Better approximation guarantees for the NSGA-II by using the current crowding distance

A recent runtime analysis (Zheng, Liu, Doerr (2022)) has shown that a variant of the NSGA-II algorithm can efficiently compute the full Pareto front of the OneMinMax problem when the population size is by a constant factor larger than the Pareto front, but that this is not possible when the population size is only equal to the Pareto front size. In this work, we analyze how well the NSGA-II with small population size approximates the Pareto front of One-MinMax. We observe experimentally and by mathematical means that already when the population size is half the Pareto front size, relatively large gaps in the Pareto front remain. The reason for this phenomenon is that the NSGA-II in the selection stage computes the crowding distance once and then repeatedly removes individuals with smallest crowding distance without updating the crowding distance after each removal. We propose an eficient way to implement the NSGA-II using the current crowding distance. In our experiments, this algorithm approximates the Pareto front much better than the previous version. We also prove that the gaps in the Pareto front are at most a constant factor larger than the theoretical minimum.