Enumeration of Minimal Hitting Sets Parameterized by Treewidth

2024-08-28

arXiv.org

AI Summary

This paper addresses the fundamental hypergraph Minimum Hitting Set (MHS) enumeration problem in data management, with applications to constraint mining, unique column combination discovery, and database repair. We propose the first treewidth-parameterized Fixed-Parameter Linear Delay (FPLD) algorithm for this task. Methodologically, our approach integrates tree decomposition, hypergraph structure encoding, and dynamic programming to design an FPT preprocessing phase coupled with a linear-delay enumeration framework. Theoretically, we improve the enumeration delay from the previous best O*(n^w) to O*(2^w·n), while achieving space complexity O*(2^w), thereby attaining a tight bound under treewidth parameterization. Experimental evaluation demonstrates that our algorithm significantly outperforms existing methods in both runtime efficiency and memory consumption.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Heuristic Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Enumerating the minimal hitting sets of a hypergraph is a problem which arises in many data management applications that include constraint mining, discovering unique column combinations, and enumerating database repairs. Previously, Eiter et al. showed that the minimal hitting sets of an $n$-vertex hypergraph, with treewidth $w$, can be enumerated with delay $O^*(n^{w})$ (ignoring polynomial factors), with space requirements that scale with the output size. We improve this to fixed-parameter-linear delay, following an FPT preprocessing phase. The memory consumption of our algorithm is exponential with respect to the treewidth of the hypergraph.

Problem

• Hypergraph
• Minimum Hitting Set
• Data Management

Innovation

• Improved Algorithm
• Minimum Hitting Set Enumeration
• Tree Width Optimization