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Covering Problems and Core Percolations on Hypergraphs.

Bruno Coelho Coutinho1, Ang-Kun Wu2, Hai-Jun Zhou3

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Summary
This summary is machine-generated.

We introduce two new core percolation models for hypergraphs, generalizing graph concepts to analyze covering problems. Real-world hypergraphs show smaller cores than expected, suggesting efficient solutions for covering problems.

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Area of Science:

  • Graph Theory
  • Network Science
  • Combinatorics

Background:

  • Percolation theory is crucial for understanding connectivity in complex systems.
  • Graph theory concepts like core percolation are well-established.
  • Hypergraphs offer a more general framework for representing complex relationships.

Purpose of the Study:

  • To generalize core percolation from graphs to hypergraphs.
  • To analyze the minimum hyperedge cover and minimum vertex cover problems in hypergraphs.
  • To investigate the computational complexity of these covering problems.

Main Methods:

  • Developed two generalizations of core percolation for hypergraphs.
  • Derived analytical solutions for uncorrelated random hypergraphs with arbitrary distributions.
  • Analyzed vertex degree and hyperedge cardinality distributions.

Main Results:

  • The two generalized core percolation models were successfully introduced.
  • Analytical solutions were obtained for random hypergraphs.
  • Real-world hypergraphs exhibited significantly smaller cores compared to null models.

Conclusions:

  • Core percolation can be effectively generalized to hypergraphs.
  • Covering problems in many real-world hypergraphs are computationally tractable.
  • Findings suggest potential for polynomial-time solutions to covering problems.