{"ID":2843980,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.07125","arxiv_id":"2511.07125","title":"Towards a Rigorous Understanding of the Population Dynamics of the NSGA-III: Tight Runtime Bounds","abstract":"Evolutionary algorithms are widely used for solving multi-objective 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 $μ$ satisfies $n+1 \\leq μ=O(\\log(n)^c(n+1))$ where $n$ denotes the problem size and $c\u003c1$ is a constant. Apart from~\\cite{opris2025multimodal}, 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 OneMinMax 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 $(2n/m + 1)^{m/2} \\leq μ\\in O(\\sqrt{\\log(n)} (2n/m + 1)^{m/2})$. 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.","short_abstract":"Evolutionary algorithms are widely used for solving multi-objective 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 understand...","url_abs":"https://arxiv.org/abs/2511.07125","url_pdf":"https://arxiv.org/pdf/2511.07125v1","authors":"[\"Andre Opris\"]","published":"2025-11-10T14:11:07Z","proceeding":"cs.NE","tasks":"[\"cs.NE\"]","methods":"[\"LoRA\"]","has_code":false}