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.
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.
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.
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.
WiFi networks have achieved remarkable success in enabling seamless communication and data exchange worldwide. The IEEE 802.11be standard, known as WiFi 7, introduces Multi-Link Operation (MLO), a groundbreaking feature that enables devices to establish multiple simultaneous connections across different bands and channels. While MLO promises substantial improvements in network throughput and latency reduction, it presents significant challenges in channel allocation, particularly in dense network environments. Current research has predominantly focused on performance analysis and throughput optimization within static WiFi 7 network configurations. In contrast, this paper addresses the dynamic channel allocation problem in dense WiFi 7 networks with MLO capabilities. We formulate this challenge as a combinatorial optimization problem, leveraging a novel network performance analysis mechanism. Given the inherent lack of prior network information, we model the problem within a Multi-Armed Bandit (MAB) framework to enable online learning of optimal channel allocations. Our proposed Best-Arm Identification-enabled Monte Carlo Tree Search (BAI-MCTS) algorithm includes rigorous theoretical analysis, providing upper bounds for both sample complexity and error probability. To further reduce sample complexity and enhance generalizability across diverse network scenarios, we put forth LLM-BAI-MCTS, an intelligent algorithm for the dynamic channel allocation problem by integrating the Large Language Model (LLM) into the BAI-MCTS algorithm. Numerical results demonstrate that the BAI-MCTS algorithm achieves a convergence rate approximately $50.44\%$ faster than the state-of-the-art algorithms when reaching $98\%$ of the optimal value. Notably, the convergence rate of the LLM-BAI-MCTS algorithm increases by over $63.32\%$ in dense networks.
Reinforcement learning (RL) is increasingly applied to real-world problems involving complex and structured decisions, such as routing, scheduling, and assortment planning. These settings challenge standard RL algorithms, which struggle to scale, generalize, and exploit structure in the presence of combinatorial action spaces. We propose Structured Reinforcement Learning (SRL), a novel actor-critic framework that embeds combinatorial optimization layers into the actor neural network. We enable end-to-end learning of the actor via Fenchel-Young losses and provide a geometric interpretation of SRL as a primal-dual algorithm in the dual of the moment polytope. Across six environments with exogenous and endogenous uncertainty, SRL matches or surpasses the performance of unstructured RL and imitation learning on static tasks and improves over these baselines by up to 92% on dynamic problems, with improved stability and convergence speed.
Combinatorial optimization (CO) problems are traditionally addressed using Operations Research (OR) methods, including metaheuristics. In this study, we introduce a demand selection problem for the Vehicle Routing Problem (VRP) with an emission quota, referred to as QVRP. The objective is to minimize the number of omitted deliveries while respecting the pollution quota. We focus on the demand selection part, called Maximum Feasible Vehicle Assignment (MFVA), while the construction of a routing for the VRP instance is solved using classical OR methods. We propose several methods for selecting the packages to omit, both from machine learning (ML) and OR. Our results show that, in this static problem setting, classical OR-based methods consistently outperform ML-based approaches.
Deep reinforcement learning (DRL) has been widely used for dynamic algorithm configuration, particularly in evolutionary computation, which benefits from the adaptive update of parameters during the algorithmic execution. However, applying DRL to algorithm configuration for multi-objective combinatorial optimization (MOCO) problems remains relatively unexplored. This paper presents a novel graph neural network (GNN) based DRL to configure multi-objective evolutionary algorithms. We model the dynamic algorithm configuration as a Markov decision process, representing the convergence of solutions in the objective space by a graph, with their embeddings learned by a GNN to enhance the state representation. Experiments on diverse MOCO challenges indicate that our method outperforms traditional and DRL-based algorithm configuration methods in terms of efficacy and adaptability. It also exhibits advantageous generalizability across objective types and problem sizes, and applicability to different evolutionary computation methods.
Recent studies exploited Large Language Models (LLMs) to autonomously generate heuristics for solving Combinatorial Optimization Problems (COPs), by prompting LLMs to first provide search directions and then derive heuristics accordingly. However, the absence of task-specific knowledge in prompts often leads LLMs to provide unspecific search directions, obstructing the derivation of well-performing heuristics. Moreover, evaluating the derived heuristics remains resource-intensive, especially for those semantically equivalent ones, often requiring omissible resource expenditure. To enable LLMs to provide specific search directions, we propose the Hercules algorithm, which leverages our designed Core Abstraction Prompting (CAP) method to abstract the core components from elite heuristics and incorporate them as prior knowledge in prompts. We theoretically prove the effectiveness of CAP in reducing unspecificity and provide empirical results in this work. To reduce computing resources required for evaluating the derived heuristics, we propose few-shot Performance Prediction Prompting (PPP), a first-of-its-kind method for the Heuristic Generation (HG) task. PPP leverages LLMs to predict the fitness values of newly derived heuristics by analyzing their semantic similarity to previously evaluated ones. We further develop two tailored mechanisms for PPP to enhance predictive accuracy and determine unreliable predictions, respectively. The use of PPP makes Hercules more resource-efficient and we name this variant Hercules-P. Extensive experiments across four HG tasks, five COPs, and eight LLMs demonstrate that Hercules outperforms the state-of-the-art LLM-based HG algorithms, while Hercules-P excels at minimizing required computing resources. In addition, we illustrate the effectiveness of CAP, PPP, and the other proposed mechanisms by conducting relevant ablation studies.
Physics-inspired computing paradigms are receiving renewed attention to enhance efficiency in compute-intensive tasks such as artificial intelligence and optimization. Similar to Hopfield neural networks, oscillatory neural networks (ONNs) minimize an Ising energy function that embeds the solutions of hard combinatorial optimization problems. Despite their success in solving unconstrained optimization problems, Ising machines still face challenges with constrained problems as they can get stuck at infeasible local minima. In this paper, we introduce a Lagrange ONN (LagONN) designed to escape infeasible states based on the theory of Lagrange multipliers. Unlike existing oscillatory Ising machines, LagONN employs additional Lagrange oscillators to guide the system towards feasible states in an augmented energy landscape and settles only when constraints are met. Taking the maximum satisfiability problem with three literals as a use case (Max-3-SAT), we harness LagONN's constraint satisfaction mechanism to find optimal solutions for random SATlib instances with up to 200 variables and 860 clauses, which provides a deterministic alternative to simulated annealing for coupled oscillators. We further discuss the potential of Lagrange oscillators to address other constraints, such as phase copying, which is useful in oscillatory Ising machines with limited connectivity.
Sampling from flat modes in discrete spaces is a crucial yet underexplored problem. Flat modes represent robust solutions and have broad applications in combinatorial optimization and discrete generative modeling. However, existing sampling algorithms often overlook the mode volume and struggle to capture flat modes effectively. To address this limitation, we propose \emph{Entropic Discrete Langevin Proposal} (EDLP), which incorporates local entropy into the sampling process through a continuous auxiliary variable under a joint distribution. The local entropy term guides the discrete sampler toward flat modes with a small overhead. We provide non-asymptotic convergence guarantees for EDLP in locally log-concave discrete distributions. Empirically, our method consistently outperforms traditional approaches across tasks that require sampling from flat basins, including Bernoulli distribution, restricted Boltzmann machines, combinatorial optimization, and binary neural networks.
Recently, deep reinforcement learning has emerged as a promising approach for solving complex combinatorial optimization problems. Broadly, deep reinforcement learning methods fall into two categories: policy-based and value-based. While value-based approaches have achieved notable success in domains such as the Arcade Learning Environment, the combinatorial optimization community has predominantly favored policy-based methods, often overlooking the potential of value-based algorithms. In this work, we conduct a comprehensive empirical evaluation of value-based algorithms, including the deep q-network and several of its advanced extensions, within the context of two complex combinatorial problems: the job-shop and the flexible job-shop scheduling problems, two fundamental challenges with multiple industrial applications. Our results challenge the assumption that policy-based methods are inherently superior for combinatorial optimization. We show that several value-based approaches can match or even outperform the widely adopted proximal policy optimization algorithm, suggesting that value-based strategies deserve greater attention from the combinatorial optimization community. Our code is openly available at: https://github.com/AJ-Correa/Unraveling-the-Rainbow.
Although LLM-based agents have attracted significant attention in domains such as software engineering and machine learning research, their role in advancing combinatorial optimization (CO) remains relatively underexplored. This gap underscores the need for a deeper understanding of their potential in tackling structured, constraint-intensive problems-a pursuit currently limited by the absence of comprehensive benchmarks for systematic investigation. To address this, we introduce CO-Bench, a benchmark suite featuring 36 real-world CO problems drawn from a broad range of domains and complexity levels. CO-Bench includes structured problem formulations and curated data to support rigorous investigation of LLM agents. We evaluate multiple agent frameworks against established human-designed algorithms, revealing key strengths and limitations of current approaches and identifying promising directions for future research. CO-Bench is publicly available at https://github.com/sunnweiwei/CO-Bench.
The NP-complete combinatorial optimization task of assigning offsets to a set of buffers with known sizes and lifetimes so as to minimize total memory usage is called dynamic storage allocation (DSA). Existing DSA implementations bypass the theoretical state-of-the-art algorithms in favor of either fast but wasteful heuristics, or memory-efficient approaches that do not scale beyond one thousand buffers. The "AI memory wall", combined with deep neural networks' static architecture, has reignited interest in DSA. We present idealloc, a low-fragmentation, high-performance DSA implementation designed for million-buffer instances. Evaluated on a novel suite of particularly hard benchmarks from several domains, idealloc ranks first against four production implementations in terms of a joint effectiveness/robustness criterion.
Vehicle Routing Problems (VRPs) are a class of NP-hard problems ubiquitous in several real-world logistics scenarios that pose significant challenges for optimization. Neural Combinatorial Optimization (NCO) has emerged as a promising alternative to classical approaches, as it can learn fast heuristics to solve VRPs. However, most research works in NCO for VRPs focus on simplified settings, which do not account for asymmetric distances and travel durations that cannot be derived by simple Euclidean distances and unrealistic data distributions, hindering real-world deployment. This work introduces RRNCO (Real Routing NCO) to bridge the gap of NCO between synthetic and real-world VRPs in the critical aspects of both data and modeling. First, we introduce a new, openly available dataset with real-world data containing a diverse dataset of locations, distances, and duration matrices from 100 cities, considering realistic settings with actual routing distances and durations obtained from Open Source Routing Machine (OSRM). Second, we propose a novel approach that efficiently processes both node and edge features through contextual gating, enabling the construction of more informed node embedding, and we finally incorporate an Adaptation Attention Free Module (AAFM) with neural adaptive bias mechanisms that effectively integrates not only distance matrices but also angular relationships between nodes, allowing our model to capture rich structural information. RRNCO achieves state-of-the-art results in real-world VRPs among NCO methods. We make our dataset and code publicly available at https://github.com/ai4co/real-routing-nco.
Domain-Independent Dynamic Programming (DIDP) is a state-space search paradigm based on dynamic programming for combinatorial optimization. In its current implementation, DIDP guides the search using user-defined dual bounds. Reinforcement learning (RL) is increasingly being applied to combinatorial optimization problems and shares several key structures with DP, being represented by the Bellman equation and state-based transition systems. We propose using reinforcement learning to obtain a heuristic function to guide the search in DIDP. We develop two RL-based guidance approaches: value-based guidance using Deep Q-Networks and policy-based guidance using Proximal Policy Optimization. Our experiments indicate that RL-based guidance significantly outperforms standard DIDP and problem-specific greedy heuristics with the same number of node expansions. Further, despite longer node evaluation times, RL guidance achieves better run-time performance than standard DIDP on three of four benchmark domains.
World models significantly enhance hierarchical understanding, improving data integration and learning efficiency. To explore the potential of the world model in the remote sensing (RS) field, this paper proposes a label-efficient remote sensing world model for multimodal data fusion (FusDreamer). The FusDreamer uses the world model as a unified representation container to abstract common and high-level knowledge, promoting interactions across different types of data, \emph{i.e.}, hyperspectral (HSI), light detection and ranging (LiDAR), and text data. Initially, a new latent diffusion fusion and multimodal generation paradigm (LaMG) is utilized for its exceptional information integration and detail retention capabilities. Subsequently, an open-world knowledge-guided consistency projection (OK-CP) module incorporates prompt representations for visually described objects and aligns language-visual features through contrastive learning. In this way, the domain gap can be bridged by fine-tuning the pre-trained world models with limited samples. Finally, an end-to-end multitask combinatorial optimization (MuCO) strategy can capture slight feature bias and constrain the diffusion process in a collaboratively learnable direction. Experiments conducted on four typical datasets indicate the effectiveness and advantages of the proposed FusDreamer. The corresponding code will be released at https://github.com/Cimy-wang/FusDreamer.
Recently, gradient-based discrete sampling has emerged as a highly efficient, general-purpose solver for various combinatorial optimization (CO) problems, achieving performance comparable to or surpassing the popular data-driven approaches. However, we identify a critical issue in these methods, which we term ''wandering in contours''. This behavior refers to sampling new different solutions that share very similar objective values for a long time, leading to computational inefficiency and suboptimal exploration of potential solutions. In this paper, we introduce a novel reheating mechanism inspired by the concept of critical temperature and specific heat in physics, aimed at overcoming this limitation. Empirically, our method demonstrates superiority over existing sampling-based and data-driven algorithms across a diverse array of CO problems.