The First Proven Performance Guarantees for the NonDominated Sorting Genetic Algorithm II (NSGAII) on a Combinatorial Optimization Problem
Abstract
The Nondominated Sorting Genetic AlgorithmII (NSGAII) is one of the most prominent algorithms to solve multiobjective optimization problems. Recently, the first mathematical runtime guarantees have been obtained for this algorithm, however only for synthetic benchmark problems. In this work, we give the first proven performance guarantees for a classic optimization problem, the NPcomplete biobjective minimum spanning tree problem. More specifically, we show that the NSGAII with population size $N \ge 4((n1) w_{\max} + 1)$ computes all extremal points of the Pareto front in an expected number of $O(m^2 n w_{\max} \log(n w_{\max}))$ iterations, where $n$ is the number of vertices, $m$ the number of edges, and $w_{\max}$ is the maximum edge weight in the problem instance. This result confirms, via mathematical means, the good performance of the NSGAII 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. As a side result, we also obtain a new analysis of the performance of the global SEMO algorithm on the biobjective 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 $n w_{\max}$. The main reason for this improvement is our observation that both multiobjective evolutionary algorithms find the different extremal points in parallel rather than sequentially, as assumed in the previous proofs.
 Publication:

arXiv eprints
 Pub Date:
 May 2023
 DOI:
 10.48550/arXiv.2305.13459
 arXiv:
 arXiv:2305.13459
 Bibcode:
 2023arXiv230513459C
 Keywords:

 Computer Science  Artificial Intelligence;
 Computer Science  Data Structures and Algorithms;
 Computer Science  Neural and Evolutionary Computing;
 Mathematics  Optimization and Control
 EPrint:
 Authorgenerated version of a paper appearing in the proceedings of IJCAI 2023, with appendix