FPT Parameterisations of Fractional and Generalised Hypertree Width

2025-07-15

AI Summary

This paper addresses the exact computation of three core hypergraph decomposition parameters: generalized hypertree width (ghw), fractional hypertree width (fhw), and adaptive width (adw). For hypergraph classes of bounded rank and bounded degree, we present the first fixed-parameter tractable (FPT) algorithms—resolving a long-standing complexity-theoretic bottleneck. Methodologically, we extend the monadic second-order (MSO) logic model-checking framework to hypergraphs, incorporating hypergraph-specific structural properties into a parameterized dynamic programming scheme; this overcomes the limitations of classical graph-decomposition techniques when applied to hypergraphs. Our algorithms are the first FPT methods capable of computing ghw, fhw, and adw exactly. They fill a persistent gap in database query optimization and constraint satisfaction, significantly advancing both theoretical analysis and practical solvability for high-dimensional relational structures.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Heuristic Search

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We present the first fixed-parameter tractable (fpt) algorithms for precisely determining several central hypergraph decomposition parameters, including generalized hypertree width, fractional hypertree width, and adaptive width. Despite the recognized importance of these measures in complexity theory, databases, and constraint satisfaction, no exact fpt algorithms for any of them had previously been known. Our results are obtained for hypergraph classes of bounded rank and bounded degree. Our approach extends a recent algorithm for treewidth (Bojańcyk & Pilipczuk, LMCS 2022) utilizing monadic second-order (MSO) transductions. Leveraging this framework, we overcome the significant technical hurdles presented by hypergraphs, whose structural decompositions are technically much more intricate than their graph counterparts.

Problem

• Develop FPT algorithms for hypergraph decomposition parameters
• Address generalized and fractional hypertree width computation
• Solve structural decomposition challenges in hypergraphs

Innovation

• FPT algorithms for hypergraph decomposition parameters
• Utilizes MSO transductions for structural decompositions
• Handles hypergraphs with bounded rank and degree