Title: Learning to Insert for Constructive Neural Vehicle Routing Solver
Venue: NeurIPS 2025
Accepted: yes
Keywords: Neural Combinatorial Optimization

Vehicle Routing Problem

Insertion-based Construction
Abstract: Neural Combinatorial Optimisation (NCO) is a promising learning-based approach for solving Vehicle Routing Problems (VRPs) without extensive manual design. While existing constructive NCO methods typically follow an appending-based paradigm that sequentially adds unvisited nodes to partial solutions, this rigid approach often leads to suboptimal results. To overcome this limitation, we explore the idea of the insertion-based paradigm and propose Learning to Construct with Insertion-based Paradigm (L2C-Insert), a novel learning-based method for constructive NCO. Unlike traditional approaches, L2C-Insert builds solutions by strategically inserting unvisited nodes at any valid position in the current partial solution, which can significantly enhance the flexibility and solution quality. The proposed framework introduces three key components: a novel model architecture for precise insertion position prediction, an efficient training scheme for model optimization, and an advanced inference technique that fully exploits the insertion paradigm's flexibility. Extensive experiments on both synthetic and real-world instances of the Travelling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) demonstrate that L2C-Insert consistently achieves superior performance across various problem sizes. The code is available at [https://github.com/CIAM-Group/L2C\_Insert](https://github.com/CIAM-Group/L2C\_Insert).


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes L2C-Insert, the first constructive Neural Combinatorial Optimisation (NCO) framework that learns insertion to address the inherent rigidity of the prevailing append operation. Its Transformer encoder-decoder jointly embeds the current node, all candidate positions, and the remaining unvisited nodes to output a masked softmax over valid insertion positions. Supervised learning is used to train the model to learn to re-insert nodes into their ground-truth positions as defined by an optimal solution , while a distance-based destroy-reconstruct loop iteratively refines solutions at inference time. Notably, a single model trained solely on 100-node synthetic data generalizes to Travelling Salesman and Capacitated VRP instances of up to 100K nodes , achieving consistently superior performance and lower optimality gaps than recent neural and hybrid baselines\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\n\nS1. The paper introduces *insertion* as the learning target, replacing the usual tail-appending step; this is a new idea for neural combinatorial optimisation and it is good to see it explored in the field.\n\nS2. The paper's insertion-based local reconstruction is a powerful method that is proven to improve the final solution.\n\nS3. The paper is well organised the method is explained clearly, and the results are backed by solid evidence.\n\n# Weaknesses\nW1 . The node to be inserted is selected using a simple nearest-neighbor heuristic, not learned by the model. This greedy choice could be a limitation, and it would be interesting to know if learning this selection was explored.\n\nW2. The paper's quality gains seem heavily dependent on the iterative destroy-reconstruct loop. On TSP100 instances, the one-shot Greedy version is notably outperformed by several NCO baselines, which makes it important to distinguish the search contribution from the policy's contribution.\n\nW3. The runtime comparisons are difficult to fairly assess, as the search iteration budget for competing iterative methods (like NeurOpt ) is not reported. Clarifying these settings is crucial for comparing against the L2C-Insert's 1000 reconstruction steps.\n\nQUESTIONS:\nPlease see the above sections.\n\nLIMITATIONS:\nWell described in the paper", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-29"}

comments: SUMMARY:
The paper proposed an insertion-based approach to Neural Combinatorial Optimisation for solving VRP (including TSP) problems. This idea is interesting as it replaced the existing appending-based approaches, and it is a natural extension of the later considering there are existing interstation-based approaches for solving VRP problems. The model used encoder-decoder architecture, and learnt three embeddings: current node, unvisited node, and position embeddings.

STRENGTHS AND WEAKNESSES:
Strength: 
The idea is novel and sensible, and the experimental results show the proposed model has promising potential. 
The presentation for the design of the model and experiments are done in a systematic and comprehensive way.  
Examples were given to better understand how insertion-based approaches work. 

Weakness:  
Computational complexity analysis on insertion-based against appending-based are not presented. 
More detailed justification on the definition of target positions in Section 3.2. 
The loss function: more details could be given.

QUESTIONS:
Why was the logarithm loss function used in Equation 3? 

Target positions: it is said: “The target position for an unvisited node is defined by the two visited nodes in the current partial solution that are adjacent to this unvisited node in the complete labelled solution”   I wonder why “two visited nodes”. 

Table 1:  I wonder why L2C-Insert methods took longer time on TSP100 compared to TSP1K and TSP10K?

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 3
IMPACT: 4
ORIGINALITY: 3
DATE: 2025-06-30

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper explores an insertion-based variant of Neural Combinatorial Optimization (NCO) for vehicle routing problems, proposing a method called L2C-Insert. In contrast to standard appending-based approaches that build solutions by adding nodes sequentially, the proposed method allows for inserting nodes at arbitrary feasible positions within partial solutions. The approach involves a custom model architecture for predicting insertion positions, a training procedure tailored to this setting, and an inference strategy that leverages the increased flexibility of the insertion paradigm. The authors evaluate their method on Travelling Salesman and Capacitated Vehicle Routing Problem instances, reporting improved results over existing NCO baselines. However, the extent and generality of the claimed improvements may warrant further scrutiny.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\nS1 The topic studied is timely and general of interest\n\n\nWeaknesses\n\nW1 There are many vague and partially wrong statements in the paper. E.g., HGS is a heuristic algorithm but is claimed to provide optimal solutions.\n\nW2 The method proposed is pitched as a construction heuristic, but is essentially used in an improvement heuristic, which misaligns the storyline with the content\n\nW3 The results are counterintuitive and need further investigation, e.g., running an HGS on a 100 customer instance should not take 4.5 hours - with the right implementation that can be downloaded from the respective git repository, it should be possible in the order of seconds. Such kind of question marks put the overall experiments in question.\n\nW4 Some benchmarks, e.g., the recent works of Tierney and coauthors are not considered in the numerical study although these outperform the used benchmarks.\n\nQUESTIONS:\nQ1 Can you specify the technical contribution? Insertion-based construction is not a new concept and the way it is applied it resembles more an improvement instead of a construction method.\n\nQ2 Can you clarify the doubts and questions listed under weaknesses with respect to the numerical experiments?\n\nLIMITATIONS:\nThe paper contains a short discussion on limitations.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces an insertion heuristic based on attention neural networks for solving large vehicle routing problems. It evaluates it on large problem instances and compares it to existing heuristics.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper is well-written and the approach is interesting. Extending NCOs with insertion capabilities is a valuable contribution. However, I find that the proposed experimental results do not allow for a fair assessment of the value of the proposed method. This is for a few reasons:\n- the methods have a different computational budget, which vary with the instance size. However, only a single point in time is shown. The authors claims that L2C-Insert can reach speed-up of up to 8x compared to the oracle heuristic benchmarks. However, the value achieved by these heuristics with the same time budget as L2C insert is never shown.\n- it is stated that L2C uses an \"accelerate[d] inference by processing only the k-nearest unvisited nodes and positions of the current node\". This is a valuable strategy. However, it is unclear why it can be applied only to L2C and not to all other benchmarks. This will lead to significant speed-ups on large instances and avoid OOM.\nThe paper is essentially an experimental one. Yet, the current experiments do not allow to fairly assess the value of the new method compared neither to existing NCOs nor HGS.\n\nHGS is the state-of-the art heuristic. It uses a variety of moves for constructing its solutions. As shown by the authors, most NCOs focus on appending nodes to partial solutions. L2C-Insert focuses on inserting nodes. Since solvers are modular, it would be very interesting to design a NCO that combines existing appending moves with L2C-Insert.\n\nQUESTIONS:\n- Why does L2C-Insert gets faster as the instance size increase?\n\nLIMITATIONS:\nSee weaknesses.", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 1, "ORIGINALITY": 2, "DATE": "2025-07-05"}