Title: Rethinking Neural Combinatorial Optimization for Vehicle Routing Problems with Different Constraint Tightness Degrees
Venue: NeurIPS 2025
Accepted: yes
Keywords: Neural Combinatorial Optimization

Vehicle Routing Problem

Constraint Tightness
Abstract: Recent neural combinatorial optimization (NCO) methods have shown promising problem-solving ability without requiring domain-specific expertise. Most existing NCO methods use training and testing data with a fixed constraint value and lack research on the effect of constraint tightness on the performance of NCO methods. This paper takes the capacity-constrained vehicle routing problem (CVRP) as an example to empirically analyze the NCO performance under different tightness degrees of the capacity constraint. Our analysis reveals that existing NCO methods overfit the capacity constraint, and they can only perform satisfactorily on a small range of the constraint values but poorly on other values. To tackle this drawback of existing NCO methods, we develop an efficient training scheme that explicitly considers varying degrees of constraint tightness and propose a multi-expert module to learn a generally adaptable solving strategy. Experimental results show that the proposed method can effectively overcome the overfitting issue, demonstrating superior performance on the CVRP and CVRP with time windows (CVRPTW) with various constraint tightness degrees. The code is available at [https://github.com/CIAM-Group/Rethinking\_Constraint\_Tightness](https://github.com/CIAM-Group/Rethinking\_Constraint\_Tightness).


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper systematically identifies and addresses a critical limitation in existing NCO models: overfitting to fixed constraint tightness. The proposed multi-expert module technique is sound, empirically effective, and practically applicable. Despite some minor shortcomings (e.g., lack of efficiency analysis), the work is of high quality and makes a meaningful contribution to the NCO literature.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. This paper, for the first time, reveals that many NCO methods tend to overfit a certain degree of constraint tightness, which is a critical yet underexplored limitation. This insight represents a meaningful and timely contribution to the field.\n\n2. The proposed multi-expert module is conceptually simple yet empirically effective, significantly enhancing the model's capacity to handle varying levels of constraint tightness. Furthermore, the authors demonstrate that this modular design can be seamlessly integrated into a broad range of backbone architectures, indicating its versatility and general applicability.\n\n3. The paper provides a comprehensive empirical evaluation, including experiments across multiple models, varying constraint levels, and different problem variants. In addition, the inclusion of ablation studies and problem similarity analyses offers strong empirical support for the paper’s central claims.\n\nWeaknesses:\n1. The proposed training approach assumes prior knowledge of the constraint tightness range $[C_{min}, C_{max}]$, but there is little discussion about how to determine this range in practical applications. \n\n2. While the proposed multi-expert module improves performance, its additional computational cost is not analyzed or quantified.\n\nQUESTIONS:\n1.\tHave you considered the impact of the node demand distribution? Most neural methods simply employ uniform node demands, which means that the capacity constraint is just counting the number of visited nodes. If the node demands were drawn from a more heterogeneous distribution, would the performance of current methods be similarly affected or further compromised?\n\n2.\tDoes the overfitting issue discussed in the context of VRPs also manifest in other combinatorial optimization problems?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis study focuses on capacitated vehicle routing problems (CVRP) and aims to improve the generalization performance of neural solvers with respect to the tightness of capacity constraints. Based on the hypothesis that existing methods fail to generalize because they are trained only on instances with a single capacity constraint, the authors propose training on instances with various capacity constraints. Experimental results demonstrate that this approach improves generalization performance regarding constraint tightness. Additionally, the introduction of a multi-expert module is shown to produce solutions with shorter route lengths.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n- The paper demonstrates that a simple approach of changing the training data from instances with a single capacity constraint to instances with multiple capacity constraints improves generalization performance with respect to capacity constraint tightness.\n\n- Based on the observation that the structure of solutions differs depending on the tightness of capacity constraints, the authors introduce a multi-expert module and show that it further shortens route lengths beyond what can be achieved by changing the training data alone.\n\n**Weaknesses**\n\n- The paper has several issues regarding clarity. First, definitions of key terminology are missing. For example, “constraint tightness,” a central concept in the paper, is not clearly defined. Second, the problem setting itself is not clearly described. In many CVRP formulations, the number of vehicles is part of the problem instance (i.e., given). In such settings, feasibility cannot be determined solely based on vehicle capacity. This affects the observation in Section 4.1 that solution characteristics change depending on constraint tightness.\n\n- The paper lacks sufficient experimental justification for why existing methods have limited generalization performance with respect to constraint tightness. For example, an analysis of whether the solution structure (e.g., the number of vehicles used, or the number of nodes visited per vehicle) changes significantly when the capacity varies would strengthen the claim that existing models are overfitting to capacity constraints.\n\n- The ablation study is incomplete. Table 4 reports results for the base model, base model + VCT, and base model + VCT + MEM, but does not include results for the base model + MEM. To clarify the contribution of VCT versus MEM, I recommend reporting the performance of the base model + MEM as well.\n\nQUESTIONS:\nI would appreciate it if the authors could answer the following questions.\n- Could you provide a formal definition of constraint tightness as used in the paper?\n\n- Besides the experimental results presented, do you have other evidence supporting the claim that existing methods overfit to capacity constraints?\n\n- Between VCT and MEM, which component contributes more to the performance improvement?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 1, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates a critical, yet often overlooked, challenge in Neural Combinatorial Optimization (NCO) for Vehicle Routing Problems (VRPs): the impact of constraint tightness on model performance. Most existing NCO methods train and test on data with fixed constraint values, leading to a strong tendency to overfit specific constraint degrees. This results in significant performance degradation when applied to problems with different (i.e., looser or tighter) constraints, limiting their real-world applicability.\n\nTaking the Capacitated Vehicle Routing Problem (CVRP) as a primary example, the paper empirically demonstrates this overfitting issue. To address this drawback, the authors propose a novel training scheme and a multi-expert module designed to learn a more generally adaptable solving strategy.\n\nThe key contributions of this work are:\n\nEmpirical Analysis of Constraint Overfitting: The paper provides a thorough empirical analysis demonstrating that existing NCO methods for CVRP overfit the capacity constraint, performing well only within a narrow range of constraint values.\n\nEfficient Training Scheme for Varying Tightness: A new training scheme is introduced that explicitly considers and incorporates varying degrees of constraint tightness during the training phase.\n\nMulti-Expert Module: A novel multi-expert module is proposed. This module learns to adapt its solving strategy based on the specific constraint tightness of a given problem instance, making the overall model more robust and generalizable.\n\nImproved Generalization: Extensive experimental results on CVRP and CVRP with Time Windows (CVRPTW) with diverse constraint tightness degrees show that the proposed method effectively overcomes the overfitting issue and achieves superior performance across a wide range of constraint values.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nQuality: The paper addresses a highly relevant and often overlooked problem in NCO: generalization across different constraint tightness degrees. The empirical analysis demonstrating overfitting in existing methods is well-conducted and clearly illustrates the problem. The proposed solution, involving both a novel training scheme and a multi-expert module, is technically sound and logically designed to tackle the identified issue. The experimental validation is comprehensive, testing on CVRP and CVRPTW with varying capacities and time window tightness, showing consistent and significant improvements over strong baselines.\n\nClarity: The paper is well-written and structured. The problem of constraint overfitting is clearly defined and motivated. The empirical analysis supporting this problem is presented concisely with illustrative figures. The proposed training scheme and the multi-expert module are explained logically, with clear diagrams (e.g., Figure 2) that aid in understanding the architecture and flow. The mathematical formulations are clear, and the overall narrative flows smoothly, making it easy to comprehend complex ideas.\n\nSignificance: This paper makes a significant contribution by highlighting and providing an effective solution for the constraint overfitting problem in NCO. This issue is a major barrier to the practical deployment of NCO models, as real-world problems rarely conform to a single, fixed constraint value. By enabling NCO models to generalize across different constraint tightness degrees, the work significantly enhances the robustness, flexibility, and real-world applicability of neural solvers for VRPs. The insight into \"rethinking\" how NCO models are trained with respect to constraints is impactful.\n\nOriginality: The core idea of systematically analyzing and then addressing the problem of NCO overfitting to specific constraint tightness degrees is highly original. While multi-task learning or conditional models exist, the specific formulation of the \"constraint tightness embedding\" and the design of the \"multi-expert module\" to adapt the solving strategy based on this embedding are novel contributions within the NCO domain. The empirical evidence of this overfitting phenomenon is also an original insight that challenges common NCO training paradigms.\n\nWeaknesses:\nWhile the paper proposes a \"multi-expert module\" to learn a generally adaptable solving strategy, a weakness is that this module, in practice, function more akin to an ensemble of specialized sub-models, each implicitly optimized for a specific range of constraint tightness. If the adaptation primarily involves switching between these specialized \"experts\" rather than a continuous, unified generalization across the entire spectrum of tightness, it could diminish the originality of achieving a single, truly adaptable policy and might be seen as a sophisticated form of training multiple models for different tightness regimes.\n\nThe specific definition of \"constraint tightness\", while effective for the Capacitated Vehicle Routing Problem (CVRP) and CVRP with Time Windows (CVRPTW), might be too problem-specific. This could limit the universality and direct applicability of the proposed multi-expert architecture and training scheme to other types of Combinatorial Optimization Problems (COPs) or even more diverse VRP variants with different or more complex constraint structures, thus potentially reducing its overall generalizability beyond the explored VRP contexts.\n\nQUESTIONS:\nQuestion: The paper proposes a \"multi-expert module\" to learn a generally adaptable solving strategy. However, could the authors elaborate on whether the model truly learns a continuous, unified generalization across the entire spectrum of constraint tightness, or if its adaptation primarily involves switching between implicitly specialized \"experts\" optimized for specific ranges of tightness?\n\nQuestion: The definition of \"constraint tightness\" is effectively applied to CVRP and CVRPTW. However, how can this specific definition and the proposed multi-expert framework be universally applied or adapted to other types of Combinatorial Optimization Problems (COPs) or more diverse VRP variants with qualitatively different or more complex constraint structures (e.g., time windows for specific deliveries, vehicle type constraints, dynamic capacities)?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
The paper studies the effect of constraint tightness on neural solvers for vehicle routing problems. It shows that existing approaches overfit to the capacity tightness of the training data and provide low quality solutions if the capacity in test differs from the one used in training. It then studies two approaches for addressing this shortcoming based on data augmentation and using capacity as a feature. The advantage of the proposed approach is shown in numerical experiments.

STRENGTHS AND WEAKNESSES:
The paper is well-written. It presents convincingly the limitations of existing NCOs in extensive experiments. I find the analysis of CVRP solutions as a continuum between OVRP and TSP solutions very intuitive, and clearly showing the effect of constraint tightness on solutions. The methodological contributions to address the shortcomings of existing approach are: (1) augmenting the data with varying constraint tightness, and (2) integrating constraint tightness as a feature when computing node probabilities. Interestingly, the first approach is already sufficient to provide significant improvements compare to the baselines.

QUESTIONS:
I find the experimental results convincing. However, I would like to see the trade-off between computational budget and solution quality more clearly presented. Specifically:
- can you show explicitly the number of training instances used in the ablation study?
- can you show the training time as a function of the constraint tightness?
- can you show the inference time between the different method as well as compared to HGS?

LIMITATIONS:
See questions.
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 2
DATE: 2025-07-05