Reinforcement Learning Enhanced LLM Agents for Complex Vehicle Routing Problems

Vehicle Routing Problems (VRPs) are fundamental combinatorial optimization problems with widespread applications in various scenarios. The advanced optimization solvers can effectively solve such problems. However, modeling complex VRP variants for solvers often requires substantial domain expertise, which limits the accessibility of advanced optimization technologies. In this paper, we propose Reinforcement Learning Enhanced LLMAgents(RLEA), a multi-agent framework designed to automate the modeling of complex VRPs. RLEA introduces a lightweight neural Planner trained with Soft Q-learning to efficiently orchestrate the actions of LLM-based agents. In addition, we equip the system with an evolutionary memory module and retrieval-augmented generation, enabling the agent to leverage both accumulated experience and external solver knowledge during program generation and refinement for solving VRPs. We evaluated 48 distinct VRP variants across various solvers. The experimental results demonstrate that RLEA outperforms the previous state-of-the-ar method, achieving a 16.67% higher success rate while significantly reducing runtime errors. These results validate that integrating reinforcement learning with LLM-based reasoning is highly effective for automated optimization modeling. The appendix is available at: https://doi.org/10.5281/zenodo.19134435.

GeoPAR: Large-Scale Multi-Agent Combinatorial Optimization with Geometry-Guided Parallel Autoregressive Learning

Multi-agent combinatorial optimization problems are notoriously challenging due to their NP-hard nature. Recent parallel autoregressive neural solvers improve inference efficiency by allowing agents to make decisions simultaneously, but their performance often degrades on large-scale instances. This is largely attributable to weak modeling of local geometric structures and the fact that conflicting task selections are handled only after action generation. To address these limitations, we propose GeoPAR, a geometry-guided parallel autoregressive reinforcement learning framework for scalable multi-agent combinatorial optimization. GeoPAR integrates three key components: (1) a projection-window sparse geometry mechanism that builds lightweight local candidate neighborhoods through multi-directional projections, (2) sparse edge-biased attention that injects these geometric relations into node representations, and (3) cache-guided conflict-aware assignment that reuses the geometric cache during decoding to suppress duplicate selections of exclusive tasks. Experiments on heterogeneous vehicle routing and open multi-depot pickup-and-delivery problems show that GeoPAR improves large-scale zero-shot generalization while substantially reducing rollout steps and maintaining efficient inference.

A hybrid quantum-classical neural network for learning to route

This work studies hybrid quantum-classical neural networks for learning routing heuristics. Specifically, this paper asks whether small quantum neural networks can replace parameter-heavy modules inside a competitive attention-based routing model while maintaining solution quality. For the capacitated vehicle routing problem, encoder feed-forward replacement emerges as the most promising design: it reduces the number of model parameters by 56.6% while keeping the hybrid model close to the classical neural baseline at small and medium instance sizes, although the gap grows for larger instances. This work also compares to classical routing algorithms, which remain highly competitive and often superior on the fixed Euclidean test sets. Our results therefore do not indicate quantum advantage or solver dominance, but identify encoder feed-forward replacement as a viable hybrid-module compression strategy for neural combinatorial optimization.

Improving Cross-Problem Vehicle Routing with Locally Augmented Preferences and Representation Disentanglement

Multi-task vehicle routing problem (VRP) solvers seek to handle multiple VRP variants within a single unified model, avoiding the need to train a separate model for every variant. In spite of recent progress, current approaches remain limited on two fronts. On the training side, reinforcement learning suffers from reward-scale disparities and shrinking advantage signals as policies improve, whereas preference optimization stagnates once sampled tours become near-identical and thus fundamentally limited by the quality of the policy's own generated solutions, leaving both paradigms with weak supervision as training progresses. On the architecture side, existing fully shared encoders entangle constraint-dependent representations across heterogeneous variants, which limits generalization. We address these gaps with two model-agnostic contributions. First, we propose Preference Optimization with Locally Augmented Refinement (POLAR), a novel training algorithm that applies a local search refinement pass to the best decoded tour before forming preference pairs, yielding much more informative pairwise margins. Second, a Progressive Layered Extraction (PLE) encoder routes each encoder layer through one shared expert and a set of task-specific experts via a gating mechanism, progressively separating common routing structure from constraint-specific encodings. Through extensive experiments on various VRP variants, we show that POLAR and PLE together elevate the current state-of-the-art among neural multi-task solvers. We reduce the average gap to reference solutions by 21.3% relative to the strongest published baseline on 16 in-distribution variants, and outperform prior neural methods on 27 out of 32 unseen variants. Ablation studies confirm the efficacy of each contribution, showing that both improve cross-problem generalization across multiple backbone model architectures.

Smart routes: a system for development and comparison of algorithms for solving vehicle routing problems with realistic constraints

The problem of route optimization with realistic constraints is becoming extremely relevant in the face of global urban population growth. While we are aware of approaches that theoretically provide an exact optimal solution, their application becomes challenging as the problem size increases because of exponential complexity. We investigate the Capacitated Vehicle Routing Problem with Time Windows (CVRPTW) and compare solutions obtaining by exact solver SCIP with heuristic algorithms such as LKH, 2-OPT, 3-OPT, the ORTools framework, and the deep learning model JAMPR. We demonstrate that for problem of size 50 deep learning and classical heuristic solutions became close to SCIP exact solution but requires less time. Additionally for problems with size 100, SCIP exact methods around 13 times slower that neural and classical heuristics with the same route cost and on around 50% worse for the first feasible solution on the same time. To conduct experiments, we developed the Smart Routes platform for solving route optimization problems, which includes exact, heuristic, and deep learning models, and facilitates convenient integration of custom algorithms and datasets.

Dynamic Multi-Depot Vehicle Routing with Online Requests: Event-Driven Transformer--DRL and Rolling-Horizon Benchmarking

This paper presents an event-driven learning and benchmarking framework for the Dynamic Multi-Depot Vehicle Routing Problem with progressively revealed requests and evolving vehicle states. Masked MLP and Transformer policies are trained through behavior cloning and proximal policy optimization. Deterministic feasibility masking prevents invalid vehicle--request assignments, while fixed-prefix/flexible-suffix route commitments protect completed, active, and near-term decisions and separately measure vehicle reassignment and resequencing. The learned policies are compared with dynamic insertion heuristics and time-limited rolling-horizon optimization. In a 20-scenario policy benchmark, all methods completed every request without invalid actions, but nearest feasible achieved the lowest mean objective and outperformed the learned policies in routing quality, waiting time, stability, makespan, and runtime. Across five independent training runs, PPO had little average effect on the MLP and improved the Transformer on average, although with greater seed variability. Under the common protocol, nearest feasible achieved the lowest combined objective and route disruption, whereas rolling horizon achieved the lowest waiting times and makespan at substantially higher computational cost. The learned policies retained millisecond-level decisions and transferred to instances with up to 80 requests without retraining, but did not outperform the strongest heuristic. No single method was best across routing efficiency, service responsiveness, stability, and online computation.

Vehicle routing problem using deep reinforcement learning - A case study about truck planning in the industry

As an important component of the supply chain industry, transportation has experienced rapid development in the past decade with the assistance of digital platforms and intelligent algorithms. Within the field of transportation research, Vehicle Routing Problem (VRP) has remained a persistent and enduring challenge. In the realm of management science, experts, and scholars from both the industrial and academic sectors have continuously explored optimization models and algorithms to effectively address routing problems, from the classical Traveling Salesman Problem to the more general Vehicle Routing Problem. These models and algorithms are applied in real-world industrial scenarios to achieve cost optimization and reduce carbon footprints. However, due to the complexity of real-world problems, numerous specific constraints are often added, and challenges such as information opacity, uncertainty, and irrational human behavior may arise. Therefore, deploying and optimizing mathematical models for VRP in practical scenarios while maintaining optimal results poses numerous challenges. This paper discusses and provides solutions for three different logistic use cases involving external truck network design. Through these industrial case study, the paper introduces how deep reinforcement learning-based vehicle routing optimization has been implemented. As a result, it can be observed that the routes optimized by reinforcement learning agent have over 10% total cost compared to baseline results. Furthermore, the paper proposes that in future research, DRL algorithms for vehicle routing problems could be generalized into more variations of VRP.

KAYROS: An Anytime and Exact Open-Source Solver for Duration-Minimization Time-Dependent Vehicle Routing. A Technical Report and a Case Study in Human-AI Engineering

Time-dependent routing recognizes that the same journey can take a different time depending on when it begins. Under duration minimization, even the departure time of each vehicle becomes a decision. Exact methods for this setting exist in the literature, but researchers and practitioners have lacked a ready-to-use open solver that combines rich piecewise-linear travel times, early feasible solutions and optimality claims. KAYROS fills this gap with two modes on one checker-consistent engine: an Iterated Local Search that streams improving solutions and a Branch-Price-and-Cut method that can issue computational optimality certificates under explicit arithmetic and search assumptions. It installs with one command and has no proprietary dependency. The public MAMUT-routing store currently contains 704 KAYROS certificates under a four-solve publication protocol, which has also led to the retraction and repair of invalid earlier claims. The report presents two complementary benchmark contributions to MAMUT-routing. The first integrates Blauth2024, a benchmark from the literature whose travel times derive from measured Uber speeds, for which KAYROS provides new best-known solutions on all 40 instances. The second proposes Poryos2026, a new benchmark of 1,080 paired static and time-dependent instances built from OpenStreetMap road networks and controlled synthetic traffic. Finally, the report describes the intensive human-AI collaboration behind this work and the verification practices that kept its outputs independently verifiable.

LaT: LLM-as-Trainer for Multi-Task Vehicle Routing Solvers

Multi-task neural solvers aim to handle multiple Vehicle Routing Problem (VRP) variants within a unified model, avoiding separate training for each constraint combination. However, VRP variants differ in optimization difficulty, while existing methods lack stage-wise feedback on their training status, making the model biased to some specific variants. Although meta-learning can support adaptive training, it typically requires bi-level optimization and additional gradient updates, increasing computational cost. To address this limitation, we propose LLM-as-Trainer (LaT), a plug-and-play training paradigm that uses a pretrained large language model as an external trainer. LaT periodically analyzes cross-task validation metrics to generate a stage-wise guidance vector. This vector is combined with the current task's constraint vector and injected into each encoder layer, providing the neural solver with additional training information during subsequent policy optimization. Experiments on 16 VRP variants show that LaT improves the solution quality of several state-of-the-art multi-task neural solvers on both trained and unseen variants, supporting the effectiveness and generality of the proposed training paradigm.

A Deep Reinforcement Learning Algorithm for the Vehicle Routing Problem with Stochastic Demands and Outsourcing

We introduce the vehicle routing problem with stochastic demands and outsourcing options (VRP-SDO), in which a logistics service provider partitions customer requests into customers outsourced to a common carrier and customers committed to its fixed fleet. The latter induces a vehicle routing problem with stochastic demands (VRP-SD), solved dynamically. Demands are revealed upon visit; residual demand may be served by other vehicles or after restocking at the depot. Work beyond the regular shift incurs overtime costs, and the unit outsourcing cost decreases with the expected outsourced demand. The objective is to minimize expected travel, overtime, and outsourcing costs. We propose an iterative two-level methodology whose first level partitions customers into committed and outsourced subsets, while the second level estimates the expected VRP-SD routing cost. To avoid solving this problem from scratch at every iteration, we learn an offline routing policy that estimates costs almost instantly for any committed subset. An iterated local search establishes the first-level partitions. We formulate the second level as a Markov decision process and solve it with a deep Q-network whose state is represented by a graph attention network aggregating customer and vehicle information by relevance to the acting vehicle. Trained offline on instances with variable customer cardinality and locations, the policy applies to any daily customer realization; online fine-tuning improves the cost approximation. Experiments show that our policy reduces routing costs by 19.6% relative to a state-of-the-art method and by at least 29.6% over classical heuristics. Our overall algorithm saves 13.7% on average over the version without the attention-based representation and generates high-quality decisions within minutes, whereas benchmarks without an offline-trained estimator require over an hour.

Learned Pairwise Deep Dual-Optimal Inequalities for Stabilizing Column Generation

Column generation (CG) is central to many large-scale optimization algorithms, including branch-price-and-cut methods for vehicle routing problems, but unstable dual solutions can substantially slow its convergence. Existing deep dual-optimal inequalities can reduce this instability by restricting the dual space. Their construction, however, typically relies on problem-specific exchange arguments that are difficult to establish for routing problems with capacity limits, time windows, and other resource constraints. We introduce learned pairwise deep dual-optimal inequalities (L-PDDOIs), a learning framework that predicts pairwise orderings between dual variables and incorporates their primal counterparts directly into the master problem. To construct training labels, the framework samples optimal dual solutions and selects pairwise order relations that hold simultaneously on a sufficiently large common subset of the samples. A classifier then assigns a score to each candidate relation. Because conflicts and redundancies among the predicted relations can impair performance, graph-based postprocessing filters and compresses the candidate set before deployment. We further introduce a recovery procedure that selectively relaxes learned inequalities and provides a certificate when the baseline CG bound has been restored. On the main test sets for the capacitated vehicle routing problem and the vehicle routing problem with time windows, direct deployment of L-PDDOIs reduces the geometric mean root CG time by 89.7% and 93.9%, respectively, while incurring mean bound losses of only 1.3% and 0.5%. The recovery procedure retains corresponding time reductions of 54.8% and 83.1%, respectively, while guaranteeing no loss in the CG bound.

Reward-Density Heuristic for Dynamic Multi-Vehicle Routing: Performance and Computational Efficiency

The Vehicle Routing Problem (VRP) and its variants represent some of the most practically consequential optimization challenges in modern logistics and urban mobility. In this study, we address a dynamic, online variant combining elements of the VRP and the Orienteering Problem (OP), in which a fleet of vehicles must maximise cumulative reward collected within a fixed time horizon while continuously replanning as new tasks arrive. We propose and evaluate a reward-density heuristic for dynamic multi-vehicle assignment, referred to as the Efficiency heuristic. We evaluate this formulation across two application domains: autonomous drone task allocation and urban taxi dispatch, across multiple fleet sizes and task scales. The proposed method is compared with four classical construction heuristics and three metaheuristic algorithms (Adaptive Large Neighbourhood Search, Genetic Algorithm, and Simulated Annealing), all evaluated under identical conditions. Across all tested configurations, the Efficiency heuristic matches the solution quality of the best metaheuristic algorithms while requiring two to three orders of magnitude less planning time, establishing Pareto dominance over all competing methods on the reward-versus-compute frontier. These findings suggest a practical design principle for real-time allocation and dispatch systems: in dynamic, time-constrained routing environments, carefully designed greedy heuristics can match the output of sophisticated search procedures at a fraction of the computational cost, making them preferable for online deployment.

Two Black Boxes, One Solver: Encoder Probing and Decoder Attribution for Neural Multi-Attribute VRP under Hard-Mask and Recourse Decoders

Neural autoregressive solvers for the Multi-Attribute Vehicle Routing Problem (MAVRP) reach competitive cost but offer no per-step justification, a problem when dispatchers must validate, accept, or compare them. We open two complementary black boxes in one protocol. On the encoder side, linear probes, spontaneous-organization metrics, rank-based richness measures, and discovered-direction analyses with intervention validation characterize how the latent represents constraint families at the graph, node, and edge level. On the decoder side, three attribution methods (gradient, integrated gradients, DeepLIFT) feed three reading angles: abductive, contrastive against the best feasible alternative, and counterfactual (smallest input change that switches the action or restores feasibility). Explanations are scored on fidelity, concentration, stability, sanity, and actionability. Across six variants combining three encoders (Attention baseline, Unimp, UnimpMoe) with two decoders (Hard-Mask, Recourse), we find that graph inductive bias improves both representational predictability and decoder sanity, that the Mixture-of-Experts encoder represents constraints in a distributed rather than axis-aligned way, and that the Recourse training regime, not merely its softer mask, produces policies that represent infeasibility usefully, exposing make-feasible counterfactuals that Hard-Mask policies fail to produce even when fed infeasible alternatives externally.

Robust Feasible Route Construction through Collaborative Partition Optimization

Large-scale Capacitated Vehicle Routing Problems (CVRPs) are commonly solved by partitioning customers into smaller routing problems that can be optimized independently. While this substantially reduces computational complexity, independently constructed routing solutions may leave some customer demand unserved even when sufficient resources exist elsewhere in the fleet. We present Collaborative Routing Constructors (CoRC), a routing framework that enables independently solved subproblems to exchange customers and vehicles during optimization rather than relying solely on a fixed partition or a subsequent global re-optimization stage. Computational experiments on AGS benchmark instances and synthetic instances containing up to 200,000 customers compare CoRC against independent routing, post-routing global re-optimization, and state-of-the-art, end-to-end routing frameworks. Across all evaluated partitioning strategies, CoRC consistently constructs feasible routing solutions where competing partition-based methods do not. Furthermore, it remains effective on problem instances for which the evaluated end-to-end routing frameworks did not produce solutions under the same computational budget. These results demonstrate that collaboration between routing subproblems provides a robust and scalable approach for feasible large-scale route construction.

Adaptive Cluster-First Route-Second Decomposition for Industrial-Scale Vehicle Routing

Large-scale capacitated vehicle routing problems (CVRPs) are commonly addressed using cluster-first route-second (CFRS) approaches that split a routing instance into smaller, computationally tractable subproblems. Existing splitting methods typically rely on fixed partitioning rules, predefined optimization objectives, or learned policies, which may perform inconsistently across instances exhibiting different spatial, demand, and operational characteristics. In this work, we propose an adaptive CFRS system that formulates a decomposition procedure as an iterative decision-making process. Motivated by the recent success of large language models (LLMs) in reasoning and tool selection, the system employs an LLM as a high-level decision maker that analyzes the evolving decomposition state and selectively applies further clustering, balancing, and refinement operators. The proposed algorithm jointly partitions customers and vehicles, enabling capacity-aware clustering while adapting partitioning decisions to the characteristics of each problem. We evaluate the approach on synthetic and benchmark-derived CVRP instances containing up to 500,000 customers. Experimental results demonstrate competitive performance on benchmark-scale instances while exhibiting improved scalability and robust routing quality on substantially larger problems. These results highlight the potential of adaptive, LLM-guided decision support as a practical approach for industrial-scale vehicle routing and large-scale logistics planning.

Optimizing Nursing Care Taxi Dispatch Leveraging Integer Linear Programming Solvers and Machine Learning

In this paper, we formulate a new vehicle dispatch optimization problem, called Nursing Care Taxi Dispatch, as a variant of the Vehicle Routing Problem, considering constraints related to wheelchair use, user compatibility, pick-up and drop-off times, and vehicle limitations. Previous neural-based methods for Vehicle Routing Problems have typically addressed a few simple constraints, while our new problem involves multiple complex constraints, resulting in having fewer destinations to select. This complexity makes it more difficult to obtain solutions that allow all nodes to be visited with a limited number of vehicles. To balance low violation rate, computational efficiency, and solution quality, we propose a supervised machine learning approach based on the Transformer architecture. We first obtain a set of high-quality solutions using an integer linear programming solver for given inputs and then train our learning model through supervised learning. Additionally, we introduce the post-processing of the paths generated by the learning model, ensuring that all constraints are satisfied. We compared each instance's objective function value (operating time), execution time, and constraint violation rate across different methods: our proposed method and some existing methods including integer linear programming and machine learning-based methods, using real-world facility data. Our method successfully produced balanced solutions regarding operating time, execution time, and constraint violation rate. Notably, we observed a decrease in the operating time for all problem sizes and regions, while keeping constraint violations to a minimum compared to existing methods. Especially, the decrease reached up to 8% for problem sizes with fewer than 30 users.

RACL: Reasoning-Agent Control Layers for Continuous Metaheuristic Learning

This paper introduces RACL, a Reasoning-Agent Control Layer for metaheuristics. RACL places a reasoning agent above an existing optimizer. The agent does not replace the optimizer and does not modify business constraints. Instead, it controls the optimizer's internal search behavior by observing operational memory, reasoning over past behavior, formulating bounded hypotheses, testing interventions, evaluating outcomes, applying guardrails, consolidating useful policies and explaining its decisions. The experiment uses vehicle routing as a testbed, but the contribution is not a new routing solver, a particular ALNS configuration or a specific set of routing rules. The contribution is the RACL method: a way for a reasoning agent to discover, validate, consolidate and explain algorithmic control rules for a metaheuristic. In the current experimental setting, RACL improves or ties the Operational Memory Policy in 21 of 21 feasible cases and improves or ties a non-reasoning Stagnation-Triggered Policy in 18 of 21 feasible cases, with an average RACL vs STP cost delta of -0.641%. In the Sevilla-9/10 runtime sample, RACL improves average cost by -8.337% versus Fixed and -1.605% versus STP without showing material computational overhead. During the proof-of-concept, Codex was used as an in-the-loop reasoning agent observing executions, interpreting logs and proposing live bounded interventions. The policy proxy was later used only to make quantitative evaluation reproducible.

Vision-Assisted Foundation Model for Solving Multi-Task Vehicle Routing Problems

Multi-task vehicle routing problems play a critical role in enhancing efficiency across various industries and service sectors. These problems consist of multiple variants that optimize routing costs while meeting diverse customer constraints. Existing multi-task VRP solvers solely utilize a graph-based modality, limiting their ability to address variants with multiple constraints. As a format to represent complex semantics, vision modality shows great potential for encoding diverse VRP constraints. This motivates us to learn patch-level semantics from the vision images, and then integrate them into a graph-based model to solve various VRP variants simultaneously. However, directly applying this approach to multi-task VRPs presents three challenges: 1) existing VRP images lack constraint representations, which are essential for multi-task VRPs, 2) the fixed receptive field of individual patches cannot effectively accommodate varying requirements across tasks, and 3) imbalanced pixel distribution among constraints may cause the model to overlook constraints with fewer pixels. In this paper, we propose a vision-assisted foundation model (VaFM) to address these challenges. In the vision modality, input images tailored to all constraints are encoded by a convolutional neural network. The obtained patch embeddings are fused with graph-based nodes to generate solutions, with an auxiliary task designed to address the pixel-imbalanced issue. The performance of VaFM is evaluated across 16 different VRP variants. The experimental results demonstrate the superiority of VaFM over state-of-the-art methods, especially for variants with complex constraints.

Beyond Objective Equivalence: Constraint Injection for LLM-Based Optimization Modeling on Vehicle Routing Problems

Large language models (LLMs) increasingly translate natural-language optimization problems into executable solver code. Yet for constraint-dense operations research (OR) problems, existing data-filtering and training pipelines largely rely on objective-equivalence signals such as differential testing and answer agreement, which a program can pass while adding spurious constraints or silently omitting required ones, whenever those constraints are non-binding on the tested instance. We propose constraint injection, which uses feasible probes to expose spurious over-constraint and one-constraint-violating probes to reveal silent constraint omission. Combined with differential testing, it forms a dual verifier. We instantiate and evaluate it on vehicle routing problems (VRPs), a representative constraint-dense combinatorial optimization testbed with coupled operational constraints. We develop VRPCoder, an 8B end-to-end model that translates natural-language VRP scenarios into Gurobi scripts, together with an expert-verified VRP benchmark suite covering 21 variants. The verifier is reused as a rejection-sampling filter during data synthesis and as a per-rollout reward in group relative policy optimization (GRPO). Across four VRP benchmarks, VRPCoder-GRPO reaches 93\% average Pass@1, outperforms Gemini-3.1-Pro Preview on three benchmarks, exceeds Claude-Sonnet-4.5 by 28 average points, and surpasses prior OR-LLMs by 78 average points.

Graph Edit Distance Formulation for the Vehicle Routing Problem: Theory and Analysis

We show that the Vehicle Routing Problem (VRP) can be reformulated as a Graph Edit Distance (GED) maximization problem. Under a simple edge-deletion cost model, minimizing total route cost is equivalent to maximizing the total weight of edges deleted from the complete instance graph. This formulation models VRP at the edge level, where solutions are defined by selected edges rather than route sequences, enabling structural analyses that are difficult in classical formulations: per-edge attribution of solution quality, decomposition of the optimality gap, characterization of solution sparsity, and identification of edges that are hard to reach by greedy construction. Theoretically, we establish a merge-decomposition theorem showing that Clarke-Wright savings equal per-merge GED increments, and an approximation-transfer theorem that turns GED approximation ratios into VRP cost bounds. Using this reformulation, we analyze 90 CVRP benchmark instances with known optimal solutions. We find that optimal routing graphs use only 5.5% of available edges, that approximately 3.0% of optimal edges are consistently not found by Clarke-Wright heuristics under repeated restarts, and that the cost gap decomposes into missed optimal edges and substituted non-optimal edges of comparable total weight. The edge-additive objective provides a natural per-edge supervision signal for future graph neural network approaches to edge prediction, suggesting a potential connection to graph neural network approaches that we leave for follow-up work.

COAgents: Multi-Agent Framework to Learn and Navigate Routing Problems Search Space

Although Vehicle Routing Problems (VRP) are essential to many real-world systems, they remain computationally intractable at scale due to their combinatorial complexity. Traditional heuristics rely on handcrafted rules for local improvements and occasional \textit{jumps} to escape local minima, but often struggle to generalize across diverse instances. We introduce \textbf{COAgents}, a cooperative multi-agent framework that models the search process as a graph: nodes represent solutions, and edges correspond to either local refinements or large perturbations for diversification (i.e., jumps). A \textit{Partial Search Graph} (PSG) is dynamically constructed during search, enabling COAgents to train a Node Selection Agent and a Move Selection Agent to guide intensification, and a Jump Agent to trigger well-timed explorations of new regions. Unlike end-to-end learning approaches, COAgents cleanly separates problem-agnostic search control from compact domain-specific encoding, facilitating adaptability across tasks. Extensive experiments on the CVRP and VRPTW benchmarks show that COAgents remains competitive with several learn-to-search baselines on CVRP and sets a new state of the art among learning-based methods on the more challenging VRPTW instances, reducing the gap to the best-known solutions by 14\% at $N\!=\!100$ and 44\% at $N\!=\!50$ relative to the strongest neural solver (POMO), and by 21\% and 40\% respectively relative to ALNS. Code is available at https://github.com/mahdims/COAgents.

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