Vehicle routing problems (VRPs) are central to combinatorial optimization with significant practical implications. Recent advancements in neural combinatorial optimization (NCO) have demonstrated promising results by leveraging neural networks to solve VRPs, yet the exploration of model scaling within this domain remains underexplored. Inspired by the success of model scaling in large language models (LLMs), this study introduces a Large Routing Model with 1 billion parameters (LRM-1B), designed to address diverse VRP scenarios. We present a comprehensive evaluation of LRM-1B across multiple problem variants, distributions, and sizes, establishing state-of-the-art results. Our findings reveal that LRM-1B not only adapts to different VRP challenges but also showcases superior performance, outperforming existing models. Additionally, we explore the scaling behavior of neural routing models from 1M to 1B parameters. Our analysis confirms power-law between multiple model factors and performance, offering critical insights into the optimal configurations for foundation neural routing solvers.
This systematic review explores the application of Large Language Models (LLMs) in Combinatorial Optimization (CO). We report our findings using the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guidelines. We conduct a literature search via Scopus and Google Scholar, examining over 2,000 publications. We assess publications against four inclusion and four exclusion criteria related to their language, research focus, publication year, and type. Eventually, we select 103 studies. We classify these studies into semantic categories and topics to provide a comprehensive overview of the field, including the tasks performed by LLMs, the architectures of LLMs, the existing datasets specifically designed for evaluating LLMs in CO, and the field of application. Finally, we identify future directions for leveraging LLMs in this field.
In unsupervised combinatorial optimization (UCO), during training, one aims to have continuous decisions that are promising in a probabilistic sense for each training instance, which enables end-to-end training on initially discrete and non-differentiable problems. At the test time, for each test instance, starting from continuous decisions, derandomization is typically applied to obtain the final deterministic decisions. Researchers have developed more and more powerful test-time derandomization schemes to enhance the empirical performance and the theoretical guarantee of UCO methods. However, we notice a misalignment between training and testing in the existing UCO methods. Consequently, lower training losses do not necessarily entail better post-derandomization performance, even for the training instances without any data distribution shift. Empirically, we indeed observe such undesirable cases. We explore a preliminary idea to better align training and testing in UCO by including a differentiable version of derandomization into training. Our empirical exploration shows that such an idea indeed improves training-test alignment, but also introduces nontrivial challenges into training.
Heuristic algorithms play a vital role in solving combinatorial optimization (CO) problems, yet traditional designs depend heavily on manual expertise and struggle to generalize across diverse instances. We introduce \textbf{HeurAgenix}, a two-stage hyper-heuristic framework powered by large language models (LLMs) that first evolves heuristics and then selects among them automatically. In the heuristic evolution phase, HeurAgenix leverages an LLM to compare seed heuristic solutions with higher-quality solutions and extract reusable evolution strategies. During problem solving, it dynamically picks the most promising heuristic for each problem state, guided by the LLM's perception ability. For flexibility, this selector can be either a state-of-the-art LLM or a fine-tuned lightweight model with lower inference cost. To mitigate the scarcity of reliable supervision caused by CO complexity, we fine-tune the lightweight heuristic selector with a dual-reward mechanism that jointly exploits singals from selection preferences and state perception, enabling robust selection under noisy annotations. Extensive experiments on canonical benchmarks show that HeurAgenix not only outperforms existing LLM-based hyper-heuristics but also matches or exceeds specialized solvers. Code is available at https://github.com/microsoft/HeurAgenix.
Combinatorial optimization (CO) problems, central to operation research and theoretical computer science, present significant computational challenges due to their NP-hard nature. While large language models (LLMs) have emerged as promising tools for CO--either by directly generating solutions or synthesizing solver-specific codes--existing approaches often neglect critical structural priors inherent to CO problems, leading to suboptimality and iterative inefficiency. Inspired by human experts' success in leveraging CO structures for algorithm design, we propose STRCMP, a novel structure-aware LLM-based algorithm discovery framework that systematically integrates structure priors to enhance solution quality and solving efficiency. Our framework combines a graph neural network (GNN) for extracting structural embeddings from CO instances with an LLM conditioned on these embeddings to identify high-performing algorithms in the form of solver-specific codes. This composite architecture ensures syntactic correctness, preserves problem topology, and aligns with natural language objectives, while an evolutionary refinement process iteratively optimizes generated algorithm. Extensive evaluations across Mixed Integer Linear Programming and Boolean Satisfiability problems, using nine benchmark datasets, demonstrate that our proposed STRCMP outperforms five strong neural and LLM-based methods by a large margin, in terms of both solution optimality and computational efficiency. The code and learned model will be publicly available upon the acceptance of the paper.
We report a higher-order neuromorphic Ising machine that exhibits superior scalability compared to architectures based on quadratization, while also achieving state-of-the-art quality and reliability in solutions with competitive time-to-solution metrics. At the core of the proposed machine is an asynchronous autoencoder architecture that captures higher-order interactions by directly manipulating Ising clauses instead of Ising spins, thereby maintaining resource complexity independent of interaction order. Asymptotic convergence to the Ising ground state is ensured by sampling the autoencoder latent space defined by the spins, based on the annealing dynamics of the Fowler-Nordheim quantum mechanical tunneling. To demonstrate the advantages of the proposed higher-order neuromorphic Ising machine, we systematically solved benchmark combinatorial optimization problems such as MAX-CUT and MAX-SAT, comparing the results to those obtained using a second-order Ising machine employing the same annealing process. Our findings indicate that the proposed architecture consistently provides higher quality solutions in shorter time frames compared to the second-order model across multiple runs. Additionally, we show that the techniques based on the sparsity of the interconnection matrix, such as graph coloring, can be effectively applied to higher-order neuromorphic Ising machines, enhancing the solution quality and the time-to-solution. The time-to-solution can be further improved through hardware co-design, as demonstrated in this paper using a field-programmable gate array (FPGA). The results presented in this paper provide further evidence that autoencoders and Fowler-Nordheim annealers are sufficient to achieve reliability and scaling of any-order neuromorphic Ising machines.
Although Large Vision Language Models (LVLMs) have demonstrated remarkable performance in image understanding tasks, their computational efficiency remains a significant challenge, particularly on resource-constrained devices due to the high cost of processing large numbers of visual tokens. Recently, training-free visual token pruning methods have gained popularity as a low-cost solution to this issue. However, existing approaches suffer from two key limitations: semantic saliency-based strategies primarily focus on high cross-attention visual tokens, often neglecting visual diversity, whereas visual diversity-based methods risk inadvertently discarding semantically important tokens, especially under high compression ratios. In this paper, we introduce GreedyPrune, a training-free plug-and-play visual token pruning algorithm designed to jointly optimize semantic saliency and visual diversity. We formalize the token pruning process as a combinatorial optimization problem and demonstrate that greedy algorithms effectively balance computational efficiency with model accuracy. Extensive experiments validate the effectiveness of our approach, showing that GreedyPrune achieves state-of-the-art accuracy across various multimodal tasks and models while significantly reducing end-to-end inference latency.
The job shop scheduling problem is an NP-hard combinatorial optimization problem relevant to manufacturing and timetabling. Traditional approaches use priority dispatching rules based on simple heuristics. Recent work has attempted to replace these with deep learning models, particularly graph neural networks (GNNs), that learn to assign priorities from data. However, training such models requires customizing numerous factors: graph representation, node features, action space, and reward functions. The lack of modular libraries for experimentation makes this research time-consuming. This work introduces JobShopLib, a modular library that allows customizing these factors and creating new components with its reinforcement learning environment. We trained several dispatchers through imitation learning to demonstrate the environment's utility. One model outperformed various graph-based dispatchers using only individual operation features, highlighting the importance of feature customization. Our GNN model achieved near state-of-the-art results on large-scale problems. These results suggest significant room for improvement in developing such models. JobShopLib provides the necessary tools for future experimentation.
Combinatorial Optimization problems are widespread in domains such as logistics, manufacturing, and drug discovery, yet their NP-hard nature makes them computationally challenging. Recent Neural Combinatorial Optimization methods leverage deep learning to learn solution strategies, trained via Supervised or Reinforcement Learning (RL). While promising, these approaches often rely on task-specific augmentations, perform poorly on out-of-distribution instances, and lack robust inference mechanisms. Moreover, existing latent space models either require labeled data or rely on pre-trained policies. In this work, we propose LGS-Net, a novel latent space model that conditions on problem instances, and introduce an efficient inference method, Latent Guided Sampling (LGS), based on Markov Chain Monte Carlo and Stochastic Approximation. We show that the iterations of our method form a time-inhomogeneous Markov Chain and provide rigorous theoretical convergence guarantees. Empirical results on benchmark routing tasks show that our method achieves state-of-the-art performance among RL-based approaches.
The Quantum Approximate Optimization Algorithm (QAOA) is a prominent variational algorithm used for solving combinatorial optimization problems such as the Max-Cut problem. A key challenge in QAOA lies in efficiently identifying suitable parameters (gamma, beta) that lead to high-quality solutions. In this paper, we propose a framework that combines Fully Informed Particle Swarm Optimization (FIPSO) with adaptive gradient correction using the Adam Optimizer to navigate the QAOA parameter space. This approach aims to avoid issues such as barren plateaus and convergence to local minima. The proposed algorithm is evaluated against two classes of graph instances, Erdos Renyi and Watts-Strogatz. Experimental results across multiple QAOA depths consistently demonstrate superior performance compared to random initialization, underscoring the effectiveness and robustness of the proposed optimization framework.
We study the problem of synthesizing non-smooth control barrier functions (CBFs) for continuous-time switched affine systems. Switched affine systems are defined by a set of affine dynamical modes, wherein the control consists of a state-based switching signal that determines the current operating mode. The control barrier functions seek to maintain the system state inside a control invariant set that excludes a given set of unsafe states. We consider CBFs that take the form of pointwise minima and maxima over a finite set of affine functions. Our approach uses ideas from nonsmooth analysis to formulate conditions for min- and max- affine control barrier functions. We show how a feedback switching law can be extracted from a given CBF. Next, we show how to automate the process of synthesizing CBFs given a system description through a tree-search algorithm inspired by branch-and-cut methods from combinatorial optimization. Finally, we demonstrate our approach on a series of interesting examples of switched affine systems.
The graph coloring problem (GCP) is a classic combinatorial optimization problem that aims to find the minimum number of colors assigned to vertices of a graph such that no two adjacent vertices receive the same color. GCP has been extensively studied by researchers from various fields, including mathematics, computer science, and biological science. Due to the NP-hard nature, many heuristic algorithms have been proposed to solve GCP. However, existing GCP algorithms focus on either small hard graphs or large-scale sparse graphs (with up to 10^7 vertices). This paper presents an efficient hybrid heuristic algorithm for GCP, named HyColor, which excels in handling large-scale sparse graphs while achieving impressive results on small dense graphs. The efficiency of HyColor comes from the following three aspects: a local decision strategy to improve the lower bound on the chromatic number; a graph-reduction strategy to reduce the working graph; and a k-core and mixed degree-based greedy heuristic for efficiently coloring graphs. HyColor is evaluated against three state-of-the-art GCP algorithms across four benchmarks, comprising three large-scale sparse graph benchmarks and one small dense graph benchmark, totaling 209 instances. The results demonstrate that HyColor consistently outperforms existing heuristic algorithms in both solution accuracy and computational efficiency for the majority of instances. Notably, HyColor achieved the best solutions in 194 instances (over 93%), with 34 of these solutions significantly surpassing those of other algorithms. Furthermore, HyColor successfully determined the chromatic number and achieved optimal coloring in 128 instances.
Combinatorial optimization problems are notoriously challenging due to their discrete structure and exponentially large solution space. Recent advances in deep reinforcement learning (DRL) have enabled the learning heuristics directly from data. However, DRL methods often suffer from limited exploration and susceptibility to local optima. On the other hand, evolutionary algorithms such as Genetic Algorithms (GAs) exhibit strong global exploration capabilities but are typically sample inefficient and computationally intensive. In this work, we propose the Evolutionary Augmentation Mechanism (EAM), a general and plug-and-play framework that synergizes the learning efficiency of DRL with the global search power of GAs. EAM operates by generating solutions from a learned policy and refining them through domain-specific genetic operations such as crossover and mutation. These evolved solutions are then selectively reinjected into the policy training loop, thereby enhancing exploration and accelerating convergence. We further provide a theoretical analysis that establishes an upper bound on the KL divergence between the evolved solution distribution and the policy distribution, ensuring stable and effective policy updates. EAM is model-agnostic and can be seamlessly integrated with state-of-the-art DRL solvers such as the Attention Model, POMO, and SymNCO. Extensive results on benchmark problems (e.g., TSP, CVRP, PCTSP, and OP) demonstrate that EAM significantly improves both solution quality and training efficiency over competitive baselines.
In complex engineering systems, the interdependencies among components or development activities are often modeled and analyzed using Design Structure Matrix (DSM). Reorganizing elements within a DSM to minimize feedback loops and enhance modularity or process efficiency constitutes a challenging combinatorial optimization (CO) problem in engineering design and operations. As problem sizes increase and dependency networks become more intricate, traditional optimization methods that solely use mathematical heuristics often fail to capture the contextual nuances and struggle to deliver effective solutions. In this study, we explore the potential of Large Language Models (LLMs) for helping solve such CO problems by leveraging their capabilities for advanced reasoning and contextual understanding. We propose a novel LLM-based framework that integrates network topology with contextual domain knowledge for iterative optimization of DSM element sequencing - a common CO problem. Experiments on various DSM cases show that our method consistently achieves faster convergence and superior solution quality compared to both stochastic and deterministic baselines. Notably, we find that incorporating contextual domain knowledge significantly enhances optimization performance regardless of the chosen LLM backbone. These findings highlight the potential of LLMs to solve complex engineering CO problems by combining semantic and mathematical reasoning. This approach paves the way towards a new paradigm in LLM-based engineering design optimization.
While Large Language Models (LLMs) have demonstrated significant advancements in reasoning and agent-based problem-solving, current evaluation methodologies fail to adequately assess their capabilities: existing benchmarks either rely on closed-ended questions prone to saturation and memorization, or subjective comparisons that lack consistency and rigor. In this work, we introduce HeuriGym, an agentic framework designed for evaluating heuristic algorithms generated by LLMs for combinatorial optimization problems, characterized by clearly defined objectives and expansive solution spaces. HeuriGym empowers LLMs to propose heuristics, receive evaluative feedback via code execution, and iteratively refine their solutions. We evaluate nine state-of-the-art models on nine problems across domains such as computer systems, logistics, and biology, exposing persistent limitations in tool use, planning, and adaptive reasoning. To quantify performance, we propose the Quality-Yield Index (QYI), a metric that captures both solution pass rate and quality. Even top models like GPT-o4-mini-high and Gemini-2.5-Pro attain QYI scores of only 0.6, well below the expert baseline of 1. Our open-source benchmark aims to guide the development of LLMs toward more effective and realistic problem-solving in scientific and engineering domains.
Quantum Annealing (QA) can efficiently solve combinatorial optimization problems whose objective functions are represented by Quadratic Unconstrained Binary Optimization (QUBO) formulations. For broader applicability of QA, quadratization methods are used to transform higher-order problems into QUBOs. However, quadratization methods for complex problems involving Machine Learning (ML) remain largely unknown. In these problems, strong nonlinearity and dense interactions prevent conventional methods from being applied. Therefore, we model target functions by the sum of rectified linear unit bases, which not only have the ability of universal approximation, but also have an equivalent quadratic-polynomial representation. In this study, the proof of concept is verified both numerically and analytically. In addition, by combining QA with the proposed quadratization, we design a new black-box optimization scheme, in which ML surrogate regressors are inputted to QA after the quadratization process.
This work proposes a simple yet effective sampling framework for combinatorial optimization (CO). Our method builds on discrete Langevin dynamics (LD), an efficient gradient-guided generative paradigm. However, we observe that directly applying LD often leads to limited exploration. To overcome this limitation, we propose the Regularized Langevin Dynamics (RLD), which enforces an expected distance between the sampled and current solutions, effectively avoiding local minima. We develop two CO solvers on top of RLD, one based on simulated annealing (SA), and the other one based on neural network (NN). Empirical results on three classic CO problems demonstrate that both of our methods can achieve comparable or better performance against the previous state-of-the-art (SOTA) SA- and NN-based solvers. In particular, our SA algorithm reduces the runtime of the previous SOTA SA method by up to 80\%, while achieving equal or superior performance. In summary, RLD offers a promising framework for enhancing both traditional heuristics and NN models to solve CO problems. Our code is available at https://github.com/Shengyu-Feng/RLD4CO.
Recent deep reinforcement learning methods have achieved remarkable success in solving multi-objective combinatorial optimization problems (MOCOPs) by decomposing them into multiple subproblems, each associated with a specific weight vector. However, these methods typically treat all subproblems equally and solve them using a single model, hindering the effective exploration of the solution space and thus leading to suboptimal performance. To overcome the limitation, we propose POCCO, a novel plug-and-play framework that enables adaptive selection of model structures for subproblems, which are subsequently optimized based on preference signals rather than explicit reward values. Specifically, we design a conditional computation block that routes subproblems to specialized neural architectures. Moreover, we propose a preference-driven optimization algorithm that learns pairwise preferences between winning and losing solutions. We evaluate the efficacy and versatility of POCCO by applying it to two state-of-the-art neural methods for MOCOPs. Experimental results across four classic MOCOP benchmarks demonstrate its significant superiority and strong generalization.
Ferroelectric polarization switching underpins the functional performance of a wide range of materials and devices, yet its dependence on complex local microstructural features renders systematic exploration by manual or grid-based spectroscopic measurements impractical. Here, we introduce a multi-objective kernel-learning workflow that infers the microstructural rules governing switching behavior directly from high-resolution imaging data. Applied to automated piezoresponse force microscopy (PFM) experiments, our framework efficiently identifies the key relationships between domain-wall configurations and local switching kinetics, revealing how specific wall geometries and defect distributions modulate polarization reversal. Post-experiment analysis projects abstract reward functions, such as switching ease and domain symmetry, onto physically interpretable descriptors including domain configuration and proximity to boundaries. This enables not only high-throughput active learning, but also mechanistic insight into the microstructural control of switching phenomena. While demonstrated for ferroelectric domain switching, our approach provides a powerful, generalizable tool for navigating complex, non-differentiable design spaces, from structure-property correlations in molecular discovery to combinatorial optimization across diverse imaging modalities.
Generating challenging instances is crucial for the evaluation and advancement of combinatorial optimization solvers. In this work, we introduce EALG (Evolutionary Adversarial Generation of Language Model-Guided Generators), a novel framework that automates the co-evolution of optimization problem instances and their corresponding heuristic solvers using large language models (LLMs). EALG leverages a mutation-based adversarial approach that dynamically evolves instance generation procedures to create increasingly difficult problems, while simultaneously synthesizing adaptive heuristic algorithms through interactions with LLMs guided by algorithmic structure. Unlike existing approaches that focus solely on static benchmark creation or manual solver design, EALG provides a seamless pipeline from instance generation to solver synthesis. Experimental results demonstrate that EALG generates significantly harder instances than current benchmarks, and its synthesized solvers generalize effectively across a broad spectrum of combinatorial tasks. This work explores a new paradigm for combinatorial optimization that integrates instance generation with solver design, resulting in state-of-the-art performance.