Title: Multi-Task Vehicle Routing Solver via Mixture of Specialized Experts under State-Decomposable MDP
Venue: NeurIPS 2025
Accepted: yes
Keywords: vehicle routing problems

combinatorial optimization problems

reinforcement learning
Abstract: Existing neural methods for multi-task vehicle routing problems (VRPs) typically learn unified solvers to handle multiple constraints simultaneously. However, they often underutilize the compositional structure of VRP variants, each derivable from a common set of basis VRP variants. This critical oversight causes unified solvers to miss out the potential benefits of basis solvers, each specialized for a basis VRP variant. To overcome this limitation, we propose a framework that enables unified solvers to perceive the shared-component nature across VRP variants by proactively reusing basis solvers, while mitigating the exponential growth of trained neural solvers. Specifically, we introduce a State-Decomposable MDP (SDMDP) that reformulates VRPs by expressing the state space as the Cartesian product of basis state spaces associated with basis VRP variants. More crucially, this formulation inherently yields the optimal basis policy for each basis VRP variant. Furthermore, a Latent Space-based SDMDP extension is developed by incorporating both the optimal basis policies and a learnable mixture function to enable the policy reuse in the latent space. Under mild assumptions, this extension provably recovers the optimal unified policy of SDMDP through the mixture function that computes the state embedding as a mapping from the basis state embeddings generated by optimal basis policies. For practical implementation, we introduce the Mixture-of-Specialized-Experts Solver (MoSES), which realizes basis policies through specialized Low-Rank Adaptation (LoRA) experts, and implements the mixture function via an adaptive gating mechanism. Extensive experiments conducted across VRP variants showcase the superiority of MoSES over prior methods.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes the State-Decomposable Markov Decision Process(SDMDP) to solve the multi-task vehicle routing problem(VRP). It decomposes the state of the problem into a combination of multiple basic state Spaces and combines the Mixture of Specialized Experts(MoSE) to solve the VRP problem and various VRP variants.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1, The SDMDP proposed in this paper fundamentally provides a unified expression of the variants of the VRP problem, which ensures that the proposed method can solve multiple variants of VRP.\n2.The effectiveness of up to 16 VRP variants was verified, demonstrating the universality of the proposed model. \n3.The proposed MoSEs can be applied to a variety of models, Meanwhile, the calculation method of hyperparameters is given.\nWeaknesses:\n1, The full text only presents the form of SDMDP, which might make it difficult for readers to understand why your SDMDP can summarize the variants of VRP. Maybe you can add specific examples at appropriate positions and explain how to construct its MDP using your model.\n2.At the experimental results level, the model with MoSES took more than twice as long as the model without MoSES to achieve an absolute Gap decrease of less than 0.4\\% on average and a relative decrease ranging from 1/10 to 1/4. It seems that the extent of performance optimization is not necessarily worth using. Perhaps expanding the scale of the problem would be more persuasive.\n3.You didn't provide in the main text how to prove Theorem 1 and Theorem 2. If you prove it in the appendix, providing guidance seems to be a better solution. Besides, the reasons for setting these two assumptions are also unclear. I'm not quite clear about what the functions of these two assumptions are respectively?\n            \nOverall, I think the proposal of SDMDP and LS-SDMDP is interesting and inspiring, but the effect of MoSES is not ideal, which limits the contribution of this article.\n\nQUESTIONS:\n1.Could you provide the specific expressions and answers of the SDMDP you proposed in the VRPTW and Generalized TSP problems? If there is not enough space for you, please answer the first one.\n2.Could you provide comparative experiments for more larger-scale problems?\n3.What are the reasons and usages for setting these two assumptions?\n\nLIMITATIONS:\nThe increase in the time cost of MoSES is really too large. It is very important to consider how to maintain the effect while reducing the time cost.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work presents a novel approach to training a multi-task NCO model for solving vehicle routing problems using a mixture of specialized experts. It introduces a new State-Decomposable MDP formulation for VRP, where the state space is decomposed into independent basis states and extended with a learnable mixture function to enable policy reuse in the latent space. Through experiments, the authors demonstrate that they can efficiently train optimal policies using specialized experts and adaptive gating mechanisms. This approach can be used in a plug-and-play manner to enhance existing state-of-the-art models.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n- Unlike other similar works, this approach pretrains the model on a single task (CVRP) and then fine-tunes it on other VRP tasks with new problem attributes.\n- The proposed State-Decomposable MDP framework is architecture-independent and can be applied to any multi-task NCO solver for VRPs.\n- Experiments show that the proposed method clearly improves upon existing state-of-the-art multi-task vehicle routing NCO solvers.\n\nWeaknesses:\n- The proposed method is applicable when all tasks share a common intersection; otherwise, maintaining a shared backbone seems impossible.\n\nQUESTIONS:\n1. Although the main idea is clear, the implementation details and the application to VRP could be explained more thoroughly. For me, only the basic state 0) for the classic VRP is clear, while the others are somewhat confusing. For example, basic state 1)-open route only introduces a binary variable - what would be the basic state for this case, and what exactly does it represent? Next, it is mentioned that 'each VRP variant can be formed by composing basic state spaces,' but isn't it mandatory that state 0) is always present?\n\n2. LoRA originally comes from fine-tuning large language models (LLMs), where low-rank matrix factorization is important due to the heavy computational cost. Did you try using simple linear task-specific experts without projecting to a low-rank space? It seems your model isn't large enough to require low-rank factorization, and avoiding that might improve the results.\n\n3. It is stated that the gated-LoRa method suppresses irrelevant features from the backbone. The backbone is trained on CVRP, and as mentioned in paragraphs 252–255, all the other basic VRP variants are derived from it. So, what exactly would be considered irrelevant features in the backbone?\n\n4. Figure 3 is quite surprising, as it doesn't indicate which activation function performs best in general. Do you have any reasonable explanation for why sigmoid works well with CaDa but completely fails with RF? This is somewhat concerning, as it appears that a universal solution for the activation function does not exist and is dependent on the architecture, which reduces the generality of your method. \n\n5. Have you considered applying the same mechanism to a unified NCO solver beyond routing problems? For such a truly universal solver, there is no clear backbone that can serve as a basis for other variants. Is it possible to apply this method to problems with no common intersection between states? This seems like a real limitation when it comes to extending this work to more general cases.\n\nLIMITATIONS:\nThe motivation for this work comes from making a unified model for solving vehicle routing problems, and the proposed solution is tailored to this class of CO problems. The approach does not seem easily extendable to other classes of CO problems or capable of making significant progress toward a foundation model for CO.\n\nThis appears to be a major limitation, as the work does not open new directions toward that goal - one that seems to be a key future milestone for the NCO research community.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-24"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe proposed paper tackles the topic of multi-task vehicle routing problem solvers. Specifically, the authors reformulate the problem as a State-Decomposable MDP (SDMDP), where the state space decomposes into basis states, each corresponding to a fundamental VRP variant (e.g., capacitated, open route). They then extend the work to latent space (LS-SDMDP), proving that the mixture of basis policies can recover optimal solutions under certain conditions. Specifically, in their method, the authors assume the most basic variant of the VRP as the capacitated problem aand each subsequent variant has an additional constraint. As such they train a fixed backbone trained on CVRPs, and use LoRA experts adapt the base embeddings to more constrained settings. They call this method the Mixture-of-Specialized-Experts Solver (MoSES) and use a weighted gating mechanism to concatenate the different expert outputs.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nThe authors suggest the decomposition of state-spaces for the various VRPs, and they then prove that this decomposition enables optimal policy reuse via latent space composition (LS-SDMDP). The approach is relatively novel, and they provide practical applications of the concept through the use of MoSES. The models implemented are not novel in themselves, but this also shows the flexibility of the approach as a general multi-task framework that allows for the use of plug-and-play backbones. The method also achieves convincing empirical results (although at the cost of increased runtime), the experiments are extensive on the datasets that are studied, and a good ablation study is performed for those datasets (this is a point I mention in the weaknesses below).\n\nWeaknesses:\n\nThe proposed method requires some base assumptions to be present. VRP variant constraints need to be easily separated into additional modeled components, and the method assumes the independence of these constraints. Any correlation between them is poorly modelled since there is minimal interaction between the experts, especially when the backbone is frozen. \n\nAdditionally, the paper does not explore the performance of the model in generated out-of-distribution datasets or in generalization to different routing problem sizes. The later is particularly important since attention-based approaches are severely constrained by small problem sizes. The method is already around 4x slower on average than the direct competitor.\n\nQUESTIONS:\n1. Seeing as the method is slower, have you tried running the direct baseline (the models you adapt) with more than 8x augmentations or using input augmentation like in [1]? The difference seems small enough that for some experiments, matching the runtime might change the model standing.\n\n2. Can experts interact with each other outside of the gated mixing? This would seem to be important for correlated tasks.\n\n3. For different problem sizes, is there any insight on when frozen vs trainable experts? Would this be due to better expert interaction with harder tasks?\n\n[1] Bdeir, Ahmad, Jonas K. Falkner, and Lars Schmidt-Thieme. \"Attention, filling in the gaps for generalization in routing problems.\" ECML, 2022.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a framework to address multi-task variants of the Vehicle Routing Problem by leveraging shared structural components. The authors introduce a State-Decomposable MDP formulation that decomposes the state space across basis VRP variants and a Latent-Space SDMDP that enables policy reuse through latent embeddings and a learnable mixture function. The MoSES uses LoRA-based specialized experts and an adaptive gating mechanism to dynamically combine expert outputs. Experimental results on 16 VRP variants suggest that this approach improves solution quality and generalization compared to prior multi-task solvers. Overall, the paper is well-motivated and presents a reasonable balance between theoretical formulation and empirical demonstration.\n\nSTRENGTHS AND WEAKNESSES:\n### **Strengths**\n\n- The idea of decomposing complex VRP tasks into shared basis variants is intuitive and offers a structured way to think about multi-task combinatorial problems.\n- The latent-space policy mixture through LS-SDMDP is a natural extension that aligns with recent trends in modular learning and policy composition.\n- MoSES is designed with practical considerations in mind (e.g., LoRA for parameter efficiency, compatibility with existing backbones), and experiments demonstrate its general applicability across many VRP types.\n- The theoretical formulation is sound and helps justify the approach, although some assumptions (e.g., deterministic mixture function) may be strong in practice.\n- The experimental section is fairly comprehensive, with comparisons to strong baselines and ablation studies.\n\n### **Weaknesses**\n\n- While the latent reuse framework is promising, the actual gains over baselines (e.g., RF-TE or CaDA) are incremental in some tasks, and come with added inference overhead due to dynamic expert aggregation.\n- The gating mechanism does not explicitly use task-specific features, which may limit its specialization capacity or interpretability, especially when handling more diverse task compositions.\n- The framework is only evaluated in the VRP domain. While the proposed MDP reformulation is general in theory, no empirical evidence is provided for transferability to other problem types.\n\nQUESTIONS:\n-  How will the framework perform by adding explicit task descriptors (e.g., constraint flags) into the gating mechanism to improve interpretability and performance?\n- If a basis expert is poorly trained, does the mixture mechanism still maintain reasonable performance? Some robustness analysis would be helpful.\n\nLIMITATIONS:\nNone", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}