Crossover for cardinality constrained optimization

Tobias Friedrich, Timo Kötzing, Aishwarya Radhakrishnan, Leon Schiller, Martin Schirneck, Georg Tennigkeit, Simon Wietheger

Abstract: In order to understand better how and why crossover can benefit optimization, we consider pseudo-Boolean functions with an upper bound B on the number of 1s allowed in the bit string (cardinality constraint). We consider the natural translation of the OneMax test function, a linear function where B bits have a weight of 1 + ε and the remaining bits have a weight of 1. The literature gives a bound of Θ(n2) for 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 both a 1 and a 0. The experimental literature proposes balanced operators, preserving the number of 1s, 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 abound of Ω(n2), just as classic bit flip mutation. Crossover and a simple island model gives O(n2/log n) (uniform crossover) and [EQUATION] (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 analyze and discuss different balanced crossover operators from the literature.