Neural Combinatorial Optimization

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Overview

Neural combinatorial optimization studies how learned models can construct, improve, or guide solutions to problems with many discrete choices and hard constraints, including vehicle routing, scheduling, packing, and graph optimization. Common approaches include graph neural networks and Transformers trained with supervised examples from solvers, reinforcement learning, self-supervised objectives, and newer diffusion or preference-based methods. Many systems combine learned policies with classical search, local improvement, or exact optimization to improve feasibility and solution quality. A central challenge is whether they generalize to larger or unfamiliar instances and offer a practical advantage over strong classical heuristics when training and inference costs are counted.

Latest papers

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137 papers17 papers in the last four weeks, up 113% on the four weeks before. 0.1% of all new papers.

May 11, 2026

Rethinking Constraint Awareness for Efficient State Embedding of Neural Routing Solver

Heavy-Encoder-Light-Decoder (HELD) neural routing solvers have emerged as a promising paradigm due to their broad applicability across multiple vehicle routing problems (VRPs). However, they typically struggle with VRP variants with complex constraints. To address this limitation, this paper systematically revisits existing neural solvers from the perspective of the generation mechanism for state embeddings (i.e., query vector prior to compatibility calculation) during decoding. We identify that current mechanisms restrict the observation space during attention computation, introducing a key bottleneck to achieving high-quality solutions. Through detailed empirical analysis, we demonstrate the necessity of preserving a global observation space. To overcome the constraint-agnostic drawback inherent to global observation spaces, we propose a simple yet powerful Constraint-Aware Residual Modulation (CARM) module. By adaptively modulating the context embedding with constraint-relevant variables, CARM effectively enhances constraint awareness, enabling the neural solver to fully leverage the global observation space and generate an efficient state embedding. Extensive experimental results across two single-task and five multi-task neural routing solvers confirm that the CARM module consistently boosts baseline performance. Notably, solvers equipped with our CARM achieve substantial improvements in scaling to large-scale instances and in generalizing to unseen VRP variants. These findings provide valuable insights for the architectural design of neural routing solvers.

Canhong Yu, Changliang Zhou, Rongsheng Chen +2

College of Computer Science and Software Engineering, Shenzhen University, Shenzhen, China · School of Automation and Intelligent Manufacturing, Southern University of Science and Technology, Shenzhen, China · Guangdong Provincial Key Laboratory of Fully Actuated System Control Theory and Technology, Southern University of Science and Technology, Shenzhen, China +1

Neural Network OptimizationCombinatorial OptimizationVehicle RoutingNeural Combinatorial Optimization
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May 10, 2026

Neural Cluster First, Route Second: Capacitated Vehicle Routing via Differentiable Optimal Transport

The Capacitated Vehicle Routing Problem (CVRP) underpins modern last-mile logistics, where routing decisions recur over the same fixed service area, like a city. In this setting, routing problems share a fixed set of potential customer locations, while active customers and demands vary between instances. We study how this spatial support can be exploited through reusable learned representations and design our method around three symmetries of the symmetric Euclidean CVRP: E(2)E(2) transformations, vehicle-route permutations, and tour reversal. We introduce Neural Cluster-First--Route-Second (CFRS), a neural extension of the Fisher--Jaikumar framework that predicts seed-selection scores and customer-to-cluster assignment costs non-autoregressively and respects the three symmetries. A differentiable entropic optimal transport layer provides capacity-aware supervision and guides discrete capacitated assignment, followed by independent traveling salesman subproblems for route recovery. Component ablations show consistent benefits from learned seed selection, while learned assignment costs perform best near the training size and classical FJ costs perform better at larger sizes under exact decoding. On the fixed-support distribution with constant capacity, a model trained on N=100N=100 achieves a 3.77%3.77\% routing gap relative to HGS at N=1000N=1000 without retraining. A shallow variant with one attention layer in each transformer achieves a 5.08%5.08\% gap at this scale, with spatial embeddings consistently improving routing quality over raw coordinates. Embedding interpolation further accommodates entirely unseen customer locations without retraining. On standard CVRP benchmarks, a separately trained model achieves a 2.73%2.73\% routing gap relative to LKH-3 at N=100N=100.

Samuel J. K. Chin, Maximilian Schiffer

MIT · TUM

Vehicle RoutingNeural Combinatorial Optimization
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May 7, 2026

LINC: Decoupling Local Consequence Scoring from Hidden Matching in Constructive Neural Routing

Constructive neural routing solvers usually score the next action by matching a decoder context to candidate embeddings, leaving deterministic one-step consequences such as travel, waiting, slack, and capacity changes implicit. We propose LINC, a decoder-side candidate decision architecture that computes these consequences explicitly. LINC uses them according to their decision role: candidate-level consequences are scored by a state-conditioned shared linear comparator, while feasible-set summaries modulate the decoder context. This preserves standard global matching while reducing the burden on the hidden state to reconstruct transition arithmetic. The Capacitated Vehicle Routing Problem with Time Windows (CVRPTW) serves as the main constrained-routing testbed, and the same interface extends to the Capacitated Vehicle Routing Problem (CVRP) and Traveling Salesman Problem (TSP). Across external benchmarks and no-retraining scale-transfer settings, LINC consistently improves strong neural baselines, with the advantage becoming more pronounced as test size moves further beyond the training scale, especially on constrained routing problems.

Shaofeng Qin, Li Wang

Beijing University of Posts and Telecommunications

Vehicle RoutingNeural Combinatorial Optimization
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May 6, 2026

Two-Stage Learned Decomposition for Scalable Routing on Multigraphs

Most neural methods for Vehicle Routing Problems (VRPs) are limited to Euclidean settings or simple graphs. In this work, we instead consider multigraphs, where parallel edges represent distinct travel options with varying trade-offs (e.g., distance vs. time). Multigraphs are highly relevant in practice, yet few neural methods are designed for them, and those that do exist face major scalability issues. We address these scalability issues with Node-Edge Policy Factorization (NEPF), which splits the routing policy into a node permutation stage and an edge selection stage. To enable the decomposition, we introduce a pre-encoding edge aggregation scheme and a non-autoregressive architecture for the edge stage, as well as a hierarchical reinforcement learning method to train the stages jointly. Our experiments across six VRP variants demonstrate that NEPF trains and runs up to orders of magnitude faster than prior neural multigraph methods and scales to considerably larger instances, while matching or improving on their solution quality.

Filip Rydin, Morteza Haghir Chehreghani, Balázs Kulcsár

Chalmers University of Technology · University of Gothenburg

Vehicle RoutingNeural Combinatorial Optimization
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May 5, 2026

ARMATA: Auto-Regressive Multi-Agent Task Assignment

Coordinating multi-agent systems over spatially distributed areas requires solving a complex hierarchical problem: first distributing areas among agents (allocation) and subsequently determining the optimal visitation order (routing). Existing methods typically decouple these stages ignoring inter-stage dependencies or rely on decentralized heuristics that lack global context. In this work, we propose a centralized, fully end-to-end auto-regressive framework that jointly generates allocation decisions and routing sequences. The core contribution of our approach is a multi-stage decoding mechanism that unifies high-level allocation and low-level routing in a single autoregressive pass while maintaining a centralized global state. This enables the model to implicitly balance workload distribution with routing efficiency, avoiding local optima common in decentralized methods. Extensive experiments demonstrate that our method significantly outperforms diverse baselines, achieving up to a 20% improvement in solution quality over industrial solvers such as Google OR-Tools, IBM CPLEX, and LKH-3, while reducing computation time from hours to seconds.

Yazan Youssef, Aboelmagd Noureldin, Sidney Givigi

Department of Electrical and Computer Engineering, Queen’s University, Kingston, ON, Canada · Department of Electrical and Computer Engineering at the Royal Military College of Canada, and the Department of Electrical and Computer Engineering at Queen’s University, both in Kingston, Canada · School of Computing, Queen’s University, Kingston, ON, Canada

Multi-Agent System OptimizationMulti-Robot Task AllocationNeural Combinatorial OptimizationMulti-Agent Resource Allocation
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Apr 30, 2026

FiLMMeD: Feature-wise Linear Modulation for Cross-Problem Multi-Depot Vehicle Routing

Solving practical multi-depot vehicle routing problems (MDVRP) is a challenging optimization task central to modern logistics, increasingly driven by e-commerce. To address the MDVRP's computational complexity, neural-based combinatorial optimization methods offer a promising scalable alternative to traditional approaches. However, neural-based methods typically rely on rigid architectures and input encodings tailored to specific problem formulations. In real-world settings, heterogeneous constraints create multiple MDVRP variants, limiting the applicability of such models. While multi-task learning (MTL) has begun to accelerate the development of unified neural-based solvers, prior works focus almost exclusively on single-depot VRPs, leaving the MDVRP unaddressed. To bridge this gap, we propose Feature-wise Linear Modulation for Cross-Problem Multi-Depot Vehicle Routing (FiLMMeD), a novel unified neural-based model for 24 different MDVRP variants. We introduce three main contributions: (1) to improve the model's generalization, we augment the standard Transformer encoder with Feature-wise Linear Modulation (FiLM), which dynamically conditions learned internal representations based on the active set of constraints; (2) we provide an initial demonstration of Preference Optimization in the MTL setting, establishing it as a superior alternative to Reinforcement Learning for future MTL works; (3) to mitigate the generalization gap caused by the introduction of multi-depot constraints, we introduce a targeted curriculum learning strategy that progressively exposes the model to increasingly more complex constraint interactions. Extensive experiments on 24 MDVRP variants (including 8 novel formulations) and 16 single-depot VRPs confirm the effectiveness of FiLMMeD, which consistently outperforms state-of-the-art baselines. Our code is available at: https://github.com/AJ-Correa/FiLMMeD/tree/main

Arthur Corrêa, Paulo Nascimento, Samuel Moniz

University of Coimbra, CEMMPRE, ARISE, Department of Mechanical Engineering, Coimbra, 3030-788, Portugal

Multi-Task LearningVehicle RoutingNeural Combinatorial Optimization
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Apr 27, 2026

Crystal structure prediction using graph neural combinatorial optimization

Crystalline materials are widely used in technological applications, yet their discovery remains a significant challenge. As their properties are driven by structure, crystal structure prediction (CSP) methods play a central role in computational approaches aiming to accelerate this process. Previously, CSP has been approached from a combinatorial optimization perspective, with the core challenge of allocating atoms on a fine grid of predefined discrete positions within a unit cell while minimizing their interaction energy. Exact mathematical optimization methods provide guaranteed solutions, but they become computationally expensive for large-scale instances, where the atomic configuration space grows rapidly, particularly in the absence of additional symmetry constraints. In this work, we introduce a neural combinatorial optimization approach to the atom allocation challenge and, subsequently, CSP, based on graph neural networks (GNNs), which can effectively sample from the distribution of feasible structures in an unsupervised manner. We leverage expander graphs to construct computational graphs over discrete positions that capture both short- and long-range interactions between atoms, and employ the Gumbel-Sinkhorn approach to enforce the desired stoichiometry of the generated structures. We demonstrate that our method outperforms classical heuristic approaches and is competitive with a commercial optimization solver across a range of chemical compositions. This enables the use of ever-expanding GPU infrastructure to tackle the inherent combinatorial challenges of CSP, paving the way for scaling beyond current capabilities.

Stavros Gerolymatos, J. Kyle Brubaker, Martin J. A. Schuetz +1

School of Computer Science and Informatics, University of Liverpool, Liverpool, England, UK · Amazon Advanced Solutions Lab, Seattle, Washington 98170, USA · Arm Inc., San Jose, California 95134

Graph Neural NetworksCrystal Structure PredictionCombinatorial OptimizationCrystal Structure GenerationNeural Combinatorial Optimization
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Apr 24, 2026

ML-Guided Primal Heuristics for Mixed Binary Quadratic Programs

Mixed Binary Quadratic Programs (MBQPs) are an important and complex set of problems in combinatorial optimization. As solving large-scale combinatorial optimization problems is challenging, primal heuristics have been developed to quickly identify high-quality solutions within a short amount of time. Recently, a growing body of research has also used machine learning to accelerate solution methods for challenging combinatorial optimization problems. Despite the increasing popularity of these ML-guided methods, a large body of work has focused on Mixed-Integer Linear Programs (MILPs). MBQPs are challenging to solve due to the combinatorial complexity coupled with nonlinearities. This work proposes ML-guided primal heuristics for Mixed Binary Quadratic Programs (MBQPs) by adapting and extending existing work on ML-guided MILP solution prediction to MBQPs. We introduce a new neural network architecture for MBQP solution prediction and a new training data collection procedure. Moreover, we extend existing loss functions in solution prediction and propose to combine contrastive and weighted cross-entropy losses. We evaluate the methods on standard and real-world MBQP benchmarks and show that the developed ML-guided methods significantly outperform existing primal heuristics and state-of-the-art solvers. Furthermore, models trained with our proposed extension with combined losses outperform other ML-based methods adapted from MILPs and improve generalization in cross-regional inference on a real-world wind farm layout optimization problem.

Weimin Huang, Natalie M. Isenberg, Ján Drgoňa +2

University of Southern California, USA · Pacific Northwest National Laboratory, USA · Johns Hopkins University, USA

Combinatorial OptimizationMixed-Integer QPNeural Combinatorial Optimization
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Apr 22, 2026

Learning to Solve the Quadratic Assignment Problem with Warm-Started MCMC Finetuning

The quadratic assignment problem (QAP) is a fundamental NP-hard task that poses significant challenges for both traditional heuristics and modern learning-based solvers. Existing QAP solvers still struggle to achieve consistently competitive performance across structurally diverse real-world instances. To bridge this performance gap, we propose PLMA, an innovative permutation learning framework. PLMA features an efficient warm-started MCMC finetuning procedure to enhance deployment-time performance, leveraging short Markov chains to anchor the adaptation to the promising regions previously explored. For rapid exploration via MCMC over the permutation space, we design an additive energy-based model (EBM) that enables an O(1)O(1)-time 2-swap Metropolis-Hastings sampling step. Moreover, the neural network used to parameterize the EBM incorporates a scalable and flexible cross-graph attention mechanism to model interactions between facilities and locations in the QAP. Extensive experiments demonstrate that PLMA consistently outperforms state-of-the-art baselines across various benchmarks. In particular, PLMA achieves a near-zero average optimality gap on QAPLIB, exhibits remarkably superior robustness on the notoriously difficult Taixxeyy instances, and also serves as an effective QAP solver in bandwidth minimization.

Yicheng Pan, Ruisong Zhou, Haijun Zou +2

School of Mathematical Science, Peking University, Beijing 100871, China · State Key Laboratory of Mathematical Sciences, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China · Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China

Markov Chain Monte CarloCombinatorial OptimizationNeural Combinatorial Optimization
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Apr 17, 2026

NCO4CVRP: Neural Combinatorial Optimization for the Capacitated Vehicle Routing Problem

Neural Combinatorial Optimization (NCO) has emerged as a powerful framework for solving combinatorial optimization problems by integrating deep learning-based models. This work focuses on improving existing inference techniques to enhance solution quality and generalization. Specifically, we modify the Random Re-Construct (RRC) approach of the Light Encoder Heavy Decoder (LEHD) model by incorporating Simulated Annealing (SA). Unlike the conventional RRC, which greedily replaces suboptimal segments, our SA-based modification introduces a probabilistic acceptance mechanism that allows the model to escape local optima and explore a more diverse solution space. Additionally, we enhance the Policy Optimization with Multiple Optima (POMO) approach by integrating Beam Search, enabling systematic exploration of multiple promising solutions while maintaining diversity in the search space. We further investigate different inference strategies, including Softmax Sampling, Greedy, Gumbel-Softmax, and Epsilon-Greedy, analyzing their impact on solution quality. Furthermore, we explore instance augmentation techniques, such as horizontal and vertical flipping and rotation-based augmentations, to improve model generalization across different CVRP instances. Our extensive experiments demonstrate that these modifications significantly reduce the optimality gap across various Capacitated Vehicle Routing Problem (CVRP) benchmarks, with Beam Search and SA-based RRC consistently yielding superior performance. By refining inference techniques and leveraging enhanced search strategies, our work contributes to the broader applicability of NCO models in real-world combinatorial optimization tasks.

Mahir Labib Dihan, Md. Ashrafur Rahman Khan, Wasif Jalal +2

Department of Computer Science and Engineering Bangladesh University of Engineering and Technology

Inference-Time SearchMetaheuristic OptimizationVehicle RoutingNeural Combinatorial Optimization
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Apr 13, 2026

Generating Hadamard matrices with transformers

We present a new method for constructing Hadamard matrices that combines transformer neural networks with local search in the PatternBoost framework. Our approach is designed for extremely sparse combinatorial search problems and is particularly effective for Hadamard matrices of Goethals--Seidel type, where Fourier methods permit fast scoring and optimisation. For orders between 100 and 200, it produces large numbers of inequivalent Hadamard matrices, and for larger orders, it succeeds where local search from random initialisation fails. The largest example found by our method has order 252. In addition to these new constructions, our experiments reveal that the transformer can discover and exploit useful hidden symmetry in the search space.

Geordie Williamson, Oded Yacobi, Paul Zinn-Justin

Combinatorial OptimizationNeural Combinatorial Optimization
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Apr 12, 2026

Enhancing Cross-Problem Vehicle Routing via Federated Learning

Vehicle routing problems (VRPs) constitute a core optimization challenge in modern logistics and supply chain management. The recent neural combinatorial optimization (NCO) has demonstrated superior efficiency over some traditional algorithms. While serving as a primary NCO approach for solving general VRPs, current cross-problem learning paradigms are still subject to performance degradation and generalizability decay, when transferring from simple VRP variants to those involving different and complex constraints. To strengthen the paradigms, this paper offers an innovative "Multi-problem Pre-train, then Single-problem Fine-tune" framework with Federated Learning (MPSF-FL). This framework exploits the common knowledge of a federated global model to foster efficient cross-problem knowledge sharing and transfer among local models for single-problem fine-tuning. In this way, local models effectively retain common VRP knowledge from up-to-date global model, while being efficiently adapted to downstream VRPs with heterogeneous complex constraints. Experimental results demonstrate that our framework not only enhances the performance in diverse VRPs, but also improves the generalizability in unseen problems.

Xiangchi Meng, Jianan Zhou, Jie Gao +4

Dalian University of Technology · Nanyang Technological University · Delft University of Technology +1

Vehicle RoutingNeural Combinatorial OptimizationTransfer LearningFederated Fine-TuningFederated Learning
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Apr 8, 2026

A First Guess is Rarely the Final Answer: Learning to Search in the Traveling Salesperson Problem

Most neural solvers for the Traveling Salesperson Problem (TSP) are trained to output a single solution, even though practitioners rarely stop there: at test time, they routinely spend extra compute on sampling or post-hoc search. This raises a natural question: can the search procedure itself be learned? Neural improvement methods take this perspective by learning a policy that applies local modifications to a candidate solution, accumulating gains over an improvement trajectory. Yet learned improvement for TSP remains comparatively immature, with existing methods still falling short of robust, scalable performance. We argue that a key reason is design mismatch: many approaches reuse state representations, architectural choices, and training recipes inherited from single-solution methods, rather than being built around the mechanics of local search. This mismatch motivates NICO-TSP (Neural Improvement for Combinatorial Optimization): a 2-opt improvement framework for TSP. NICO-TSP represents the current tour with exactly nn edge tokens aligned with the neighborhood operator, scores 2-opt moves directly without tour positional encodings, and trains via a two-stage procedure: imitation learning to short-horizon optimal trajectories, followed by critic-free group-based reinforcement learning over longer rollouts. Under compute-matched evaluations that measure improvement as a function of both search steps and wall-clock time, NICO-TSP delivers consistently stronger and markedly more step-efficient improvement than prior learned and heuristic search baselines, generalizes far more reliably to larger out-of-distribution instances, and serves both as a competitive replacement for classical local search and as a powerful test-time refinement module for constructive solvers.

Andoni Irazusta Garmendia

University of the Basque Country (UPV/EHU)

Traveling Salesperson ProblemNeural Combinatorial OptimizationCombinatorial OptimizationLocal SearchRL for Combinatorial Optimization
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Apr 7, 2026

Neural Network Pruning via QUBO Optimization

Neural network pruning can be formulated as a combinatorial optimization problem, yet most existing approaches rely on greedy heuristics that ignore complex interactions between filters. Formal optimization methods such as Quadratic Unconstrained Binary Optimization (QUBO) provide a principled alternative but have so far underperformed due to oversimplified objective formulations based on metrics like the L1-norm. In this work, we propose a unified Hybrid QUBO framework that bridges heuristic importance estimation with global combinatorial optimization. Our formulation integrates gradient-aware sensitivity metrics - specifically first-order Taylor and second-order Fisher information - into the linear term, while utilizing data-driven activation similarity in the quadratic term. This allows the QUBO objective to jointly capture individual filter relevance and inter-filter functional redundancy. We further introduce a dynamic capacity-driven search to strictly enforce target sparsity without distorting the optimization landscape. Finally, we employ a two-stage pipeline featuring a Tensor-Train (TT) Refinement stage - a gradient-free optimizer that fine-tunes the QUBO-derived solution directly against the true evaluation metric. Experiments on the SIDD image denoising dataset demonstrate that the proposed Hybrid QUBO significantly outperforms both greedy Taylor pruning and traditional L1-based QUBO, with TT Refinement providing further consistent gains at appropriate combinatorial scales. This highlights the potential of hybrid combinatorial formulations for robust, scalable, and interpretable neural network compression.

Osama Orabi, Artur Zagitov, Hadi Salloum +3

Department of Computer Science and Engineering, Innopolis University, 420500 Innopolis, Russia · Q Deep, Laboratory of Quantum Computing, 420502 Innopolis, Russia · Research Center of the Artificial Intelligence Institute, Innopolis University, 420500 Innopolis, Russia +2

Neural Network PruningQuadratic Unconstrained Binary OptimizationNeural Combinatorial Optimization
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Apr 6, 2026

Vehicle-as-Prompt: A Unified Deep Reinforcement Learning Framework for Heterogeneous Fleet Vehicle Routing Problem

Unlike traditional homogeneous routing problems, the Heterogeneous Fleet Vehicle Routing Problem (HFVRP) involves heterogeneous fixed costs, variable travel costs, and capacity constraints, rendering solution quality highly sensitive to vehicle selection. Furthermore, real-world logistics applications often impose additional complex constraints, markedly increasing computational complexity. However, most existing Deep Reinforcement Learning (DRL)-based methods are restricted to homogeneous scenarios, leading to suboptimal performance when applied to HFVRP and its complex variants. To bridge this gap, we investigate HFVRP under complex constraints and develop a unified DRL framework capable of solving the problem across various variant settings. We introduce the Vehicle-as-Prompt (VaP) mechanism, which formulates the problem as a single-stage autoregressive decision process. Building on this, we propose VaP-CSMV, a framework featuring a cross-semantic encoder and a multi-view decoder that effectively addresses various problem variants and captures the complex mapping relationships between vehicle heterogeneity and customer node attributes. Extensive experimental results demonstrate that VaP-CSMV significantly outperforms existing state-of-the-art DRL-based neural solvers and achieves competitive solution quality compared to traditional heuristic solvers, while reducing inference time to mere seconds. Furthermore, the framework exhibits strong zero-shot generalization capabilities on large-scale and previously unseen problem variants, while ablation studies validate the vital contribution of each component.

Shihong Huang, Shengjie Wang, Lei Gao +4

Polytechnic Institute, Zhejiang University, 310015, Hangzhou, China · S.F. Technology Co., Ltd., 518054, Shenzhen, China · School of Management, Zhejiang University, 310058, Hangzhou, China

Vehicle RoutingRL for Combinatorial OptimizationNeural Combinatorial OptimizationReinforcement Learning
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Mar 28, 2026

Aligning LLMs with Graph Neural Solvers for Combinatorial Optimization

Recent research has demonstrated the effectiveness of large language models (LLMs) in solving combinatorial optimization problems (COPs) by representing tasks and instances in natural language. However, purely language-based approaches struggle to accurately capture complex relational structures inherent in many COPs, rendering them less effective at addressing medium-sized or larger instances. To address these limitations, we propose AlignOPT, a novel approach that aligns LLMs with graph neural solvers to learn a more generalizable neural COP heuristic. Specifically, AlignOPT leverages the semantic understanding capabilities of LLMs to encode textual descriptions of COPs and their instances, while concurrently exploiting graph neural solvers to explicitly model the underlying graph structures of COP instances. Our approach facilitates a robust integration and alignment between linguistic semantics and structural representations, enabling more accurate and scalable COP solutions. Experimental results demonstrate that AlignOPT achieves state-of-the-art results across diverse COPs, underscoring its effectiveness in aligning semantic and structural representations. In particular, AlignOPT demonstrates strong generalization, effectively extending to previously unseen COP instances.

Shaodi Feng, Zhuoyi Lin, Yaoxin Wu +4

National Yang Ming Chiao Tung University · Institute for Infocomm Research (I2R), A*STAR · Eindhoven University of Technology Eindhoven +3

Combinatorial OptimizationNeural Combinatorial OptimizationLarge Language Model-Guided Optimization
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Mar 26, 2026

Offline Decision Transformers for Neural Combinatorial Optimization: Surpassing Heuristics on the Traveling Salesman Problem

Combinatorial optimization problems like the Traveling Salesman Problem are critical in industry yet NP-hard. Neural Combinatorial Optimization has shown promise, but its reliance on online reinforcement learning (RL) hampers deployment and underutilizes decades of algorithmic knowledge. We address these limitations by applying the offline RL framework, Decision Transformer, to learn superior strategies directly from datasets of heuristic solutions; it aims to not only to imitate but to synthesize and outperform them. Concretely, we (i) integrate a Pointer Network to handle the instance-dependent, variable action space of node selection, and (ii) employ expectile regression for optimistic conditioning of Return-to-Go, which is crucial for instances with widely varying optimal values. Experiments show that our method consistently produces higher-quality tours than the four classical heuristics it is trained on, demonstrating the potential of offline RL to unlock and exceed the performance embedded in existing domain knowledge.

Hironori Ohigashi, Shinichiro Hamada

Panasonic Connect Co., Ltd., Japan

Pointer NetworksNeural Combinatorial OptimizationTraveling Salesperson ProblemDecision TransformerOffline RL
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Mar 25, 2026

A Deep Reinforcement Learning (DRL)-Based Transformer Method for Solving the Open Shop Scheduling Problem

The open shop scheduling problem (OSSP) arises in many industrial and service settings but remains computationally challenging as the number of jobs and machines increases. While exact methods quickly become intractable, classical dispatching rules and metaheuristics may require substantial tuning to maintain solution quality at large scales. This study develops a Transformer-based scheduling policy for OSSP using an encoder-decoder architecture with multi-head attention. The model is trained on Taillard benchmark instances (4x4, 5x5, 7x7, and 10x10) using only the processing-time matrix as input and produces feasible schedules with makespans typically within 15-30% of best-known values. To evaluate scalability, the trained policy is applied without retraining to randomly generated instances from 40x40 to 100x100 and compared against classical dispatching heuristics, including SPT, LPT, MWKR, and EST. Across these large instances, the Transformer achieved average gaps of 12.89-15.12% relative to a standard lower bound. Compared with EST, the Transformer remained competitive, typically within a modest margin, while substantially outperforming SPT and LPT. These results indicate that a Transformer policy trained on small OSSP instances can generalize to substantially larger problems and provide a feature-light, learning-based alternative to classical dispatching rules.

Faezeh Ardali, Mwembezi A. Nyelele, Gerald M. Knapp

Louisiana State University, Baton Rouge, LA, USA · University of Minnesota Duluth, Duluth, MN, USA

Combinatorial OptimizationNeural Combinatorial OptimizationRL for Combinatorial OptimizationTransformer-Based RL
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Mar 24, 2026

A Learning Method with Gap-Aware Generation for Heterogeneous DAG Scheduling

Efficient scheduling of directed acyclic graphs (DAGs) is a core problem in large-scale data-intensive computing systems, where query plans, data-processing workloads, and computation graphs consist of dependent tasks competing for limited heterogeneous resource pools. In practice, achieving high-performance execution requires schedulers to adapt across environments with varying resource pools and task types, while generating schedules under tight runtime budgets. We propose WeCAN, an end-to-end reinforcement learning framework for heterogeneous DAG scheduling that addresses task-pool compatibility coefficients and generation-induced optimality gaps. It adopts a two-stage single-pass design: a single forward pass produces task-pool scores and global parameters, followed by a generation map that constructs schedules without repeated network calls. Its weighted cross-attention encoder models task-pool interactions gated by compatibility coefficients, and is size-agnostic to environment fluctuations. Moreover, widely used list-scheduling maps can incur generation-induced optimality gaps from restricted reachability. We introduce an order-space analysis that characterizes the reachable set of generation maps via feasible schedule orders, explains the mechanism behind generation-induced gaps, and yields sufficient conditions for gap elimination. Guided by these conditions, we design a skip-extended realization with an analytically parameterized decreasing skip rule, which enlarges the reachable order set while preserving single-pass efficiency. Experiments on real-world TPC-H query DAGs, resource-intensive workload datasets, and ML-compiler computation graphs demonstrate improved makespan over strong baselines, with inference time comparable to classical heuristics and faster than multi-round neural schedulers.

Ruisong Zhou, Haijun Zou, Li Zhou +2

School of Mathematical Science, Peking University, Beijing 100871, China · State Key Laboratory of Mathematical Sciences, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China · Theory Lab, Central Research Institute, 2012 Labs, Huawei Technologies Co., Ltd, China +1

Neural Combinatorial OptimizationRL for Combinatorial OptimizationHeterogeneous Computing
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Mar 19, 2026

CAMO: A Conditional Neural Solver for the Multi-objective Multiple Traveling Salesman Problem

Robotic systems often require a team of robots to collectively visit multiple targets while optimizing competing objectives, such as total travel cost and makespan. This setting can be formulated as the Multi-Objective Multiple Traveling Salesman Problem (MOMTSP). Although learning-based methods have shown strong performance on the single-agent TSP and multi-objective TSP variants, they rarely address the combined challenges of multi-agent coordination and multi-objective trade-offs, which introduce dual sources of complexity. To bridge this gap, we propose CAMO, a conditional neural solver for MOMTSP that generalizes across varying numbers of targets, agents, and preference vectors, and yields high-quality approximations to the Pareto front (PF). Specifically, CAMO consists of a conditional encoder to fuse preferences into instance representations, enabling explicit control over multi-objective trade-offs, and a collaborative decoder that coordinates all agents by alternating agent selection and node selection to construct multi-agent tours autoregressively. To further improve generalization, we train CAMO with a REINFORCE-based objective over a mixed distribution of problem sizes. Extensive experiments show that CAMO outperforms both neural and conventional heuristics, achieving a closer approximation of PFs. In addition, ablation results validate the contributions of CAMO's key components, and real-world tests on a mobile robot platform demonstrate its practical applicability.

Fengxiaoxiao Li, Xiao Mao, Mingfeng Fan +4

National University of Singapore, Singapore. · China Unicom Shenzhen Branch, Shenzhen, China. · Central South University, Changsha, China.

Neural Combinatorial OptimizationMulti-Objective OptimizationMulti-Robot Coordination
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Mar 16, 2026

Towards Solving Polynomial-Objective Integer Programming with Hypergraph Neural Networks

Complex real-world optimization problems often involve both discrete decisions and nonlinear relationships between variables. Many such problems can be modeled as polynomial-objective integer programs, encompassing cases with quadratic and higher-degree variable interactions. Nonlinearity makes them more challenging than their linear counterparts. In this paper, we propose a hypergraph neural network (HNN) based method to solve polynomial-objective integer programming (POIP). Besides presenting a high-degree-term-aware hypergraph representation to capture both high-degree information and variable-constraint interdependencies, we also propose a hypergraph neural network, which integrates convolution between variables and high-degree terms alongside convolution between variables and constraints, to predict solution values. Finally, a search process initialized from the predicted solutions is performed to further refine the results. Comprehensive experiments across a range of benchmarks demonstrate that our method consistently outperforms both existing learning-based approaches and state-of-the-art solvers, delivering superior solution quality with favorable efficiency. Note that our experiments involve both polynomial objectives and constraints, demonstrating our HNN's versatility for general POIP problems and highlighting its advancement over the existing literature.

Minshuo Li, Yaoxin Wu, Pavel Troubil +2

Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, the Netherlands · Eindhoven Artificial Intelligence Systems Institute, Eindhoven University of Technology, Eindhoven, the Netherlands · Department of Industrial Engineering and Innovation Sciences, Eindhoven University of Technology, Eindhoven, the Netherlands +1

Combinatorial OptimizationNeural Combinatorial OptimizationHypergraph Neural Networks
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Mar 12, 2026

UniHetCO: A Unified Heterogeneous Representation for Multi-Problem Learning in Unsupervised Neural Combinatorial Optimization

Unsupervised neural combinatorial optimization (NCO) offers an appealing alternative to supervised approaches by training learning-based solvers without ground-truth solutions, directly minimizing instance objectives and constraint violations. Yet for graph node subset-selection problems (e.g., Maximum Clique and Maximum Independent Set), existing unsupervised methods are typically specialized to a single problem class and rely on problem-specific surrogate losses, which hinders learning across classes within a unified framework. In this work, we propose UniHetCO, a unified heterogeneous graph representation for constrained quadratic programming-based combinatorial optimization that encodes problem structure, objective terms, and linear constraints in a single input. This formulation enables training a single model across multiple problem classes with a unified label-free objective. To improve stability under multi-problem learning, we employ a gradient-norm-based dynamic weighting scheme that alleviates gradient imbalance among classes. Experiments on multiple datasets and four constrained problem classes demonstrate competitive performance with state-of-the-art unsupervised NCO baselines, strong cross-problem adaptation potential, and effective warm starts for a commercial classical solver under tight time limits.

Kien X. Nguyen, Ilya Safro

Department of Computer and Information Sciences, University of Delaware, Newark DE 19716, USA

Combinatorial OptimizationNeural Combinatorial OptimizationConstrained OptimizationHeterogeneous GNNs
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Mar 9, 2026

Cluster-Aware Attention-Based Deep Reinforcement Learning for Pickup and Delivery Problems

The Pickup and Delivery Problem (PDP) is a fundamental and challenging variant of the Vehicle Routing Problem, characterized by tightly coupled pickup--delivery pairs, precedence constraints, and spatial layouts that often exhibit clustering. Existing deep reinforcement learning (DRL) approaches either model all nodes on a flat graph, relying on implicit learning to enforce constraints, or achieve strong performance through inference-time collaborative search at the cost of substantial latency. In this paper, we propose CAADRL (Cluster-Aware Attention-based Deep Reinforcement Learning), a DRL framework that explicitly exploits the multi-scale structure of PDP instances via cluster-aware encoding and hierarchical decoding. The encoder builds on a Transformer and combines global self-attention with intra-cluster attention over depot, pickup, and delivery nodes, producing embeddings that are both globally informative and locally role-aware. Based on these embeddings, we introduce a Dynamic Dual-Decoder with a learnable gate that balances intra-cluster routing and inter-cluster transitions at each step. The policy is trained end-to-end with a POMO-style policy gradient scheme using multiple symmetric rollouts per instance. Experiments on synthetic clustered and uniform PDP benchmarks show that CAADRL matches or improves upon strong state-of-the-art baselines on clustered instances and remains highly competitive on uniform instances, particularly as problem size increases. Crucially, our method achieves these results with substantially lower inference time than neural collaborative-search baselines, suggesting that explicitly modeling cluster structure provides an effective and efficient inductive bias for neural PDP solvers.

Wentao Wang, Lifeng Han, Guangyu Zou

Leicester International Institute, Dalian University of Technology, Panjin, 124221, Liaoning, China. · Department of Electronic and Information Technology, Dalian University of Technology, Panjin, 124221, Liaoning, China.

Vehicle RoutingRL for Combinatorial OptimizationNeural Combinatorial OptimizationCombinatorial OptimizationAttention Mechanisms
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Mar 8, 2026

Constraints Matrix Diffusion based Generative Neural Solver for Vehicle Routing Problems

Over the past decade, neural network solvers powered by generative artificial intelligence have garnered significant attention in the domain of vehicle routing problems (VRPs), owing to their exceptional computational efficiency and superior reasoning capabilities. In particular, autoregressive solvers integrated with reinforcement learning have emerged as a prominent trend. However, much of the existing work emphasizes large-scale generalization of neural approaches while neglecting the limited robustness of attention-based methods across heterogeneous distributions of problem parameters. Their improvements over heuristic search remain largely restricted to hand-curated, fixed-distribution benchmarks. Furthermore, these architectures tend to degrade significantly when node representations are highly similar or when tasks involve long decision horizons. To address the aforementioned limitations, we propose a novel fusion neural network framework that employs a discrete noise graph diffusion model to learn the underlying constraints of vehicle routing problems and generate a constraint assignment matrix. This matrix is subsequently integrated adaptively into the feature representation learning and decision process of the autoregressive solver, serving as a graph structure mask that facilitates the formation of solutions characterized by both global vision and local feature integration. To the best of our knowledge, this work represents the first comprehensive experimental investigation of neural network model solvers across a 378-combinatorial space spanning four distinct dimensions within the CVRPlib public dataset. Extensive experimental evaluations demonstrate that our proposed fusion model effectively captures and leverages problem constraints, achieving state-of-the-art performance across multiple benchmark datasets.

Zhenwei Wang, Tiehua Zhang, Ning Xue +3

School of Computer Science, University of Nottingham Ningbo China, Ningbo, 315100, China · School of Computer Science and Technology, Tongji University, Shanghai, 200092, China · Computational Optimisation and Learning Lab, School of Computer Science, University of Nottingham, NG8 1BB Nottingham, U.K. +1

Vehicle RoutingNeural Combinatorial Optimization
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Mar 6, 2026

Learning to Solve Orienteering Problem with Time Windows and Variable Profits

The orienteering problem with time windows and variable profits (OPTWVP) is common in many real-world applications and involves continuous time variables. Current approaches fail to develop an efficient solver for this orienteering problem variant with discrete and continuous variables. In this paper, we propose a learning-based two-stage DEcoupled discrete-Continuous optimization with Service-time-guided Trajectory (DeCoST), which aims to effectively decouple the discrete and continuous decision variables in the OPTWVP problem, while enabling efficient and learnable coordination between them. In the first stage, a parallel decoding structure is employed to predict the path and the initial service time allocation. The second stage optimizes the service times through a linear programming (LP) formulation and provides a long-horizon learning of structure estimation. We rigorously prove the global optimality of the second-stage solution. Experiments on OPTWVP instances demonstrate that DeCoST outperforms both state-of-the-art constructive solvers and the latest meta-heuristic algorithms in terms of solution quality and computational efficiency, achieving up to 6.6x inference speedup on instances with fewer than 500 nodes. Moreover, the proposed framework is compatible with various constructive solvers and consistently enhances the solution quality for OPTWVP.

Songqun Gao, Zanxi Ruan, Patrick Floor +3

Department of Industrial Engineering, Università di Trento, Trento, Italy · Interdepartmental Robotics Labs (IDRA), Trento, Italy · Department of Engineering for Innovation Medicine, Università di Verona, Verona, Italy +2

Learning to OptimizeCombinatorial OptimizationNeural Combinatorial Optimization
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Mar 3, 2026

RADAR: Learning to Route with Asymmetry-aware DistAnce Representations

Recent neural solvers have achieved strong performance on vehicle routing problems (VRPs), yet they mainly assume symmetric Euclidean distances, restricting applicability to real-world scenarios. A core challenge is encoding the relational features in asymmetric distance matrices of VRPs. Early attempts directly encoded these matrices but often failed to produce compact embeddings and generalized poorly at scale. In this paper, we propose RADAR, a scalable neural framework that augments existing neural VRP solvers with the ability to handle asymmetric inputs. RADAR addresses asymmetry from both static and dynamic perspectives. It leverages Singular Value Decomposition (SVD) on the asymmetric distance matrix to initialize compact and generalizable embeddings that inherently encode the static asymmetry in the inbound and outbound costs of each node. To further model dynamic asymmetry in embedding interactions during encoding, it replaces the standard softmax with Sinkhorn normalization that imposes joint row and column distance awareness in attention weights. Extensive experiments on synthetic and real-world benchmarks across various VRPs show that RADAR outperforms strong baselines on both in-distribution and out-of-distribution instances, demonstrating robust generalization and superior performance in solving asymmetric VRPs.

Hang Yi, Ziwei Huang, Yining Ma +1

Singapore Management University · Massachusetts Institute of Technology

Vehicle RoutingNeural Combinatorial Optimization
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Mar 2, 2026

Can Computational Reducibility Lead to Transferable Models for Graph Combinatorial Optimization?

A key challenge in developing unified neural solvers for combinatorial optimization (CO) is the efficient generalization of models from a given set of tasks to new tasks unseen during initial training. To address this, we first establish a new GNN encoder, which uses a GCON module as a form of expressive message passing together with energy-based unsupervised loss functions. This model achieves highly competitive performance across multiple CO tasks when trained individually on each task. We then leverage knowledge from the computational reducibility literature to propose pretraining and fine-tuning strategies that transfer effectively (a) between MVC, MIS and MaxClique, and (b) in a multi-task learning setting that additionally incorporates MaxCut, MDS and graph coloring. Additionally, in a leave-one-out, multi-task learning setting, we observe that pretraining on all but one task almost always leads to faster convergence on the remaining task when fine-tuning, while avoiding negative transfer. Our findings indicate that learning common representations across multiple graph CO problems is viable through the use of expressive message passing coupled with pretraining strategies that are informed by the polynomial reducibility literature, thereby taking an important step towards enabling the development of foundational models for neural CO. We provide an open source implementation of our work at https://github.com/semihcanturk/COPT-MT.

Semih Cantürk, Thomas Sabourin, Frederik Wenkel +2

Dept. of Computer Science and Operations Research, Université de Montréal · Mila – Quebec AI Institute · Dept. of Mathematics & Statistics, Université de Montréal +2

Neural Combinatorial OptimizationTransfer LearningGraph Neural NetworksMulti-Task Optimization
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Feb 28, 2026

Efficient Decoder Scaling Strategy for Neural Routing Solvers

Construction-based neural routing solvers, typically composed of an encoder and a decoder, have emerged as a promising approach for solving vehicle routing problems. While recent studies suggest that shifting parameters from the encoder to the decoder enhances performance, most works restrict the decoder size to 1-3M parameters, leaving the effects of scaling largely unexplored. To address this gap, we conduct a systematic study comparing two distinct strategies: scaling depth versus scaling width. We synthesize these strategies to construct a suite of 12 model configurations, spanning a parameter range from 1M to ~150M, and extensively evaluate their scaling behaviors across three critical dimensions: parameter efficiency, data efficiency, and compute efficiency. Our empirical results reveal that parameter count is insufficient to accurately predict the model performance, highlighting the critical and distinct roles of model depth (layer count) and width (embedding dimension). Crucially, we demonstrate that scaling depth yields superior performance gains to scaling width. Based on these findings, we provide and experimentally validate a set of design principles for the efficient allocation of parameters and compute resources to enhance the model performance.

Qing Luo, Fu Luo, Ke Li +1

Guangdong Provincial Key Laboratory of Fully Actuated System Control Theory and Technology, School of Automation and Intelligent Manufacturing, Southern University of Science and Technology, Shenzhen, China. · Department of Computer Science, City University of Hong Kong, Hong Kong SAR, China.

Neural Combinatorial OptimizationNeural Scaling LawsVehicle Routing
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Feb 25, 2026

Survey on Neural Routing Solvers

Neural routing solvers (NRSs) that leverage deep learning to tackle vehicle routing problems have demonstrated notable potential for practical applications. By learning implicit heuristic rules from data, NRSs replace the handcrafted counterparts in classic heuristic frameworks, thereby reducing reliance on costly manual design and trial-and-error adjustments. This survey makes two main contributions: (1) The heuristic nature of NRSs is highlighted, and existing NRSs are reviewed from the perspective of heuristics. A hierarchical taxonomy based on heuristic principles is further introduced. (2) A generalization-focused evaluation pipeline is proposed to address limitations of the conventional pipeline. Comparative benchmarking of representative NRSs across both pipelines uncovers a series of previously unreported gaps in current research.

Yunpeng Ba, Xi Lin, Changliang Zhou +7

Guangdong Provincial Key Laboratory of Fully Actuated System Control Theory and Technology, School of Automation and Intelligent Manufacturing, Southern University of Science and Technology, Shenzhen, China. · the School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China. · the Institute of Big Data Science and Industry, Shanxi University, Taiyuan, China. +1

Neural Combinatorial OptimizationVehicle RoutingAutomated Heuristic Design
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Feb 25, 2026

Mamba Meets Scheduling: Learning to Solve Flexible Job Shop Scheduling with Efficient Sequence Modeling

The Flexible Job Shop Problem (FJSP) is a well-studied combinatorial optimization problem with extensive applications for manufacturing and production scheduling. It involves assigning jobs to various machines to optimize criteria, such as minimizing total completion time. Current learning-based methods in this domain often rely on localized feature extraction models, limiting their capacity to capture overarching dependencies spanning operations and machines. This paper introduces an innovative architecture that harnesses Mamba, a state-space model with linear computational complexity, to facilitate comprehensive sequence modeling tailored for FJSP. In contrast to prevalent graph-attention-based frameworks that are computationally intensive for FJSP, we show our model is more efficient. Specifically, the proposed model possesses an encoder and a decoder. The encoder incorporates a dual Mamba block to extract operation and machine features separately. Additionally, we introduce an efficient cross-attention decoder to learn interactive embeddings of operations and machines. Our experimental results demonstrate that our method achieves faster solving speed and surpasses the performance of state-of-the-art learning-based methods for FJSP across various benchmarks.

Zhi Cao, Cong Zhang, Yaoxin Wu +2

Dalian University of Technology, Dalian, China · Nanyang Technological University, Singapore · Eindhoven University of Technology, Eindhoven, the Netherlands

Flexible Job Shop SchedulingJob Shop SchedulingCombinatorial OptimizationNeural Combinatorial Optimization
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