Crossover for Cardinality Constrained Optimization

To understand better how and why crossover can benefit constrained optimization, we consider pseudo-Boolean functions with an upper bound B on the number of 1-bits allowed in the length-n bit string (i.e., a cardinality constraint). We investigate the natural translation of the OneMax test function to this setting, a linear function where B bits have a weight of 1+ 1/n and the remaining bits have a weight of 1. Friedrich et al. [TCS 2020] gave a bound of Θ (n2) for the expected running time of the (1+1) EA on this function. Part of the difficulty when optimizing this problem lies in having to improve individuals meeting the cardinality constraint by flipping a 1 and a 0 simultaneously. The experimental literature proposes balanced operators, preserving the number of 1-bits, as a remedy. We show that a balanced mutation operator optimizes the problem in O(n log n) if n-B = O(1). However, if n-B = Θ (n), we show a bound of Ω (n2), just as for classic bit mutation. Crossover together with a simple island model gives running times of O(n2 / log n) (uniform crossover) and \(O(n\sqrt {n})\) (3-ary majority vote crossover). For balanced uniform crossover with Hamming-distance maximization for diversity, we show a bound of O(n log n). As an additional contribution, we present an extensive analysis of different balanced crossover operators from the literature.