fitness landscape analysis

This study investigates how landscape characteristics of multi-objective combinatorial optimization problems influence algorithm performance prediction. We propose a feature extraction framework based on the Compressed Pareto Local Optima Subnetwork (C-PLOS-net) to systematically quantify key terrain properties—such as ruggedness and objective correlation—in rmnk-landscapes, and evaluate PLS, GSEMO, and NSGA-II across diverse landscapes using resolution and hypervolume metrics. Our key contribution is a novel algorithm–landscape co-analysis paradigm: for each algorithm, we identify a tailored set of landscape features whose importance dynamically varies with the number of objectives and landscape complexity. Empirical results reveal intrinsic “algorithm–landscape” matching mechanisms, offering interpretable, landscape-driven guidance for algorithm selection and design in multi-objective optimization.

When is local search both effective and efficient for combinatorial optimization? Focusing on fitness landscapes induced by binary constraint satisfaction problems (CSPs), we observe that even on unimodal landscapes—such as semismooth or fully unimodal pseudo-Boolean functions—classical algorithms like steepest ascent may require superpolynomial time, revealing a fundamental efficiency bottleneck. Method: We introduce *conditional smoothness*, a novel landscape class that captures the broadest unimodal structure under which widely used local search algorithms—including randomized ascent, simulated annealing, and Kernighan–Lin—admit polynomial-time convergence. Our analysis integrates partial order modeling, Boolean function theory, local search complexity theory, and structural reasoning over hypercube graphs. Contribution/Results: We prove that all aforementioned algorithms converge in polynomial time on conditionally smooth landscapes, whereas steepest ascent remains superpolynomial. This work provides the first fine-grained theoretical characterization of the precise boundary for efficient local search applicability.

This paper addresses the lack of systematic characterization of fitness landscapes in Large Language Model (LLM)-assisted Automated Algorithm Search (LAS). We propose a graph-based landscape representation method, covering six algorithmic tasks and six mainstream LLMs. Our empirical analysis—first of its kind—reveals that LAS landscapes exhibit high multimodality and strong task- and model-dependent structural heterogeneity: heuristic design yields dense high-performance clusters, whereas symbolic regression shows sparse, discrete optima. We further quantify how population size modulates the exploration–exploitation trade-off. The study formally establishes landscape ruggedness and intrinsic multimodality, providing both interpretable theoretical foundations and practical hyperparameter tuning guidelines for designing effective LAS strategies.

Coevolutionary algorithms are widely applied in hardware design, game strategy optimization, and vulnerability repair, yet their pathological behaviors—such as gradient vanishing, relative overgeneralization, and mediocre stagnation—lead to unpredictable performance and lack rigorous theoretical guarantees. Method: This work establishes the first rigorous runtime analysis framework for population-based competitive coevolutionary algorithms, focusing on bilinear minimax optimization. It integrates probabilistic modeling, Markov chain analysis, and population dynamics to characterize convergence behavior. Contribution/Results: We precisely identify the phase transition between polynomially solvable and exponentially hard regimes: proving that a class of simple coevolutionary algorithms converges in polynomial expected time under specific conditions, while rigorously demonstrating that, with high probability, exponential time is required in other settings. This work fills a fundamental gap in the theoretical analysis of coevolutionary solvability and provides the first formal criterion for assessing algorithmic reliability.

The theoretical runtime performance of multi-objective evolutionary algorithms (MOEAs) on discrete optimization problems remains poorly understood. Method: We conduct rigorous mathematical runtime analysis, integrate structural characterizations of Pareto fronts, and leverage recent structural lemmas specific to NSGA-III, establishing near-tight upper bounds on expected optimization time for classic MOEAs—including SEMO, global SEMO, SMS-EMOA, and NSGA-III—on four canonical benchmark problems (e.g., OneMinMax). Contribution/Results: Our analysis breaks the previously known quadratic dependence of runtime bounds on Pareto front size, achieving instead a linear dependence. This is the first such result extended to arbitrary numbers of objectives (high-dimensional objective spaces). The derived bounds involve only low-degree polynomial factors in bit-string length and number of objectives. These findings demonstrate that mainstream MOEAs exhibit significantly stronger scalability under multi-objective extensions than previously recognized, thereby establishing new theoretical benchmarks for algorithm design and analysis.

This work addresses the challenge of modeling fitness progress in evolutionary strategies when operating far from the global optimum, where traditional assumptions often fail. To capture the search dynamics in complex optimization problems—such as hyperparameter tuning—the paper introduces a homogenous progress model, positing that the fitness improvement of offspring over parents follows a stationary distribution. The authors innovatively develop an analytical framework tailored to steady-state $(\mu+1)$-ES, establishing for the first time provable upper and lower bounds on its expected rate of progress, thereby overcoming limitations of conventional approaches in handling intergenerational dependencies. Leveraging probabilistic modeling and asymptotic analysis, they rigorously derive a tight bound under the condition $Z \sim \mathcal{N}(-\delta, 1)$ with $\mu \le e^\delta$: $\mathcal{R}_\mu = \frac{\log^{1 + o(1)} \mu}{\mu} \mathcal{R}_1$.

This work addresses the inefficiency of traditional genetic algorithms in solving optimization problems due to their reliance on random mutation and recombination, which lack goal-directedness. The authors formulate the problem through the lens of query complexity and propose objective-guided mutation and recombination operators informed by the optimization target. Leveraging reinforcement learning and formal language theory, they analyze the theoretical properties of these operators. For the first time, the study mathematically characterizes the mechanism of goal-directed genetic operators and demonstrates the necessity of population diversity for certain classes of optimization problems. A general model of genetic algorithms is established, enabling the design of a tight algorithm for a specific problem class, and proving that the synergy among generation, mutation, and recombination is essential for efficient optimization.

This work proposes a robust landscape equivalence relation that simultaneously incorporates invariance under permutation, translation, and rotation, enabling a systematic classification of all pseudo-Boolean optimization functions—including non-injective cases—in dimensions one through three. By combining combinatorial enumeration with exhaustive verification, the study constructs for the first time a complete set of 12,007 invariant landscape classes, substantially fewer than those obtained under permutation invariance alone. The analysis reveals that non-injective functions dominate landscape diversity and elucidates intricate relationships among neutrality, deception, and the performance of hill-climbing algorithms. These findings provide a foundational resource for benchmark design and theoretical investigations in discrete optimization.

This study systematically investigates the consistency and divergence among four state-of-the-art landscape representation methods—ELA, DeepELA, TransOptAS, and DoE2Vec—in characterizing the structural properties of problem instances from the MA-BBOB benchmark suite within black-box optimization. Through unsupervised clustering, coverage stability assessment, cross-representation similarity analysis, and empirical performance evaluation using differential evolution and particle swarm optimization, the work reveals significant discrepancies and complementary strengths among the representations: ELA and TransOptAS yield compact structures, DeepELA exhibits balanced behavior, and DoE2Vec demonstrates strong semantic alignment albeit with fragmentation. The study introduces a multi-perspective landscape analysis framework, demonstrating that no single representation suffices for comprehensive algorithm performance prediction, thereby establishing a new paradigm for landscape-informed algorithm selection.

This work addresses the long-standing lack of theoretical foundations in online algorithm selection (OAS), particularly the absence of non-artificial problem instances demonstrating asymptotic speedups and principled switching strategies. The study introduces OneMax—a natural benchmark problem—and designs a practical switching strategy between the $(1+\lambda)$ EA and the $(1+(\lambda,\lambda))$ GA. By integrating fixed-start and fixed-target analytical perspectives, it reveals complementary strengths of the two algorithms across different optimization phases. Through rigorous probabilistic analysis and runtime complexity theory, the proposed strategy achieves an expected optimization time of $O(n \log \log n)$, significantly improving upon the best-known bound of $\Theta\left(n \sqrt{ \frac{ \log n \log \log \log n}{ \log \log n}}\right)$ for either algorithm used in isolation. This constitutes the first non-artificial theoretical evidence of asymptotic acceleration in OAS.