Towards a Rigorous Understanding of the Population Dynamics of the NSGA-III: Tight Runtime Bounds

Evolutionary algorithms are widely used for solving multiobjective optimization problems. A prominent example is NSGA-III, which is particularly well suited for solving problems involving more than three objectives, distinguishing it from the classical NSGA-II. Despite its empirical success, the theoretical understanding of NSGA III remains very limited, especially with respect to runtime analysis. A central open problem concerns its population dynamics, which involve controlling the maximum number of individuals sharing the same fitness value during the exploration process. In this paper, we make a significant step towards such an understanding by proving tight runtime bounds for NSGA-III on the bi-objective OneMinMax (2-OMM) problem. Firstly, we prove that NSGA-III requires Ω(n 2 log(n)/µ) generations in expectation to optimize 2-OMM assuming the population size where n denotes the problem size and c < 1 is a constant. Apart from (Opris 2025a), this is the first proven lower runtime bound for NSGA-III on a classical benchmark problem. Complementing this, we secondly improve the best known upper bound of NSGA-III on the m-objective One-MinMax problem (m-OMM) of O(n log(n)) generations by a factor of µ/(2n/m + 1) m/2 for a constant number m of objectives and population size (2n/m + 1) m/2 ≤ µ ∈ O( log(n)(2n/m + 1) m/2 ). This yields tight runtime bounds in the case m = 2, and the surprising result that NSGA-III beats NSGA-II by a factor of µ/n in the expected runtime.