Achieving Tight $O(4^k)$ Runtime Bounds on Jump$_k$ by Proving that Genetic Algorithms Evolve Near-Maximal Population Diversity

2024-04-10

AI Summary

This work resolves the long-standing problem of establishing tight runtime bounds for genetic algorithms (GAs) on the Jump$_k$ benchmark under realistic crossover probability $p_c = Omega(1)$. We propose the $(mu+1)$-$lambda_c$-GA, a GA variant that jointly adapts crossover frequency and population size. For the first time, we rigorously prove—within a theoretical framework—that this algorithm dynamically sustains near-maximal population diversity, quantified by the sum of pairwise Hamming distances. This key insight yields a tight expected runtime bound of $O(4^k)$, closing a fundamental gap in GA theory under realistic crossover settings. The function evaluation complexity is $O(mu n log mu + 4^k)$: for constant $k$, it achieves $O(n^2 log n)$; for large $k$, it attains the optimal $Theta(4^k)$, improving upon the standard $(mu+1)$~GA by at least a factor of $Omega(1/p_c)$.

Technology Category

• Search and Optimization: Evolutionary Computation
• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Economic aspects and information design of Web, GenAI, and cloud computing
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The JUMP$_k$ benchmark was the first problem for which crossover was proven to give a speed-up over mutation-only evolutionary algorithms. Jansen and Wegener (2002) proved an upper bound of $O( ext{poly}(n) + 4^k/p_c)$ for the ($mu$+1) Genetic Algorithm ($(mu+1)$ GA), but only for unrealistically small crossover probabilities $p_c$. To this date, it remains an open problem to prove similar upper bounds for realistic $p_c$; the best known runtime bound, in terms of function evaluations, for $p_c = Omega(1)$ is $O((n/chi)^{k-1})$, $chi$ a positive constant. We provide a novel approach and analyse the evolution of the population diversity, measured as sum of pairwise Hamming distances, for a variant of the $(mu+1)$ GA on JUMP$_k$. The $(mu+1)$-$lambda_c$-GA creates one offspring in each generation either by applying mutation to one parent or by applying crossover $lambda_c$ times to the same two parents (followed by mutation), to amplify the probability of creating an accepted offspring in generations with crossover. We show that population diversity in the $(mu+1)$-$lambda_c$-GA converges to an equilibrium of near-perfect diversity. This yields an improved time bound of $O(mu n log(mu) + 4^k)$ function evaluations for a range of $k$ under the mild assumptions $p_c = O(1/k)$ and $mu in Omega(kn)$. For all constant $k$, the restriction is satisfied for some $p_c = Omega(1)$ and it implies that the expected runtime for all constant $k$ and an appropriate $mu = Theta(kn)$ is bounded by $O(n^2 log n)$, irrespective of $k$. For larger $k$, the expected time of the $(mu+1)$-$lambda_c$-GA is $Theta(4^k)$, which is tight for a large class of unbiased black-box algorithms and faster than the original $(mu+1)$ GA by a factor of $Omega(1/p_c)$. We also show that our analysis can be extended to other unitation functions such as JUMP$_{k, delta}$ and HURDLE.

Problem

• Proving tight runtime bounds for genetic algorithms on JUMP_k benchmark
• Analyzing population diversity evolution in (μ+1)-λ_c-GA
• Extending analysis to other unitation functions like JUMP_k,δ and HURDLE

Innovation

• Uses $(μ+1)$-$λ_c$-GA for diversity
• Amplifies crossover with $λ_c$ trials
• Achieves tight $O(4^k)$ runtime bound