering more algorithmic flexibility. In particular, it provides a natural way to incorporate preferences and constraints into the computation of decompositions. A prototype implementation and preliminary experiments confirm that this novel framework can indeed have a practical impact on query evaluation.

Soft and Constrained Hypertree Width

TL;DR

This work introduces soft hypertree width (shw), a tractable relaxation of hypertree width that preserves fixed- decidability while potentially yielding smaller decompositions. By grounding shw in candidate tree decompositions (CTDs) and dropping the HD-specific

Abstract

Hypertree decompositions provide a way to evaluate Conjunctive Queries (CQs) in polynomial time, where the exponent of this polynomial is determined by the width of the decomposition. In theory, the goal of efficient CQ evaluation therefore has to be a minimisation of the width. However, in practical settings, it turns out that there are also other properties of a decomposition that influence the performance of query evaluation. It is therefore of interest to restrict the computation of decompositions by constraints and to guide this computation by preferences. To this end, we propose a novel framework based on candidate tree decompositions, which allows us to introduce soft hypertree width (shw). This width measure is a relaxation of hypertree width (hw); it is never greater than hw and, in some cases, shw may actually be lower than hw. Most importantly, shw preserves the tractability of deciding if a given CQ is below some fixed bound, while offering more algorithmic flexibility. In particular, it provides a natural way to incorporate preferences and constraints into the computation of decompositions. A prototype implementation and preliminary experiments confirm that this novel framework can indeed have a practical impact on query evaluation.

Paper Structure

This paper contains 30 sections, 12 theorems, 17 equations, 16 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Structure
  4. Preliminaries
  5. Revisiting Candidate Tree Decompositions
  6. Soft Hypertree Width
  7. Even Softer Hypertree Width
  8. Constrained Hypertree Decompositions
  9. Tractable Constrained Decompositions
  10. Constraints in Other Approaches for Computing Decompositions
  11. Experiments
  12. Results
  13. Conclusion
  14. Further Details on Section \ref{['sec:softhw']}
  15. Institutional Robber and Marshals Games
  16. ...and 15 more sections

Key Result

Theorem 1

Let $k\geq 1$. Deciding, for given hypergraph $H$, whether $\mathit{shw}\xspace(H) \leq k$ holds, is feasible in polynomial time in the size of $H$. The problem even lies in the highly parallelisable class LogCFL.

Figures (16)

  • Figure 1: (a) A hypergraph $H_2$ and (b) its soft hypertree decomposition.
  • Figure 2: Illustrations for \ref{['ex:softer']}
  • Figure 3: Performance over a $\mathsf{ConCov}\xspace$-$\mathit{shw}\xspace$ 2 TPC-DS query using PostgreSQL as a backend.
  • Figure 4: Performance over two $\mathsf{ConCov}\xspace$-$\mathit{shw}\xspace$ 2 Hetionet queries using PostgreSQL as a backend.
  • Figure 5: Illustration of why monotone strategies for the IRMG can be as powerful as monotone strategies for the Robber and Marshals Game.
  • ...and 11 more figures

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Definition 3: The set $\mathsf{Soft}\xspace_{H,k}\xspace$
  • Definition 4: Soft Hypertree Width
  • Theorem 1
  • Theorem 2
  • Example 1
  • Definition 5
  • Definition 6
  • Lemma 3
  • ...and 14 more