Markov Chain Selection Hyper-heuristic for the Optimisation of Constrained Magic Squares

Abstract

A square matrix of size n × n, containing each of the numbers (1, . . . , n2) in which every row, column and both diagonals has the same total is referred to as a magic square. The problem can be formulated as an optimisation problem where the task is to minimise the deviation from the magic square constraints and is tackled here by using hyper-heuristics. Hyper-heuristics have recently attracted the attention of the artificial intelligence, operations research, engineering and computer science communities where the aim is to design and develop high level strategies as general solvers which are applicable to a range of different problem domains. There are two main types of hyper-heuristics in the literature: methodologies to select and to generate heuristics and both types of approaches search the space of heuristics rather than solutions. In this study, we describe a Markov chain selection hyper-heuristic as an effective solution methodology for optimising constrained magic squares. The empirical results show that the proposed hyper-heuristic is able to outperform the current state-of-the-art method.