Friedrich, Tobias; Kötzing, Timo; Radhakrishnan, Aishwarya; Schiller, Leon; Schirneck, Martin; Tennigkeit, Georg; Wietheger, Simon Crossover for Cardinality Constrained OptimizationACM Transactions on Evolutionary Learning and Optimization 2023: 1–32
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 \(\Theta(n^2)\) 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 \(n-B = \Theta(n)\), we show a bound of \(\Omega(n^2)\), just as for classic bit mutation. Crossover together with a simple island model gives running times of \(O(n^2 / \log n)\) (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.
