An Analysis of NK Landscapes: Interaction Structure, Statistical Properties, and Expected Number of Local Optima

Abstract

Simulated landscapes have been used for decades to evaluate search strategies whose goal is to find the landscape location with maximum fitness. Understanding properties of landscapes is important for understanding search difficulty. This paper presents a novel and transparent characterization of $N\!K$ landscapes and derives an analytic expression representing the expected number of local optima. We prove that $N\!K$ landscapes can be represented by parametric linear interaction models where model coefficients have meaningful interpretations. We derive the statistical properties of the model coefficients, providing insight into how the $N\!K$ algorithm parses importance to main effects and interactions. An important insight derived from the linear model representation is that the rank of the linear model defined by the $N\!K$ algorithm is correlated with the number of local optima, a strong determinant of landscape complexity, and search difficulty. We show that the maximal rank for an $N\!K$ landscape is achieved through epistatic interactions that form partially balanced incomplete block designs. Finally, an analytic expression representing the expected number of local optima on the landscape is derived, providing a way to quickly compute the expected number of local optima for very large landscapes.