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

Diversity optimization is the class of optimization problems in which we aim to find a diverse set of good solutions. One of the frequently-used approaches to solve such problems is to use evolutionary algorithms that evolve a desired diverse population. This approach is called evolutionary diversity optimization (EDO). In this paper, we analyze EDO on a three-objective function LOTZ$_k$, which is a modification of the two-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$_D$ algorithm (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 in $O(kn^2\log(n))$ expected iterations (which is asymptotically faster than the upper bound on the runtime until it finds a Pareto-optimal population), and for the second measure we show an upper bound of $O(k^2n^3\log(n))$ expected iterations. We complement our theoretical analysis with an empirical study, which shows a very similar behavior for both diversity measures. The results of experiments suggest that our bounds for the total imbalance measure are tight, while the bounds for the imbalances vector are too pessimistic.