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Finite-size scaling in random K-satisfiability problems.

Sang Hoon Lee1, Meesoon Ha, Chanil Jeon

  • 1Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 15, 2011
PubMed
Summary
This summary is machine-generated.

This study analyzes phase transitions in random K-satisfiability problems using stochastic-local-search algorithms. We found the density of unsatisfied clauses indicates transitions between solvable and unsolvable states, with finite-size scaling exponents confirmed by simulations.

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Area of Science:

  • Computer Science
  • Statistical Physics
  • Computational Complexity

Background:

  • Random K-satisfiability problems are fundamental in computational complexity.
  • Stochastic-local-search algorithms are widely used for solving constraint satisfaction problems.
  • Understanding phase transitions is crucial for algorithm performance analysis.

Purpose of the Study:

  • To comprehensively analyze phase transitions in random K-satisfiability problems.
  • To investigate the role of the finite-size scaling (FSS) exponent in these transitions.
  • To connect FSS theory of nonequilibrium absorbing phase transitions to K-SAT problem solvability.

Main Methods:

  • Application of finite-size scaling (FSS) theory from nonequilibrium absorbing phase transitions.
  • Analysis of the density of unsatisfied clauses as an indicator of phase transitions.
  • Numerical simulations for K-satisfiability problems with 2 ≤ K ≤ 3.
  • Utilizing a solution clustering (percolation-type) argument.

Main Results:

  • The density of unsatisfied clauses effectively distinguishes between solvable (absorbing) and unsolvable (active) phases.
  • The noise parameter and constraint density were identified as key factors influencing phase transitions.
  • Two conjectured values for the FSS exponent were numerically confirmed for 2 ≤ K ≤ 3.

Conclusions:

  • The FSS theory provides a robust framework for understanding phase transitions in random K-SAT.
  • The density of unsatisfied clauses is a reliable order parameter for K-SAT solvability.
  • The study offers insights into the behavior of stochastic-local-search algorithms for K-SAT problems.