# Runtime Analysis of the $(1+(\lambda,\lambda))$ Genetic Algorithm on Random Satisfiable 3-CNF Formulas

## TL;DR

This paper provides a rigorous runtime analysis of the $(1+(mbda,mbda))$ genetic algorithm on random 3-SAT instances, demonstrating its efficiency despite weaker fitness-distance correlation, with insights into population size management.

## Contribution

It extends the theoretical understanding of the $(1+(mbda,mbda))$ GA to more complex problems like 3-SAT, showing its robustness and proposing solutions for population size issues.

## Key findings

- GA achieves runtimes better than (n  log n) on random 3-SAT.
- Self-adjusting GA can be improved with an upper population size limit.
- Performance remains good on sparse instances with proper population control.

## Abstract

The $(1+(\lambda,\lambda))$ genetic algorithm, first proposed at GECCO 2013, showed a surprisingly good performance on so me optimization problems. The theoretical analysis so far was restricted to the OneMax test function, where this GA profited from the perfect fitness-distance correlation. In this work, we conduct a rigorous runtime analysis of this GA on random 3-SAT instances in the planted solution model having at least logarithmic average degree, which are known to have a weaker fitness distance correlation. We prove that this GA with fixed not too large population size again obtains runtimes better than $\Theta(n \log n)$, which is a lower bound for most evolutionary algorithms on pseudo-Boolean problems with unique optimum. However, the self-adjusting version of the GA risks reaching population sizes at which the intermediate selection of the GA, due to the weaker fitness-distance correlation, is not able to distinguish a profitable offspring from others. We show that this problem can be overcome by equipping the self-adjusting GA with an upper limit for the population size. Apart from sparse instances, this limit can be chosen in a way that the asymptotic performance does not worsen compared to the idealistic OneMax case. Overall, this work shows that the $(1+(\lambda,\lambda))$ GA can provably have a good performance on combinatorial search and optimization problems also in the presence of a weaker fitness-distance correlation.