Soft and Constrained Hypertree Width (2412.11669v2)

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.

• The paper introduces Soft Hypertree Width, relaxing traditional hypertree conditions to enable flexible, efficient query evaluation.
• It leverages candidate tree decompositions to maintain polynomial tractability while accommodating constraints and database preferences.
• The framework’s integration of constraints yields significant performance gains in query decomposition, bridging theoretical insights and practical optimization.

Soft and Constrained Hypertree Width: A Framework for Flexible Query Optimization

This paper introduces the concept of Soft Hypertree Width (SHW) and the associated framework for evaluating Conjunctive Queries (CQs) via soft hypertree decompositions. It presents a relaxation of the traditional hypertree width (HW), which maintains computational tractability and offers additional algorithmic flexibility. This flexibility is achieved by dropping the special condition inherent in HW and introducing a framework that accommodates constraints and preferences for decomposition computation. The proposed framework is validated both theoretically and through empirical experiments, suggesting practical implications on query evaluation.

Key Contributions

1. Soft Hypertree Width (SHW): SHW is a novel width measure for hypergraphs that extends the utility of hypertree width by relaxing the special condition and allowing greater flexibility in decomposition computations. This relaxation is seen as particularly useful in query evaluation settings, where the structure of the decomposition significantly impacts performance and efficiency.
2. Tractability with Candidate Tree Decompositions: The paper leverages candidate tree decompositions (CTDs) to systematically derive soft hypertree decompositions. The authors illustrate that the introduction of SHW retains the tractability of HW, with the decision problem remaining within polynomial time for a fixed width. Notably, SHW allows computing decompositions via CTDs efficiently without sacrificing calculability, unlike more complex notions like generalized hypertree width (GHW).
3. Algorithmic Flexibility: Through CTDs, the framework accommodates various constraints and preferences, such as connected covers, shallow cyclicity, and partition clustering, effectively guiding the decompositions towards more optimal structures for specific applications. This flexibility is heralded as a significant advance over traditional approaches, aligning theoretical tractability with practical desiderata.
4. Iterative Soft Hypertree Widths: The authors extend SHW into a hierarchy of measures, denoted as $(shw_i)_{i \geq 0}$, where $shw^0$ is SHW and converges towards GHW. This hierarchy bridges the gap between SHW and GHW, offering potential for refined analysis of CQ tractability.
5. Integrated Constraint Handling: A formal methodology is proposed for integrating constraints and preferences into the SHW computation. The paper shows that these can be seamlessly woven into CTDs, ensuring that properties such as tractability are preserved while also enabling more specialized and application-specific decompositions, an important property for distributed and networked databases.

Theoretical Implications

Theoretically, the introduction of SHW and the broader family of $shw_i$ measures set a new landscape for the paper of hypergraph decompositions. By mapping SHW within the spectrum from HW to GHW through finely-tuned iterations, the work generates a nuanced decomposition landscape. This aspect could spur further research into tailored decomposition strategies and inspire novel CQ tractability results by exploring other hypergraph properties within this framework.

Practical Implications

Practically, the incorporation of constraints and preferences in query plans presents a substantial improvement over purely theoretical measures. Constraints like connected covers directly address common inefficiencies in database systems by preventing suboptimal operations such as Cartesian products. Furthermore, by leveraging efficient algorithms that exploit candidate bags, the framework is poised to handle real-world complexities where purely mathematical elegance meets operational feasibility.

Experimental Validation

Preliminary empirical results underline the effectiveness of this approach, demonstrating tangible performance gains in query evaluation when soft hypertree decompositions are employed. This experimentation reinforces the theoretical insights with practical outcomes, suggesting that the approach has significant potential in real-world database performance optimization.

Future Directions

Several avenues for future research and development arise from this work:

1. Expansion of the Constraints Framework: Further exploration of constraints including those driven by practical database configurations or distributed computation environments can broaden the applicability of the framework.
2. Refinement of Cost Functions: Tailoring cost functions for specific database schemata and workloads could refine the decomposition computation to align even more closely with practical performance metrics.
3. Investigation of SHW Hierarchy: Deeper theoretical analysis of the $shw_i$ hierarchy could unveil further insights into the relationship between different hypertree widths and their impact on query tractability.
4. Optimization of Real-world Database Systems: Integrating these theoretical advances into existing database management systems could lead to significant enhancements in terms of query planning and evaluation strategies.

In summary, the introduction of Soft Hypertree Width and the associated framework presents a promising direction for both theoretical exploration and practical application in the context of efficient query evaluation, offering a blend of computational tractability and real-world adaptability that could transform the landscape of database query optimization.