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Determining the existence of partite, 3-uniform hypergraphs with specific degrees is NP-complete. However, a polynomial algorithm exists for almost-regular degree sequences, and a new sampling method shows promise for hypergraph analysis and statistical testing.

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Area of Science:

  • Combinatorics and Graph Theory
  • Computational Complexity
  • Statistical Modeling

Background:

  • Partite, 3-uniform hypergraphs are defined by specific constraints on their vertex classes and hyperedges.
  • The degree sequence problem investigates the existence of hypergraphs with given degree sequences.
  • Sampling uniformly from hypergraphs with prescribed properties is a challenging computational task.

Purpose of the Study:

  • To analyze the computational complexity of the degree sequence problem for partite, 3-uniform hypergraphs.
  • To develop an efficient algorithm for a specific class of degree sequences.
  • To introduce a novel sampling method for partite, 3-uniform hypergraphs and explore its applications in statistical testing.

Main Methods:

  • Proving NP-completeness for the general degree sequence decision problem.
  • Developing a polynomial-time algorithm for almost-regular degree sequences.
  • Implementing a Parallel Tempering method for sampling hypergraphs based on degree sequence deviation.
  • Applying the method to chi-squared (χ2) testing of contingency tables.

Main Results:

  • The decision problem for partite, 3-uniform hypergraph existence is NP-complete.
  • A polynomial-time algorithm is provided for almost-regular degree sequences.
  • The Parallel Tempering method effectively samples hypergraphs and demonstrates superior sensitivity in χ2 testing compared to the standard method, particularly for small datasets.

Conclusions:

  • The study establishes the computational hardness of a fundamental problem in hypergraph theory.
  • Efficient solutions are provided for specific cases, enhancing practical applicability.
  • The proposed sampling technique offers a powerful new tool for hypergraph analysis and statistical inference, outperforming traditional methods in certain scenarios.