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Multi-objective evolutionary algorithms (MOEAs) are among the most widely and successfully applied optimizers for multi-objective problems. However, to store many optimal trade-offs (the Pareto optima) simultaneously, MOEAs are typically run with a large population of solution candidates. This slows down the algorithm and renders the choice of the population size a crucial design decision. In this work, we aim to overcome these difficulties by proposing the dynamic NSGA-II, a variant of the well-known NSGA-II that starts with a small initial population and doubles it after a user-specified number ? of function evaluations, up to a maximum size of ?ₘₐₓ. We prove that the dynamic NSGA-II with optimal parameters computes the Pareto front of the OneMinMax benchmark of size ? with high probability in O(? log² ?) function evaluations, which is considerably faster than the Θ(?² log ?) runtime of the static NSGA-II with optimal parameters. For the OneJumpZeroJump benchmark with gap size ?, we show a runtime of O(?^? log² ?), improving upon the known runtime of Θ(?^(? + 1)). We also propose a variant that uses the initial population size for a longer period and achieves slightly better performance. Finally, we show that a simple concurrent-run strategy turns our dynamic NSGA-II variants into parameter-less algorithms that exceed the above runtimes only by a logarithmic factor and hence still outperform the static NSGA-II by a factor of ͠Ω(?).
Evolutionary Algorithms (EAs) are currently the most popular tool for solving multi-objective optimization problems. Balancing exploration and exploitation is fundamental to the performance of Multi-Objective EAs (MOEAs). Achieving this requires maintaining a set of high-quality solutions for effective exploitation while simultaneously enabling exploration of diverse regions of the search space. In MOEAs, these two roles are typically fulfilled by a single evolutionary population. However, high-quality solutions that are promising for exploitation may lack the potential to guide the search toward different regions of the search space. Conversely, solutions with high exploratory potential that can lead the search toward more promising regions may themselves be of low quality. To address this issue, this paper proposes a dual-population MOEA with Exploration and Exploitation Decoupled (MOEA/EED): the exploration population uses a simple aging mechanism, while the exploitation population preserves the currently optimal solutions. We theoretically prove the benefits of MOEA/EED, i.e., it achieves the upper bounds, O(n^(k+1) · min{1, (e ln(3e) / k)^k}) and O((5e ln(3e))^(n/5)), on the expected running time for solving two commonly studied bi-objective problems OneJumpZeroJump and bi-objective RealRoyalRoad, which are significantly better (even exponentially smaller) than that of the popular SMS-EMOA, NSGA-II, and their non-elitist variants. Lastly, we empirically show its effectiveness on a wide range of practical optimization problems.
In heuristic search, a well-known technique to speed up search while providing a suboptimality guarantee is to multiply the heuristic function by a weight w > 1. In this paper, we study the theoretical and practical implications of using such a technique in bi-objective heuristic search, a natural academic exercise that has remained unexplored. We introduce Weighted BOA∗ϵ (WBOA∗ ϵ ), a weighted version of the BOA* algorithm, which uses two real parameters: a weight w for the heuristic and an approximation factor ϵ. Higher values of w and ϵ allow for faster computation of approximate Pareto-optimal solution sets. We prove that WBOA∗ ϵ returns a representative solution set containing (w − 1, ϵ)-approximate solutions. We empirically compare it to A*pex, the state-of-the-art approximate bi-objective search algorithm. We find that WBOA∗ ϵ is competitive with A*pex: when using perfect heuristic functions, in road maps WBOA∗ϵ is faster for higher approximation factors, while in grid maps WBOA∗ϵ dominates A*pex. With imperfect heuristic functions, WBOA∗ϵ performs better than A*pex for lower approximation factors, while the opposite is true for larger approximation factors.
Treewidth is a fundamental graph invariant that quantifies how tree-like a given graph is. It is extensively used with dynamic programming to design fixed-parameter tractable algorithms for many NP-hard graph combinatorial optimization problems. However, despite broad theoretical applicability, treewidth dynamic programming (TDP) does not scale in practice beyond graphs with very small treewidth. Rather than applying TDP as a standalone technique, we demonstrate that TDP can serve as a broadly applicable enhancer for a wide range of graph combinatorial optimization algorithms. Our framework leverages the concept of treewidth modulators, which refer to vertex sets whose removal significantly reduces the treewidth. We further propose an empirically efficient procedure for generating such treewidth modulators. To enhance an algorithm ?, we use ? to heuristically make decisions on the modulators vertices, after which the remaining decisions outside the treewidth modulators become scalable for TDP. To demonstrate the general applicability of our proposed framework. We experimented with three classic graph combinatorial optimization models: Maximum Independent Set, Minimum Vertex Cover, and Max Cut. We apply TDP to enhance algorithms across diverse paradigms, including evolutionary search, greedy heuristics, and graph-neural-network-based heuristics. For all combinations of optimization models and base algorithms, TDP significantly improves performance over the original methods. In many settings, TDP-enhanced greedy heuristics are competitive with, and sometimes clearly outperform, state-of-the-art commercial solvers.
Problems defined on binary decision spaces have been intensively studied in the theory of multi-objective evolutionary algorithms (MOEAs). In contrast, no mathematical runtime analyses exist so far for MOEAs dealing with decision variables that take a finite number r > 2 of values, despite the prevalence of such problems in practice. In this work, we begin to fill this research gap. We analyze how the classic SEMO algorithm with unit-strength local mutation computes the Pareto front of an r-valued counterpart of the classic ONEMINMAX benchmark. For the expected number of function evaluations until the Pareto front is covered by the population of this MOEA, we prove an upper bound of O(n²r² log n) and a near-tight lower bound of Ω(n²r(r + log n)). We can close the small remaining gap between these two bounds by considering a variant of the algorithm that accepts only strictly better solutions; for this variant, we show an upper bound of O(n²r(r + log n)), matching our lower bound, which also holds for this variant. Our results suggest that classic MOEAs encounter no significant additional difficulties when dealing with multi-valued decision variables. However, significantly more advanced tools may be required to obtain tight bounds for algorithms with more complex population dynamics.
NeuroEvolution of Augmenting Topologies (NEAT) is a widely used neuroevolution algorithm for learning neural network architectures and weights for control tasks. However, standard offline optimisation searches for connection strengths directly, which can scale poorly in high-dimensional weight spaces and more difficult continuous control problems. Hybrid methods that combine neuroevolution with online learning can address this challenge, but their theoretical properties remain underexplored. This paper gives the first regret analysis for a general NeuroEvolutionary Online Learning (NEOL) framework, which decouples learning into two timescales: an outer loop for architecture search and an inner loop for online weight adaptation via reward-modulated plasticity. Under mild conditions, we prove that NEOL achieves sublinear regret. Empirically, under fixed interaction budgets on four standard control benchmarks, a NEAT-based NEOL implementation achieves higher final fitness and lower variance than pure NEAT, and is competitive with strong reinforcement learning (RL) baselines on several tasks. The results are supported by Wilcoxon rank-sum tests and ablation studies. Overall, the findings show that online plasticity can improve the sample efficiency and robustness of two-timescale neuroevolution. Code is available at https://github.com/boobaa2001/NeuroEvolution_Online_Learning_NEOL
Finding an arbitrary number of shortest paths in a graph is a fundamental problem in Graph Theory and Artificial Intelligence with numerous applications to real-world problems. While some algorithms produce paths which might contain loops, the most interesting and challenging problem variant consists of finding simple (or loopless) paths. This problem has been addressed with algorithms which always conduct an arbitrary number of Dijkstra searches. While the current state-of-the-art leverages the notion of sidetrack edge, a new development upon sidetrack edges has been recently introduced which can be used to produce non-simple paths very efficiently, centroids. In this paper, centroids are used to compute an arbitrary number of simple paths with some important benefits: the expansion of a single centroid delivers an arbitrary number of paths; only a single Dijkstra search is required to complete the task; the same algorithm can be easily coupled with heuristics that improve search efficiency. Experimental results across various domains show substantial improvements over the current state of the art, both in runtime and memory usage.
In recent years, theoretical understanding has rapidly advanced regarding how popular multi-objective evolutionary algorithms (MOEAs) can optimize many-objective problems. However, the benefits of using crossover in many-objective optimization are theoretically not understood, except for specifically designed benchmark functions tuned to particular crossover operators, and still lag significantly behind its practical use. In this paper, we build upon this line of research and present a theoretical runtime analysis of the widely used NSGA-III algorithm on the classical m-objective m-OneJumpZeroJump function (m-OJZJ for short). Our results demonstrate that NSGA-III with crossover optimizes m-OJZJ asymptotically faster than NSGA-III without crossover for any number m of objectives for huge parameter regimes. We complement our analysis by providing a lower runtime bound on 4-OJZJ when crossover is turned off.
Parallel tempering, or the replica exchange method, is a Markov Chain Monte Carlo (MCMC) sampling technique for finding low-energy states in complex landscapes by executing multiple processes at different temperatures and allowing for states to migrate between parallel processes based on the Metropolis criterion. Despite a growing interest in the technique as a randomized search heuristic, it is not clear when and why parallel tempering is effective. We conduct a first runtime analysis of the parallel tempering algorithm on a simple quadratic unconstrained binary optimization problem that induces a rugged energy landscape with tunable parameters. We prove that a single-temperature Metropolis process fails to efficiently optimize the problem at any temperature. In sharp contrast, a two-state parallel tempering approach using a low and a high temperature typically solves the problem in O(n^2 log n) iterations. The low-temperature process ensures stability, as it is unlikely to accept worsenings, whereas the high-temperature process facilitates exploration by traversing energy barriers. Improvements are discovered through exploration in the high-temperature process and transferred to the low-temperature process, where they are permanently retained. This effective interplay between processes demonstrates the power of parallel tempering and lends credence to its design.
Multi-objective evolutionary algorithms (MOEAs) are popular tools for multi-objective optimization (MOO), and have been successfully applied to many real-world MOO problems. However, the theoretical study has lagged behind their practical success and remains largely confined to synthetic pseudo-Boolean functions. To close this gap, this paper--drawing inspiration from a class of popular continuous problems with real-world relevance--introduces a multi-objective benchmark defined on an integer space, featuring an analyzable landscape and the presence of local optima. We conduct a running time analysis on the proposed benchmark and derive several theoretical results. Specifically, we prove that a widely-studied MOEA, GSEMO, using unit-step mutation can be trapped in local optimal regions and fail to identify the Pareto front. Fortunately, we find that this difficulty can be overcome either by incorporating an ageing mechanism or using heavy-tailed mutations that allow multi-valued changes along each dimension of an individual. In addition, we demonstrate the extendability of the proposed benchmark to more complex landscapes with numerous local optima, resembling well-established problems in the field (e.g., those from the ZDT and DTLZ suites). We hope this work is a step forward for the theoretical study of MOEAs on problems that are closely related to those commonly investigated in empirical research.
Many real-world applications require solving families of expensive multi-objective optimization problems~(EMOPs) under varying operational conditions. This can be formulated as parametric expensive multi-objective optimization problems (P-EMOPs) where each task parameter defines a distinct optimization instance. Current multi-objective Bayesian optimization methods have been widely used for finding finite sets of Pareto optimal solutions for each task. However, P-EMOPs present a fundamental challenge: the continuous task parameter space can contain infinite distinct problems, each requiring separate expensive evaluations. To address this, we propose learning an inverse model to amortize the multi-objective optimization cost across the continuous task-preference space, enabling direct solution prediction for any query without the need for expensive re-evaluation. This paper introduces a novel parametric multi-objective Bayesian optimizer that learns this inverse model by alternating between (1) generative solution sampling via conditional generative models and (2) acquisition-driven search leveraging inter-task synergies. This approach enables effective optimization across multiple tasks and finally achieves direct solution prediction for unseen parameterized EMOPs without re-evaluations. We theoretically justify the faster convergence by leveraging inter-task synergies through task-aware Gaussian processes. Based on that, empirical studies in synthetic and real-world benchmarks further verify the effectiveness of the proposed parametric optimizer.
Finding the maximum common induced subgraph (MCIS) between two graphs is a well-known NP-hard problem. While sequential MCIS algorithms have been extensively studied, parallel computing has emerged as an important direction for further performance enhancement with the advancement of computing resources. In this paper, we propose a parallel MCIS framework integrating a dynamic task decomposition method guided by search information and a novel pruning strategy based on shared information. The experimental results demonstrate that the algorithm enhanced by our framework achieves superior performance over both state-of-the-art sequential and existing parallel algorithms. Extensive results further demonstrate the wide scalability and generality of our framework, and the effectiveness of our strategies.