Runtime Analysis for State-of-the-Art Multi-objective Evolutionary Algorithms on the Subset Selection Problem

Renzhong Deng, Weijie Zheng, Mingfeng Li, Jie Liu, Benjamin Doerr

Abstract: In the last few years, the mathematical runtime analysis of randomized search heuristics has made a huge step forward by developing the methods to analyze the most prominent multi-objective evolutionary algorithms (MOEAs) as opposed to previously only simplistic algorithms. These results confirmed that many previous results extend to state-of-the-art MOEAs, but also showed that algorithms like the NSGA-II can have unexpected difficulties on problems easily solved by simple MOEAs. We continue this line of research by analyzing how the NSGA-II and the SMS-EMOA (also with a recently proposed stochastic population update) solve the NP-hard subset selection problem. For these two state-of-the-art algorithms, we prove performance guarantees that agree with those previously shown for the POSS algorithm, a variant of the simplistic GSEMO, namely that they compute \((1-e^{-\gamma })\)-approximate solutions in expected time \(O(k^2n)\). Our experiments confirm these findings. This work is the first runtime analysis of state-of-the-art MOEAs for the subset selection problem, and also the first runtime analysis of SMS-EMOA on a combinatorial problem.