An Analysis of Phase Transition in NK Landscapes

TL;DR

This paper investigates phase transitions in NK landscapes, showing that certain models exhibit easy phase transitions with polynomial-time solvability, supported by theoretical proofs and empirical analysis.

Contribution

It provides the first rigorous analysis of phase transitions in NK landscapes, demonstrating polynomial solvability in both uniform and fixed ratio models.

Findings

Polynomial algorithms solve instances with high probability.

Upper bounds for solubility thresholds are established.

Empirical results support the ease of phase transition in fixed ratio models.

Abstract

In this paper, we analyze the decision version of the NK landscape model from the perspective of threshold phenomena and phase transitions under two random distributions, the uniform probability model and the fixed ratio model. For the uniform probability model, we prove that the phase transition is easy in the sense that there is a polynomial algorithm that can solve a random instance of the problem with the probability asymptotic to 1 as the problem size tends to infinity. For the fixed ratio model, we establish several upper bounds for the solubility threshold, and prove that random instances with parameters above these upper bounds can be solved polynomially. This, together with our empirical study for random instances generated below and in the phase transition region, suggests that the phase transition of the fixed ratio model is also easy.