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Effect of constraint relaxation on the minimum vertex cover problem in random graphs
Aki Dote1,2, Koji Hukushima1,3
1Graduate School of Arts and Sciences, The University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8902, Japan.
Physical Review. E
|May 17, 2024
Summary
Constraint relaxation in random graphs affects minimum vertex cover solutions. This study uses statistical mechanics to analyze how relaxing constraints impacts solution properties and critical temperatures.
Area of Science:
- Statistical mechanics
- Graph theory
- Computational complexity
Background:
- The minimum vertex cover problem seeks the smallest set of vertices covering all graph edges.
- Erdős-Rényi random graphs are a fundamental model in network science.
- Constraint relaxation is a technique to analyze complex optimization problems.
Purpose of the Study:
- To investigate the impact of constraint relaxation on the minimum vertex cover problem in random graphs.
- To analyze the interplay between covering edges and satisfying constraints.
- To understand how relaxation affects solution properties and phase transitions.
Main Methods:
- Statistical-mechanical analysis using the replica method.
- Application of the cavity method.
- Penalty-method formulation for constraint relaxation.
Main Results:
- Constraint relaxation introduces degeneracies in vertex and edge states, influencing cover and penalty ratios.
- Replica-symmetric (RS) approximation accuracy for minimum cover ratio is improved in the replica symmetry-breaking (RSB) region.
- RS/RSB boundary for ground states expands, and critical temperature decreases with constraint relaxation.
Conclusions:
- Constraint relaxation significantly alters the landscape of the minimum vertex cover problem in random graphs.
- The study provides a more accurate RS approximation in the RSB region.
- Relaxation effectively lowers the critical temperature, expanding the region of complex ground states.
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