Runtime Analysis of Evolutionary Diversity Optimization on the Multi-objective (LeadingOnes, TrailingZeros) Problem

The diversity optimization is the class of optimization problems, in which we aim at finding a diverse set of good solutions. One of the frequently used approaches to solve such problems is to use evolutionary algorithms which evolve a desired diverse population. This approach is called evolutionary diversity optimization (EDO). In this paper, we analyse EDO on a 3-objective function LOTZ_k, which is a modification of the 2-objective benchmark function (LeadingOnes, TrailingZeros). We prove that the GSEMO computes a set of all Pareto-optimal solutions in O(kn^3) expected iterations. We also analyze the runtime of the GSEMO_D (a modification of the GSEMO for diversity optimization) until it finds a population with the best possible diversity for two different diversity measures, the total imbalance and the sorted imbalances vector. For the first measure we show that the GSEMO_D optimizes it asymptotically faster than it finds a Pareto-optimal population, in O(kn^2log(n)) expected iterations, and for the second measure we show an upper bound of O(k^2n^3log(n)) expected iterations. We complement our theoretical analysis with an empirical study, which shows a very similar behavior for both diversity measures that is close to the theory predictions.