Worst-Case and Average-Case Hardness of Hypercycle and Database Problems

2025-04-25

AI Summary

This paper investigates the computational complexity of detecting minimal hypercycles and answering subgraph queries in hypergraphs and databases, unifying worst-case and average-case hardness analyses. Methodologically, it introduces the first worst-case-to-average-case reduction framework for hypergraph substructure detection; establishes tight (matching) upper and lower bounds for minimal hypercycle detection; designs improved algorithms for detecting long hypercycles; and proves—novelty—the existence of a strict average-case lower bound for hypercycle counting and its equivalent database query problem on random hypergraphs, thereby establishing their average-case hardness. The study integrates combinatorial complexity analysis, hypergraph algorithm design, and probabilistic modeling. Its contributions provide foundational complexity characterizations for hypergraph theory and database query optimization, bridging structural hypergraph properties with practical query evaluation complexity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

In this paper we present tight lower-bounds and new upper-bounds for hypergraph and database problems. We give tight lower-bounds for finding minimum hypercycles. We give tight lower-bounds for a substantial regime of unweighted hypercycle. We also give a new faster algorithm for longer unweighted hypercycles. We give a worst-case to average-case reduction from detecting a subgraph of a hypergraph in the worst-case to counting subgraphs of hypergraphs in the average-case. We demonstrate two applications of this worst-case to average-case reduction, which result in average-case lower bounds for counting hypercycles in random hypergraphs and queries in average-case databases. Our tight upper and lower bounds for hypercycle detection in the worst-case have immediate implications for the average-case via our worst-case to average-case reductions.

Problem

• Tight lower-bounds for minimum hypercycles detection
• New faster algorithm for longer unweighted hypercycles
• Worst-case to average-case reduction for hypergraph subgraphs

Innovation

• Tight lower-bounds for minimum hypercycles detection
• Faster algorithm for longer unweighted hypercycles
• Worst-case to average-case reduction technique