Evolutionary Algorithms for Cardinality-Constrained Ising Models

The Ising model is a famous model of ferromagnetism, in which atoms can have one of two spins and atoms that are neighboured prefer to have the same spin. Ising models have been studied in evolutionary computation due to their inherent symmetry that poses a challenge for evolutionary algorithms. Here we study the performance of evolutionary algorithms on a variant of the Ising model in which the number of atoms with a specific spin is fixed. These cardinality constraints are motivated by problems in materials science in which the Ising model represents chemical species of the atom and the frequency of spins is constrained by the chemical composition of the alloy being modelled. Under cardinality constraints, mutating spins independently becomes infeasible, thus we design and analyse different mutation operators of increasing complexity that swap different atoms to maintain feasibility. We prove that randomised local search with a naive swap operator finds an optimal configuration in Theta(n(4)) expected worst case time. This time is drastically reduced by using more sophisticated operators such as identifying and swapping clusters of atoms with the same spin. We show that the most effective operator only requires O(n) iterations to find an optimal configuration.