Abstract
A local optima network (LON) encodes local optima connectivity in the fitness landscape of a combinatorial optimisation problem. Recently, LONs have been studied for their fractal dimension. Fractal dimension is a complexity index where a non-integer dimension can be assigned to a pattern. This paper investigates the fractal nature of LONs and how that nature relates to metaheuristic performance on the underlying problem. We use visual analysis, correlation analysis, and machine learning techniques to demonstrate that relationships exist and that fractal features of LONs can contribute to explaining and predicting algorithm performance. The results show that the extent of multifractality and high fractal dimensions in the LON can contribute in this way when placed in regression models with other predictors. Features are also individually correlated with search performance, and visual analysis of LONs shows insight into this relationship.
Highlights
Fractals are patterns which contain parts resembling the whole (Mandelbrot 1972)
Each box contains values for local optima network (LON) associated with a particular quadratic assignment problem library (QAPLIB) instance class—those are indicated on the x-axis labels
Provided in the Figures as accompanying text for each box is the performance of iterated local search on the quadratic assignment problem (QAP) instances associated with those LONs; this is the performance metric p(ILS)
Summary
Fractals are patterns which contain parts resembling the whole (Mandelbrot 1972). Fitness landscapes of some combinatorial optimisation problems have been viewed under a fractal lens (Weinberger and Stadler 1993). Fitness landscapes are both a lucid metaphor and a mathematical object; they contain the set of solutions to an optimisation problem, the fitnesses of those solutions (these can be visualised as the heights), and a function for measuring adjacency between solutions. The study of fitness landscape architecture provides insight about reactions between metaheuristic algorithms and problems. This can serve as a springboard for more informed algorithm design or selection
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