Runtime Analysis of Cartesian Genetic Programming in Evolving Boolean Functions

Cartesian Genetic Programming (CGP) is among the practical and popular forms of Genetic Programming as it uses a graph-based representation of programs. This paper presents a first runtime analysis of CGP in evolving Boolean functions using complete training sets. We prove an asymptotic bound $O(n D^5)$ for the expected number of fitness evaluations of CGP to construct a conjunction of $n$ inputs using at most $D \geq n-1$ binary gates, a minimal function set, and even with a strict survival selection. When the non-strict selection is used, the bound is improved to $O(n D^4)$. Our analysis reveals interesting characteristics of CGP induced search, which have been only observed empirically. In particular, enabling the acceptance of equally good solutions, including those with connected gates non-contributing to fitness, can lead to a speedup, and consequently a better asymptotic time bound. In contrast to conjunctions, we also prove a negative result which shows that CGP requires exponential time to evolve an exclusive disjunction. Experiments evolving conjunctions complement our theoretical findings. The use of incomplete training sets is found to further reduce the average number of fitness evaluations while maintaining a good level of generalisation.

SPEA2$^+$: Improved Density Estimation in SPEA2 with Provable Runtime Guarantees

The Strength Pareto Evolutionary Algorithm 2 (SPEA2) is a popular and prominent evolutionary algorithm for solving multi-objective optimisation problems. Despite its popularity, theoretical analyses of SPEA2 have only appeared recently. Moreover, these analyses focus exclusively on how SPEA2 handles non-dominated solutions and disregard the algorithmic components responsible for handling dominated solutions. We conduct a first runtime analysis of SPEA2 for which these components are analysed. We prove that, unlike other prominent algorithms, including NSGA-II, NSGA-III and SMS-EMOA under the same setting of constant population size and duplicate elimination, SPEA2 is unable to cover the Pareto front of the OneTrapZeroTrap benchmark efficiently. Our results indicate that using k-th nearest-neighbour distance in the fitness assignment provides an insufficient signal to maintain diversity among dominated individuals. To address this issue, we propose an improved variant, SPEA2$^+$, that considers all pairwise distances. The new algorithm achieves the same performance guarantees as the other prominent algorithms on OneTrapZeroTrap, while matching the performance of the original SPEA2 on simpler problems. Experimental results complement our theoretical findings.

Parent Selection Mechanisms in Elitist Crossover-Based Algorithms

Parent selection methods are widely used in evolutionary computation to accelerate the optimization process, yet their theoretical benefits are still poorly understood. In this paper, we address this gap by incorporating different parent selection strategies into the $(μ+1)$ genetic algorithm (GA). We show that, with an appropriately chosen population size and a parent selection strategy that selects a pair of maximally distant parents with probability $Ω(1)$ for crossover, the resulting algorithm solves the Jump$_k$ problem in $O(k4^kn\log(n))$ expected time. This bound is significantly smaller than the best known bound of $O(nμ\log(μ)+n\log(n)+n^{k-1})$ for any $(μ+1)$~GA using no explicit diversity-preserving mechanism and a constant crossover probability. To establish this result, we introduce a novel diversity metric that captures both the maximum distance between pairs of individuals in the population and the number of pairs achieving this distance. The crucial point of our analysis is that it relies on crossover as a mechanism for creating and maintaining diversity throughout the run, rather than using crossover only in the final step to combine already diversified individuals, as it has been done in many previous works. The insights provided by our analysis contribute to a deeper theoretical understanding of the role of crossover in the population dynamics of genetic algorithms.

Towards a Rigorous Understanding of the Population Dynamics of the NSGA-III: Tight Runtime Bounds

Evolutionary algorithms are widely used for solving multi-objective optimization problems. A prominent example is NSGA-III, which is particularly well suited for solving problems involving more than three objectives, distinguishing it from the classical NSGA-II. Despite its empirical success, the theoretical understanding of NSGA III remains very limited, especially with respect to runtime analysis. A central open problem concerns its population dynamics, which involve controlling the maximum number of individuals sharing the same fitness value during the exploration process. In this paper, we make a significant step towards such an understanding by proving tight runtime bounds for NSGA-III on the bi-objective OneMinMax ($2$-OMM) problem. Firstly, we prove that NSGA-III requires $Ω(n^2 \log(n) / μ)$ generations in expectation to optimize $2$-OMM assuming the population size $μ$ satisfies $n+1 \leq μ=O(\log(n)^c(n+1))$ where $n$ denotes the problem size and $c<1$ is a constant. Apart from~\cite{opris2025multimodal}, this is the first proven lower runtime bound for NSGA-III on a classical benchmark problem. Complementing this, we secondly improve the best known upper bound of NSGA-III on the $m$-objective OneMinMax problem ($m$-OMM) of $O(n \log(n))$ generations by a factor of $μ/(2n/m + 1)^{m/2}$ for a constant number $m$ of objectives and population size $(2n/m + 1)^{m/2} \leq μ\in O(\sqrt{\log(n)} (2n/m + 1)^{m/2})$. This yields tight runtime bounds in the case $m = 2$, and the surprising result that NSGA-III beats NSGA-II by a factor of $μ/n$ in the expected runtime.

A First Runtime Analysis of NSGA-III on a Many-Objective Multimodal Problem: Provable Exponential Speedup via Stochastic Population Update

The NSGA-III is a prominent algorithm in evolutionary many-objective optimization. It is well-suited for optimizing functions with more than three objectives, setting it apart from the classic NSGA-II. However, theoretical insights about NSGA-III of when and why it performs well are still in its early development. This paper addresses this point and conducts a rigorous runtime analysis of NSGA-III on the many-objective \textsc{OneJumpZeroJump} benchmark (\textsc{OjZj} for short), providing the first runtime bounds where the number of objectives is constant. We show that NSGA-III finds the Pareto front of \textsc{OjZj} in time $O(n^{k+d/2}+ \mu n \ln(n))$ where $n$ is the problem size, $d$ is the number of objectives, $k$ is the gap size, a problem specific parameter, if its population size $\mu \in 2^{O(n)}$ is at least $(2n/d+1)^{d/2}$. Notably, NSGA-III is faster than NSGA-II by a factor of $\mu/n^{d/2}$ for some $\mu \in \omega(n^{d/2})$. We also show that a stochastic population update, proposed by Bian et al., provably guarantees a speedup of order $\Theta((k/b)^{k-1})$ in the runtime where $b>0$ is a constant. To our knowledge, this is the first rigorous runtime analysis of NSGA-III on \textsc{OjZj}. Proving these bounds requires a much deeper understanding of the population dynamics of NSGA-III than previous papers achieved.

Tight Runtime Guarantees From Understanding the Population Dynamics of the GSEMO Multi-Objective Evolutionary Algorithm

The global simple evolutionary multi-objective optimizer (GSEMO) is a simple, yet often effective multi-objective evolutionary algorithm (MOEA). By only maintaining non-dominated solutions, it has a variable population size that automatically adjusts to the needs of the optimization process. The downside of the dynamic population size is that the population dynamics of this algorithm are harder to understand, resulting, e.g., in the fact that only sporadic tight runtime analyses exist. In this work, we significantly enhance our understanding of the dynamics of the GSEMO, in particular, for the classic CountingOnesCountingZeros (COCZ) benchmark. From this, we prove a lower bound of order $\Omega(n^2 \log n)$, for the first time matching the seminal upper bounds known for over twenty years. We also show that the GSEMO finds any constant fraction of the Pareto front in time $O(n^2)$, improving over the previous estimate of $O(n^2 \log n)$ for the time to find the first Pareto optimum. Our methods extend to other classic benchmarks and yield, e.g., the first $\Omega(n^{k+1})$ lower bound for the OJZJ benchmark in the case that the gap parameter is $k \in \{2,3\}$. We are therefore optimistic that our new methods will be useful in future mathematical analyses of MOEAs.

A Many Objective Problem Where Crossover is Provably Indispensable

This paper addresses theory in evolutionary multiobjective optimisation (EMO) and focuses on the role of crossover operators in many-objective optimisation. The advantages of using crossover are hardly understood and rigorous runtime analyses with crossover are lagging far behind its use in practice, specifically in the case of more than two objectives. We present a many-objective problem class together with a theoretical runtime analysis of the widely used NSGA-III to demonstrate that crossover can yield an exponential speedup on the runtime. In particular, this algorithm can find the Pareto set in expected polynomial time when using crossover while without crossover it requires exponential time to even find a single Pareto-optimal point. To our knowledge, this is the first rigorous runtime analysis in many-objective optimisation demonstrating an exponential performance gap when using crossover for more than two objectives.

Theoretical Analysis of Quality Diversity Algorithms for a Classical Path Planning Problem

Quality diversity (QD) algorithms have shown to provide sets of high quality solutions for challenging problems in robotics, games, and combinatorial optimisation. So far, theoretical foundational explaining their good behaviour in practice lack far behind their practical success. We contribute to the theoretical understanding of these algorithms and study the behaviour of QD algorithms for a classical planning problem seeking several solutions. We study the all-pairs-shortest-paths (APSP) problem which gives a natural formulation of the behavioural space based on all pairs of nodes of the given input graph that can be used by Map-Elites QD algorithms. Our results show that Map-Elites QD algorithms are able to compute a shortest path for each pair of nodes efficiently in parallel. Furthermore, we examine parent selection techniques for crossover that exhibit significant speed ups compared to the standard QD approach.

Illustrating the Efficiency of Popular Evolutionary Multi-Objective Algorithms Using Runtime Analysis

Runtime analysis has recently been applied to popular evolutionary multi-objective (EMO) algorithms like NSGA-II in order to establish a rigorous theoretical foundation. However, most analyses showed that these algorithms have the same performance guarantee as the simple (G)SEMO algorithm. To our knowledge, there are no runtime analyses showing an advantage of a popular EMO algorithm over the simple algorithm for deterministic problems. We propose such a problem and use it to showcase the superiority of popular EMO algorithms over (G)SEMO: OneTrapZeroTrap is a straightforward generalization of the well-known Trap function to two objectives. We prove that, while GSEMO requires at least $n^n$ expected fitness evaluations to optimise OneTrapZeroTrap, popular EMO algorithms NSGA-II, NSGA-III and SMS-EMOA, all enhanced with a mild diversity mechanism of avoiding genotype duplication, only require $O(n \log n)$ expected fitness evaluations. Our analysis reveals the importance of the key components in each of these sophisticated algorithms and contributes to a better understanding of their capabilities.

Runtime Analyses of NSGA-III on Many-Objective Problems

NSGA-II and NSGA-III are two of the most popular evolutionary multi-objective algorithms used in practice. While NSGA-II is used for few objectives such as 2 and 3, NSGA-III is designed to deal with a larger number of objectives. In a recent breakthrough, Wietheger and Doerr (IJCAI 2023) gave the first runtime analysis for NSGA-III on the 3-objective OneMinMax problem, showing that this state-of-the-art algorithm can be analyzed rigorously. We advance this new line of research by presenting the first runtime analyses of NSGA-III on the popular many-objective benchmark problems mLOTZ, mOMM, and mCOCZ, for an arbitrary constant number $m$ of objectives. Our analysis provides ways to set the important parameters of the algorithm: the number of reference points and the population size, so that a good performance can be guaranteed. We show how these parameters should be scaled with the problem dimension, the number of objectives and the fitness range. To our knowledge, these are the first runtime analyses for NSGA-III for more than 3 objectives.