Relaxation in graph coloring and satisfiability problems

TL;DR

This paper investigates the relaxation dynamics of graph coloring and satisfiability problems using Monte Carlo simulations, revealing phase transitions in relaxation behavior that occur before the problems become unsolvable.

Contribution

It provides new insights into the relaxation processes and phase transitions in graph coloring and satisfiability problems through simulation analysis.

Findings

Transition from exponential to power-law relaxation

Identification of freezing behavior in relaxation dynamics

Relaxation transitions occur before solvability thresholds

Abstract

Using T=0 Monte Carlo simulation, we study the relaxation of graph coloring (K-COL) and satisfiability (K-SAT), two hard problems that have recently been shown to possess a phase transition in solvability as a parameter is varied. A change from exponentially fast to power law relaxation, and a transition to freezing behavior are found. These changes take place for smaller values of the parameter than the solvability transition. Results for the coloring problem for colorable and clustered graphs and for the fraction of persistent spins for satisfiability are also presented.