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Minimum vertex cover problems on random hypergraphs: replica symmetric solution and a leaf removal algorithm.

Satoshi Takabe1, Koji Hukushima1

  • 1Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|July 15, 2014
PubMed
Summary
This summary is machine-generated.

This study explores minimum vertex-cover problems on random hypergraphs. A phase transition is identified, impacting replica symmetry and approximation algorithm performance.

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Area of Science:

  • Statistical mechanics
  • Theoretical computer science
  • Graph theory

Background:

  • Minimum vertex-cover is a fundamental NP-hard problem.
  • Random hypergraphs offer a complex framework for studying combinatorial problems.
  • Approximation algorithms are crucial for tackling NP-hard problems.

Purpose of the Study:

  • Investigate the minimum vertex-cover problem on random α-uniform hypergraphs.
  • Analyze the behavior of replica methods and leaf removal algorithms.
  • Identify the relationship between replica symmetry and algorithm performance.

Main Methods:

  • Application of the replica method from statistical mechanics.
  • Utilizing a leaf removal algorithm.
  • Numerical examination of core percolation using finite-size scaling.

Main Results:

  • A phase transition occurs at the critical average degree e/(α-1).
  • Below the critical degree, replica symmetry holds, and the leaf removal algorithm is exact.
  • Above the critical degree, replica symmetry breaks down, and the algorithm fails due to core emergence.

Conclusions:

  • The study reveals a strong connection between replica symmetry and the effectiveness of approximation algorithms.
  • The findings provide insights into the computational complexity of vertex-cover on random hypergraphs.
  • Phase transitions significantly influence the solvability of combinatorial optimization problems.