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Ground-state entropy of the random vertex-cover problem
1Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China.
Summary
This study estimates the entropy of minimum vertex covers in random graphs using a novel cavity method approach. The technique overcomes iterative divergence issues in spin-glass theory, enabling calculation of ground state entropies.
Area of Science:
- Statistical mechanics
- Combinatorial optimization
- Graph theory
Background:
- Counting ground states in spin-glass and NP-complete problems is computationally challenging.
- Existing methods like zero-temperature first-step replica-symmetry-breaking (1RSB) spin-glass theory face iterative divergence.
- The stability of 1RSB mean-field theory is a limitation for certain systems.
Purpose of the Study:
- To estimate the entropy of minimum vertex covers in random graphs.
- To develop a method that overcomes iterative divergence in spin-glass calculations.
- To provide a framework for computing ground state and metastable state entropies in complex systems.
Main Methods:
- Utilizes iterative equations derived from the cavity method of statistical mechanics.
- Updates cavity entropy contributions and cavity magnetizations for each vertex during iteration.
- Applies to random graphs and extends to other spin-glass systems.
Main Results:
- Successfully estimates the entropy of minimum vertex covers for random graphs.
- Circumvents the iterative divergence problem inherent in 1RSB spin-glass theory.
- Demonstrates applicability even when 1RSB mean-field theory is unstable.
Conclusions:
- The cavity method provides a robust approach for calculating ground state entropy in complex optimization problems.
- This method offers an advancement over traditional spin-glass theories, particularly in overcoming numerical instabilities.
- The technique is extensible to a broader range of random-graph spin-glass systems for analyzing ground and metastable states.
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