{"type":"ScholarlyArticle","title":"The First Proven Performance Guarantees for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) on a Combinatorial Optimization Problem","authors":[{"name":"Sacha Cerf","slug":"sacha-cerf","given":"Sacha","family":"Cerf","affiliation":"École Polytechnique , Institut Polytechnique de Paris , Palaiseau , France"},{"name":"Benjamin Doerr","slug":"benjamin-doerr","given":"Benjamin","family":"Doerr","affiliation":"Laboratoire d'Informatique (LIX) , CNRS ,"},{"name":"Benjamin Hebras","slug":"benjamin-hebras","given":"Benjamin","family":"Hebras","affiliation":"École Polytechnique , Institut Polytechnique de Paris , Palaiseau , France"},{"name":"Yakob Kahane","slug":"yakob-kahane","given":"Yakob","family":"Kahane","affiliation":"École Polytechnique , Institut Polytechnique de Paris , Palaiseau , France; visiting École Poytechnique , France."},{"name":"Simon Wietheger","slug":"simon-wietheger","given":"Simon","family":"Wietheger","affiliation":"Hasso Plattner Institute , University of Potsdam , Germany"}],"abstract":"The Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is one of the most prominent algorithms to solve multi-objective optimization problems. Recently, the first mathematical runtime guarantees have been obtained for this algorithm, however only for synthetic benchmark problems. \r\n\r\nIn this work, we give the first proven performance guarantees for a classic optimization problem, the NP-complete bi-objective minimum spanning tree problem. More specifically, we show that the NSGA-II with population size N >= 4((n-1) wmax + 1) computes all extremal points of the Pareto front in an expected number of O(m^2 n wmax log(n wmax)) iterations, where n is the number of vertices, m the number of edges, and wmax is the maximum edge weight in the problem instance. This result confirms, via mathematical means, the good performance of the NSGA-II observed empirically. It also shows that mathematical analyses of this algorithm are not only possible for synthetic benchmark problems, but also for more complex combinatorial optimization problems. \r\n  \r\n  As a side result, we also obtain a new analysis of the performance of the  global SEMO algorithm on the bi-objective minimum spanning tree problem, which improves the previous best result by a factor of |F|, the number of extremal points of the Pareto front, a set that can be as large as n wmax. The main reason for this improvement is our observation that both multi-objective evolutionary algorithms find the different extremal points in parallel rather than sequentially, as assumed in the previous proofs.","doi":"10.24963/ijcai.2023/613","url":"https://ijcai.org/proceedings/2023/613","pdf":"https://ijcai.org/proceedings/2023/0613.pdf","markdown":"https://ijcai.org/proceedings/2023/613.md","bibtex":"https://ijcai.org/proceedings/2023/bibtex/613","volume":{"slug":"2023","title":"Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2023","url":"https://ijcai.org/proceedings/2023/","year":2023,"location":"Macao, SAR","dates":"19-25 August 2023","editors":"Edith Elkind","isbn":"978-1-956792-03-4","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2023"},"section":"Search","track":"Main Track","number":613,"pages":{"first":5522,"last":5530},"keywords":["Search: S: Evolutionary computation","Search: S: Heuristic search"],"videos":[],"references":[{"position":1,"raw":"[Antipov and Doerr, 2021] Denis Antipov and Benjamin Doerr. 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