Neural Deconstruction Search for Vehicle Routing Problems
Authors: André Hottung
Paula Wong-Chung
Kevin Tierney
Venue: Accepted by TMLR ()

Abstract: Autoregressive construction approaches generate solutions to vehicle routing problems in a step-by-step fashion, leading to high-quality solutions that are nearing the performance achieved by handcrafted operations research techniques. In this work, we challenge the conventional paradigm of sequential solution construction and introduce an iterative search framework where solutions are instead deconstructed by a neural policy. Throughout the search, the neural policy collaborates with a simple greedy insertion algorithm to rebuild the deconstructed solutions. Our approach matches or surpasses the performance of state-of-the-art operations research methods across three challenging vehicle routing problems of various problem sizes.

Published in Transactions on Machine Learning Research (05/2025)
Neural Deconstruction Search for Vehicle Routing Problems
André Hottung
Bielefeld University, Germany
Paula Wong- Chung University of British Columbia, Canada
Kevin Tierney
kevin, www.reeey@oniibbceeeeele........
Bielefeld University, Germany
Reviewed on OpenReview: https: opwpr//veeeereeee.e.mmm/m net/ forun? id= bCmEPiLtuq
Abstract
Autoregressive construction approaches generate solutions vehicle routing problems in step-by-ste by-sep fashion, leading high-quality solutions that are nearing the performance achieved by handcrafted operations research techniques. In this work, challenge the conventional paradigm of sequential solution construction and introduce iterative search framework where solutions are instoad deconstructed by neural policy. Throughout the search, the neural policy collaborates with & simple greedy insertion algorithm to rebuild the deconstructed solutions. Our approach matches or surpasses the performance of state- of-the-art operations research methods across three challenging vehicle routing problems of various problem sizes.
1 Introduction
Methods that can learn to solve complex optimization problems have the potential to transform decision- making processes across virtually all domains. It. is therefore unsurprising that learning-based optimization approaches have garnered significant attention and yielded substantial advancements (Bello et al.) 2016 Kool et al.] 2019: Kwon et. al.) (0200). Notably, reinforcement learning (RL) approaches are particularly promising because they do not rely on a pre-defined training set of representative solutions and can develop new strategics from scratch for novel optimization problems. These methods generally construct solutions incrementally through sequential decision-making process and have been successfully applied various vehicle routing problems.
Despite recent progress, lcarning-basse methods for combinatorial optimization (CO) problems usually fall short of outperforming the shate-of-the-at techniques from the operations research (OR) community. For instance, while some new construction approaches for the capacitated vehicle routing problem (CVRP) have surpassed the LKH3 solver [Helsgaun 2000), they still struggle to match the performance of the state-of- the-art HGS solver (Vidal et al.) 2012), particularly for larger instances with over 100 nodes. One reason for this is their inability explore as many solutions as traditional approaches within the same amount of time. Given the limitations of current construction approaches, we propose challenging the traditional paradigm of sequential solution construction by intruing novel iterative search framework, neural deconstruction search (NDS), which instead deconstruets solutions using neural policy.
NDS is an itcrative scarch method designed to enhance a given solution through a two-phase process involving deconstruction and reconstruction along the lines of large neighborhood search (LNS) (Shaw 1998) and ruin- and-recreate (Schrimpf et al. 2000) paradigms. The deconstruction phase employs a deep neural network (DNN) to determine the customers to be removed from the tours of the current solution. This is achieved through a sequential decision-making process, in which nodes are removed one at a time based on the network's guidance. The reconstruction phase utilizes straightforward greedy insertion algorithm, which

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inserts customers in the order given by the neural network at the locally optimal positions. Note that NDS is trained using reinforcement learning, which makes it adaptable to problems for which reference solutions are available for training.
The overall concept of modifying a solution by first removing some solution components and then conducting a rebuilding step has been successfully used in various vehicle routing problem methods. Non-learning- methods that use this concept include the rip-up and reroute method from DDess & Smith (1981), LNS from Shaw 19988) and the ruin and recreate method from [Schrimpf al. (2000). Learning- based methods have also harnessed this paradigm. The local rewriting method from Chen & Tian (2019), neural large neighborhood search from Hottung & Tierney (2020), and the random reconstruction technique introduced in Luo et al. (2023) employ a DNN during the reconstruction phase. The approaches from Li et al. (2021) and Falkner Schmidt- Thieme (023) both generate different subproblems for a given solution and then use a DNN to choose which subproblem should be considered in the reconstruction phase.
NDS has been designed with the goal of achieving a fast. search procedure without sacrifieing the high- quality search guidance of DNN. For medium-sized CVRP instances with 500 customers, state-of-the- OR approaches such as SISRs (Christiaens & Vanden Berghel 2020) can examine upwards of 270k solutions per second, however neural combinatorial optimization approaches, like POMO (Kwon et al. 2020), can only create around 10k per second. In contrast, NDS can process 120k solutions per second, significantly more than existing neural construction techniques. When combined with a powerful deconstruction DNN, NDS is able to outperform state-of-the-art OR approaches like SISRs and HGS in similar wall-clock time.
We evaluate NDS on several challenging problems, including the CVRP, the vehicle routing problem with time windows (VRPTW), and the prizo-collecting vehicle routing problem (PCVRP). NDS demonstrates substantial performance gains compared to existing learned construction methods and matches surpasses seate-of-the-ar OR methods across various routing problems of different sizes. To the best of our knowledge, NDS is the first learning-base based approach that achieves this milestone.
In summarry we provide the following contributions:
We propose to use a learned deconstruction policy combination with a simple greedy insertion algorithm.
We introduce a novel training procedure designed to learn effective deconstruction policies.
We present a network architecture optimized for encoding the current solution.
We develop high- perforerform search algorithm specifically designed leverage the parallel com- puting capabilitics of GPUs.
2 Literature Review
The introduction of the pointer network architecture by Vinyals et al. (2015) marked the first autoregressive, deep learning-basedl approach for solving routing problems. In their ini- tial work, the authors employ supervised learning to train the models, demonstrating its application to the traveling salesperson problem (TSP) with 50 nodes. Building on this, Bello et al. (2016) propose us- ing reinforcement learning to train pointer networks, showcasing its effectiveness in addressing larger TSP instances.
Construction Methods
For the more complex CVRP, the first learning- baaedd construction methods were introduced by Nazari al.) (2018) and Kool et al. (2019). Recognizing the symmetries inherent in many combinatorial optimization problems, Kwon et al. (2000) develop POMO, method that leverages these symmetries to improve explo- ration of the solution space during both training and testing. Extending this concept, Kim et al. (2022) propose general-purpost symmetric learning framework.
Various tochniques have been proposed to enhance performance in neural combinatorial optimization. For instance, Hottung et al. (2022) introduce efficient active search, which updates a subset of model paraneters during inference. Choo et al. 2022) propose SGBS, combining Monte Carlo tree search with beam search
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to guide the search process more effectively. Additionally, Drakulic et al. (2023 and Luo et al. (2023)
focus on improving out-of-distribution generalization by re-encoding the remaining subproblem after each construction step. To enhance solution diversity during sampling, Grinsztaju et al. (2023) and Hottung et al. (2025) explore approaches that learn a set of policies, rather than a single policy.
Instead of constructing solutions autoregressively, some approaches predict heat maps that highlight promis- ing solution components (e.g., arcs in graph), which are then used in post-hoc searches to construct solutions (Joshi et al. 2019 Fu et al. 2021 Kool et al., 2022b Min et al. 2023). Other approaches focus
more complex variants of routing problems, such as the VRPTW (Falkner & Schmidt-Thiemi
2020 Kool
ct al. 2022a
Berto et al. 2024b ca, or the min-max hetcrogeneous CVRP (Berto et al. 2024a).
Improvement Methods Improvement methods focus on iteratively refining a given starting solution. In addition to the ruin-and-rocreate approaches discussed in the introduction, several other methods aim to enhance solution quality through iterative adjustments. For instance, Ma al. (2021) propose learning to iteratively improve solutions by performing local modifications. Similarly, several works have guided the k-opt hcuristic for vehicle routing problems (Wu et al.)
2019 da Costa et al.. 2020 although they are
constrained by fixed, small k. More recently, Ma et al. (2023 introduced method capable of handling
any k. Furthermore, Ye et al. (2024a) and Kim et al. (2024) integrate learning-based approaches with ant
colony optimization to allow for a more extensive search phase, Additionally, several divide-and-conquer
methods have been developed to address large-scale routing problems (Kim et al.
2021
Li et al. 2021
Ye
et al. 2024b Zheng et al. 2024].
3 Vehicle Routing Problems
Vehicle routing problems (VRPs) represent a broad class of combinatorial optimization problems that are fundamental in logistics and transportation. These problems involve determining optimal routes for a fleet of vehicles to serve a set of customers while satisfying specific constraints. The standard VRP generalizes the well-known TSP and is also NP-hard. In this paper, we consider three key VRP variants: the CVRP, the VRPTW, and the PCVRP. Each variant introduces additional constraints and objectives, making them suitable for different real-world applications. We briefly introduce all three variants below, while a more detailed discussion can be found in Toth & Vigo 2014).
Capacitated Vehicle Routing Problem The CVRP is one of the most studied VRP variants, where each customer has a specific demand that must be met by a fleet of vehicles with limited capacity. Formally, the problem is defined on a complete graph where a depot and a set of N customers, denotod as .y.....Cy are represented as nodes with associated coordinates in two-dimensional Euclidean space. Each customer a has demand Qi, and each vehicle has a maximum capacity Q. The objective is to determine a set of routes that collectively serve all customers while minimizing the total travel cost. We calculate the travel costs as the sum of Euclidean distances between visited nodes. Each route must start and end at the depot, and the sum of customer demands on any route must not exceed Q. An instance of the problem is denoted by I, which encapsulates the locations the depot and customers, their demands, and the vehicle capacity. A solution 8 for an instance consists of a set of routes, where each route defines the sequence of customer visits assigned one vehicle.
Vehicle Routing Problem with Time Windows The VRPTW extends the CVRP by incorporating time constraints customer deliveries. Each customer is assigned time window [a,, bi], specifying the carliest and latest allowable delivery times. A vehicle may arrive carly but must wait until the time window opens. Additionally, each customer has a service time, representing the duration required to complete the delivery before the vehicle can proceed. The objective is to minimize total travel time while ensuring that all deliveries occur within the specified time windows. In this paper, the travel time between two nodes is identical to the Euclidean distance between those nodes. Like the CVRP, all routes start and end at the central depot, and vehicle capacity coints must be respected.
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Autoregressive Neural Deconstruction
Sequential Greedy Insertion
Remove Customers
CJ Css
Encode Solution
Select Customer
C3
Figure Improving solution via neural deconstruction.
Prize- CrizeeCollecting Vehicle Routing Problem The PCVRP is variant of the VRP in which not all customers need to be visited. Unlike the CVRP and VRPTW, where all customers must be served, the PCVRP assigns a prize to each customer, and the objective is to maximize the difference between the total prize collected and the travel costs. Vehicles still start and end al a central depot and must respect capacity constraints. This variant is particularly useful in scenarios where serving every customer is not mandatory but must be balanced against operational costs.
4 Neural Deconstruction Search
NDS is an improvement method for vehicle routing problems. During the scarch, new candidate solution are generated by applying a neural deconstruction policy followed by a greedy reconstruction algorithm the current solution. An example for generating new candidate solution for the CVRP is shown Figure [ First, the static instance information 1 and the current solution s are given to neural policy which then selects customers for removal from the tours of in sequential decision making process. The selected customers are then removed from the tours of resulting in a (partial) solution in which the selected customers are not part of any tour and are thus unvisited. Finally, the selected customers are reinserted into tours using a simple sequential greedy insertion algorithm that inserts customers in the order defined by the policy.
Note that the number of dccision steps (and hence the number of model calls) depends solely on the number of customers that removed. Since keep that number fixed across different instances sizes, NDS is able to generate candidate solutions much faster than construction approaches that build solutions from scratch. Furthermore, NDS is built for efficient GPU and CPU interaction by using batched rollouts.
One key requirement for the learned policies is that they contain a sufficient degree of randomness during the deconstruction process. During the search, a good solution is often deconstructed hundreds times before an improving solution is found. is hence vital that the learned policy able to produce multiple diverse deconstruction instructions for a single solution. We encourage diverse output generation by providing the policy with additional seed input as proposed in Hottung et al. (2025). During training, the model learns to condition its output v, resulting in the selection of different customer sets for different seed values at test time.
4.1 Deconstruction Policy
For solution deconstruction, neural policy employed sequentially select customers for removal from given solution. We model this selection process Markov decision process. Given solution for VRP instance I, policy network тe, parameterized by 0, is used to select M customers for removal. At each step m {1, f1....,M), MY, an action am E [1....,N] is chosen according to the probability distribution wo (am 1, v, a1,m-1), where Om corresponds lo selecting customer Ca,,, to is a random seed, and @1:m-1 are the previous actions. We condition the policy on random seed v to encourage more diverse rollouts as explained Hottung et al. (2025). Each seed is a randomly generated binary vectors of dimension d, (we set d, 10 in all experiments). Finally, after all M customers are selected the reward can be computed discussed in the following sections.
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In lines 9 to 18, the solution s is improved over / iterations. At the start of each iteration, the policy mo is used to sample K rollouts 7I, .2...,, 7K, using K different, random seed vectors Uo...., th. Each rollout Published in Transactions on Machine Learning Research (05/2025)
4.2 Training
The deconstruction policy in NDS is trained using reinforcement learning. During the training process, solutions are repeatedly deconstructed and reconstructed, aiming to discover deconstruction policy that facilitates the reconstruction of high-quality solutions. Algorithm | outlines our training procedure. is important to implement the algorithm a way that allows processing batches of instances in parallel to ensure efficient training. However, for clarity, the pseudocode presented describes the training process for a single instance a time.
A Algorithm NDS Training
procedure TRAiN(teeaaiions per instance rollouts per solution K, improvement steps J)
2: Initialize policy network Wig
while Termination criteria not reached do
4: { GENERATEINSTANCE()
5:
8 < CENERATESTARRUSOLTTONN
6:
for j I.....J J do
7:
+ IMPROOEEENTTSTEPEE...
> Improve solution using procedure shown in Figure 2
8:
end for
9:
for i 1..... do {n, r....., xx) e ROLLOUTPOLLCY(mpple
10:
Sample K rollouts
11:
8k REMOOVCCUSTOMRRNSS TR)
Vk E (I....,K. K}
12:
4 GREEDDYINSSRCCTON(EE... nx)
Vk (l...., K]
13:
7k max(OBI(s) OBI(s().0)
Vk (l,...,
b Calculate reward
14:
b Σx=
Calculate baseline
15:
k* arg max
16:
9i + (ru. b) log To (o
Calculate gradients
17:
+ x-
> Update with best found solution
18:
end for 8 + A E/1 9
19:
b Optimizer step with accumulated gradients
20:
end while
21:
end procedure
The main training loop runs until a termination criterion (such as the number of processed instances) is met. In each iteration of the loop, & new instance and its corresponding solution are generated in lines 4-8. The solution then repeatedly deconstructed and reconstrueted for I iterations (lines 9-18), during which gradients computed based on the rewards obtained. After completing T iterations, the gradients are accumulated, and the network parameters are updated using the learning rate a. The following section provides more detailed explanation of this process.
At the start of each iteration of the training loop, a new instance / and its corresponding solution are generated. The instance is sampled from the same distribution as the test instances. In line 5, an initial solution is constructed using a simple procedure: for an instance with N customers, we generate N tours, each containing one customer. In lines 6-8, this initial solution is iteratively improved through J/ improvement steps of the NDS scarch procedure. This scarch procedure is detailed in Section 4.4 By improving before the training rollouts, we ensure that the training focuses on non-trivial solutions.
is a sequence of M actions that specifies the indices of customers to be removed from the tours in solution s. Each rollout. 7k is individually applied to deconstruct solution s by removing the specified customers, yielding K deconstructed solutions si,...,5k. 5k. These deconstructed solutions are then repaired using the greedy insertion algorithm, which is described in more detail below. Next, the reward rı is calculated for each rollout Tki based on the difference in cost between the original solution and the reconstructed solution s Importantly, the reward is constrained to be non-ncgative, encouraging the lcarning of risk-taking policies. In lines 14 to 16, the gradients are computed using the REINFORCE method. The overall probability of generating a rollout Tı is given by Te(Tk 1, s, 0k) = IIMM Ty Ta(am s, Uk a1m-1). The baseline b is set as the average cost of all rollouts. Gradients are only calculated with respect to the best cest-performing rollout, denoted k., to encourage diversity in the solutions as proposed by Grinsstajj et al. (2023). Finally, at the end of each iteration, the solution is replaced by the reconstructed solution with the highest reward.
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Greedy Insertion The greedy insertion procedure reintegrates the customers removed by the policy, inserting them one by one into either existing or new tours. Specifically, if M customers have been removed, the procedure performs M iterations, where in each iteration, a single customer Con is inserted. At each iteration m, the cost of inserting customer Ca., at every fcasible position in all tours is evaluated. Throughout this process, various constraints, such as vehicle capacity limits, are taken into account. Tf least one feasible insertion point is found within any of the existing tours, the customer cam, is placod at the position that incurs the least additional cost. If no feasible insertion is available, a new tour created for customer Com.
The order in which removed customers are reinserted significantly impacts the overall performance. We reinsert customers either in the order determined by the neural network or at random. Allowing the network to control the reinsertion order gives it control over the reconstruction process, enabling it to find ordering strategies that lead to better reconstructed solutions. If customers are ordered at random, a deconstructed solution should be reconstructed multiple times using different insertion orders. This can provide a more stable learning signal during training.
4.3 Model Architecture
We design a transformer-based architecture that consists of an encoder and a decoder. The encoder used to generate embeddings for all nodes based on the instance and the current solution s. The decoder used to decode sequence of actions based these embeddings in an iterative fashion.
4.3.1 Encoder
The encoder processes the static node features Dy for each of the N + 1 nodes in an instance 1, where ro corresponds to the depot and 21...., ... Ey correspond customer nodes. Additionally, incorporates the current solution s. For the CVRP, the depot feature vector xn includes the depot coordinates, while the customer feature vectors z, (i 2 1) include the customer coordinates and demands 9:. For the VRPTW, the customer feature vectors are further extended to include the earliest a, and latest arrival times b,, as well the service duration. For the PCVRP, the CVRP customer feature set is additionally extended to include customer prizes.
Initially, each imput vector x; is mapped to a 128-dimensional node embedding h; via a linear transformation, using distinct parameters for the depot and customer nodes. The embeddings .o.....hy are then processod through several layers. First, two attention layers encode static instance information. Next, the current solution is integrated via a message passing layer, followed by tour encoding layer. The message passing layer facilitates information exchange between consecutive nodes in the solution, while the tour encoding layer computes embeddings for each tour. These two layers are explained in more detail below. Finally, two additional attention layers refine the representations. The attention mechanisms employed are consistent with those used in prior work (e.g., Kwon et al. 2020))) ) and detailed descriptions are omitted here for brevity.
Message Passing Layer The message passing layer updates the embedding of customer by incor- porating information from its immediate neighbors (i.e., nodes that are visited before and after e in the solution s). Specifically, the embedding h; of customer e is updated as follows:
h4 Norm (h+ + FF (ReLU (w3 [h:: W - (pre(()) + 2 beest(())))
In this equation, piev(i) and next(i) represent the indices of the nodes immediately preceding and following G in the solution s. The weight matrices W and 2 are used to transform the embeddings of these neighboring nodes, while W is applied to the concalenated vector of h4 and the aggregated embeddings from the neighbors. The ReLU activation function introduces non-linearity into the transformation. The output of this transformation processod through food-forward network, which consists of two linear layers with a ReLU activation function in between. The resulting output, combined with the original embedding he via a skip connection, is then normalized using instance normalization.

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Improvement Step
8,
Rollouts
Set
Remove customers
%
Greedy Insertion
Accept?
8:
DNN
Set
Remove customers
Greedy Insertion
Accept?
Set K Gup a... Gr
Remove customers
Greedy Insertion
82.4
Accept? +
-
Figure 2: Improvement step of NDS.
Tour Encoding Layer The tour encoding layer updates the embedding of each customer &; by incorporat- ing information from the tour they are part of. To this end, tour embedding is first computed using mean aggregation of the embeddings of all customers within the same tour, and this aggregated tour embedding is then used to update the individual customer embeddings. Specifically, the embedding h2 of customer is updated as follows:
h FF Σ JET()) k()))).
where T(i) denotes the set of customers in the same tour as customer and W is weight matrix. This layer captures important information about which customers belong to the same tour in the current solution, without considering their specific positions within the tour.
4.3.2 Decoder
Given the node embeddings generated by the encoder, the decoder is responsible for sequentially selecting customers for removal. The overall architecture of our decoder is identical to that of Hottung al. (2025). which utilizes multi-head attention mechanism (Vaswani et al. 2017) followed by a pointer mechanism [Vinyals et al. 2015). This architecture has been widely used in many routing problem methods (Kool et al.
2019 Kwon et al. 2020).
Our approach differs from previous works in that we account for the already selected customers at each decision step. This contrasts with construction-based methods, where each decision is independent of prior selections. To achieve this, we integrate gated recurrent unit (GRU) (Cho 2014), which is used to compute the query for the multi-head attention mechanism. At each decision step, the GRU takes the embedding of the previously selected customer as input, updating its internal state to incorporate past decisions.
4.4 Search
At test. time, we leverage the learned policy within a search framework that supports batched rollouts, enabling fast exccution. Importantly, this framework is problem-agnostic, meaning it contains no problem- specific components, allowing it to be applied to broader range of problems than those evaluated in this paper.
Our search framework consists of two main components: the improvement step function (illustrated in Fig- ure 2 and the high-level augmented simulated annealing (ASA) algorithm (Algorit 2 The improvement step function aims to enhance a given solution by iteratively applying the policy model through a series of dcconstruction and reconstruction steps. The ASA algorithm intogrates this function and supports batched execution for improved performance on the GPU. It important note that we parallelize solely on the GPU, requiring only single CPU core during test time.

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Algorithm 2 Auginented Simulated Annealing
1: procedure SEARcn(Instance 1, Number of iterations mazlter, number of augmentations A, number of rollouts
K, start lemperature Xdaart, lemperature decay rate Xderay, trained policy network ne, threshold factor 6)
2: A x A
3: (11.1...... 1/.....,, K.a + CHEATTAAUGMPENTATIONS((
Sa < GENERATESTARISOLUUOONNI
Va E (1....,A)
5:
for iter = I,..., maxlee do
S. IMPROVEMENTŠTEPPE T6, K) 7: costa + OnJ(sa) Va (1....,AA
Va E (I....,A)
cost" 4- min(coslo......o costa)
9:
thresh + cost" + 6x
10:
for = I....; A do
11:
if costa > threeh then
12:
S.. f- RANDOMCHOICC(((S e (so.... OBI(S') < thresh))
13:
end if end for À < REDUCETEEMPERATUURE deccay)
14:
15:
16: end for
17:
end procedure
Improvement Step The improvement step, the core component the overall search algorithm, is depicted in Figure 2 Note that the improvement step procodure is designed limit the data transfer between GPU and CPU while exploring large number of different solutions. The process begins with an initial solution So that is passed to the policy DNN, which generates K rollouts, each consisting of M actions that specify the customers to be removed. Once the policy DNN completes its execution, these rollouts are sequentially applied to produce new candidate solutions. Specifically, the solution Su is first deconstructed based on the actions from the first rollout (yielding 50) and then reconstructed into %6. A After reconstruction, a simulated annealing (SA) based acceptance criterion is used to determine whether & or 8p should be retained, resulting in 81. This process is repeated in each subsequent iteration. After iterations, the final solution 8к is returned, representing the outcome of K consecutive deconstruction and reconstruction operations. By performing these iterations sequentially; the solution so is significantly modified, often leading to notable cost improvements between the initial imput Su and the final output sk. while only needing to transfer data from the GPU to the CPU single time.
Augmented Simulated Annealing We introduced a novel simulated annealing (SA) algorithm to con- duct a high-level search specifically designed for GPU-based parallelization. While parallel SA algorithms have been proposed in prior work, (Ferreiro et al. 2013 Jeong & Kim 1990; Oubaçoğlu & Özdamar 2001), their main concern is on the information exchange between CPU cores. In contrast, our approach focuses on executing parallel rollouts of the policy network on the GPU.
At high level, the ASA technique, shown in Algorithm 2 modifies solutions over multiple iterations using temperature-based acceptance criterion. This criterion allows worsening solutions to be accepted with a certain probability, which depends on the current temperature. The temperature À is manually set at the start of the search (line 2) and is gradually reduced after each iteration (line 15), resulting in a decreasing probability of accepting worsening solutions during the improvement step (line 6). For a detailed discussion on SA, we refer the reader to Gendreau et al. (2010).
To enable parallel search for single instance, employ the augmentation technique introduced in Kwon et al. (2020). which creates set of augmentations L L2 I'.....,',, for an instance l. The search is then conducted in parallel for these augmentations. After cach modification by the improvement step procedure (line 6). solutions can be exchanged between different augmentations. Specifically, we iterate over all augmentation instances (lines 10 to 14) and replace solutions that surpass a certain cost threshold with randomly selected solutions whose costs fall below the threshold. This threshold is calculated based on the cost of the current best solution and the temperature, adjusted by factor 8 > 1, as shown in line The goal is replace solutions that are unlikely to surpass the quality of the current best solution, given the current temperature.
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5
Experiments
We evaluate NDS three VRP variants with 100 to 2000 customers and compare to state-of-the-art learning- based and traditional OR methods. Additionally, we provide ablation experiments for the individual com- ponents of NDS and evaluate the generalization across different instance distributions. All experiments are conducted on a rescarch cluster utilizing a single Nvidia A100 GPU per run. Our implementation of NDS is available at betpp://githing wtt.ccoo///oo..oo www.//ottttt///DD.....
5.1 Problem Instances
CVRP We use the instance generator from Kool et al. (2019) to create scenarios with uniformly distributed customer locations, and the generator from Queiroga et al. (2022) for generating more realistic instances with clustcred customer locations to better simulate real-world conditions.
VRPTW We use the instance generator from Quciroga et al. 2022) to generate customer locations and demands, while time windows are generated following the methodology outlined by Solomon 1987
To generate PCVRP instances, we use the instance generator from Queiroga et al. 2022) to create customer locations and demands. Customer prize values are generated al random, with higher prizes assigned to customers with greater demand, reflecting the increased resources required to service them.
PCVRP
5.2 Search Performance
Baselines We compare NDS to several heuristic solvers, including HGS (Vidal 2022 SISRs Christiaens
& Vanden Berghe 2020), and LKH3 (Helsgaun 2017). Additionally, we include VyP (Wuua et al. 2024]
(version 0.9.0), which an open-source extension of HGS for other VRP variants. For the CVRP, we further
compare NDS to the state-of-the-art learning- baased methods, SGBS- EAS (Choo al.) 2022). NeuOpt (Ma)
ct al. 2023 BQ (Drakulic et al. 2023), LEHD (Luo et al.) 2023). UDC (Zheng ct al. 2024 and GLOP
Ye et al. 2024b). We run all baselines ourselves the same Lest instances. More details on the baseline settings can be found in Appendix A
NDS Training For each problem and problem size, we perform separate training run. Training consists of 2000 epochs for settings with 1000 or fewer customers. For the 2000 customer setting, we resumee training from the 1000 customer model checkpoint at 1500 epochs and train for an additional 500 epochs. In each epoch, we process 1500 instances, with each instance undergoing 100 iterations, 128 rollouts, and 10 initial improvement steps. The learning rate set to 10-4 and 15 customers are selected per deconstruction step across all problem sizes. The training durations are approximately 5, 8, 15, and 8 days for the problem sizes 100, 500, 1000 and 2000, respectively. The training curves are presented in Appendix B while visualizations of policy rollouts are available in Appendix
Evaluation Setup At test time, we limit the runtime to 5, 60, 120, and 240 seconds of wall time per instance for HGS, SISRs, and NDS to ensure fair comparison, as these methods process test instances sequentially. For most learning- approaches we report results for two different termination criteria, i.e., the original termination criteria proposed in the respective paper and a new wall-time- termination criteria. For approaches that process instances in batches, an adjusted time limit is applied to each batch as whole, ensuring that the total time required to solve the complete set matches that of approaches using per-instance time limit. Note that this experimental setup favors batch-based approaches over NDS, processing instances in batches is computationally more efficient, but not feasible in all applications. All approaches are restricted to using a single CPU core. For the CVRP, we use the test instances from Kool et al. (2019) for 100 (10,000 instances), Drakulic el. al. (2023) for 5-500 (128 instances), and Ye et al. (2024b) for 1000 and 2 (100 instances each). For the VRPTW and PCVRP, we generate new test sets consisting of 10,000 instances for N=100 and 250 instances for scttings with more than 100 customers. To allow for a fair comparison, we ensure that all reinforcement learning based approaches are trained on instances from the same distribution that also used for sampling the test instances.
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Table 1: Performance on test data. The gap is calculated relative to HGS for the CVRP and relative to PyVRP-HGS for the VRPTW and PCVRP. Runtime is reported on per-instance basis in seconds. The best results (i.e., those with the lowest objective function value) are shown in bold, and the second-best are underlined.
N=100
N=500
N=1000
N=2000
Method
Obj.
Gap
Time
Obj.
Gap
Time
Obj.
Gap
Time
Obj.
Gap
Time
HGS SISRs
15.57
5
36.66
60
41.51
121
57.38
241
15.62
0.32%
5
36.65
0.01%
60
11.14
-0.83%
120
56.01
-2.27%
240
LKH3
15.64
0.50%
41
37.25
1.66%
174
42.16
1.61%
408
58.12
1.35%
1448
SGBS-EAS
Limit
15.59
0.20%
2
Time Limit
15.59
0.17%
5
NeuOpt
10k Ites.
15.66
0.59%
CVRP
Time Limit
15.59
0.15%
BQ
Beam widch 18
15.81
1.55%
37.64
2.68%
10
43.53
4.86%
61.75
7.61%
323
width
15.74
1.13%
37.51
2.32%
23
43.32
1.36%
164
LEHD
15.63
0.41%
37.10
1.21%
28
42.41
2.17%
159
59.45
3.60%
1476
Time Limit
15.61
0.30%
5
37.04
1.04%
60
42.47
2.31%
121
60.11
4.76%
246
UDC
250 Stages
37.71
2.86%
13
42.77
3.04%
21
Time Limis
37.63
2.67%
60
42.65
2.76%
121
GLOP (LKH3)
45.90
10.58%
63.02
9.82%
NDS
15.57
0.04%
5
36.57
-0.20%
60
41.11
-0.90%
120
56.00
-2.34%
240
PyVRP-HGS
12.98
5
49.01
60
90.35
120
173.46
240
PCVRP VRPTW
SISRs
13.00
0.20%
5
48.09
-1.87%
60
87.68
-2.98%
120
167.49
-3.49%
240
SGBS-EAS
Default
13.15
1.35%
3
Time Limis
13.13
1.22%
5
NDS
12.95
-0.19%
5
47.94 -217%%
60
87.54
-3.14%
120
167.48
-3.50%
240
PyVRP-HGS
10.11
5
44.97
60
84.91
120
165.56
240
SISRs
9.91
-1.66%
5
13.22
-3.90%
60
81.12
-4.55%
120
158.17
-4.54%
240
NDS
9.90
-2.07%
5
43.12
-4.12%
60
80.99
-4.71%
121
158.09
-4.60%
211
NDS Test Configuration For NDS, the starting temperature start is set to 0.1 and decays exponentially 0.001 throughout the scarch. The threshold factor 8 fixed at 15. During the improvement step, 200 rollouts are performed per instance, and each deconstructed solution is reconstructed 5 times (1x based on the selected order of the DNN and 4x using random customer order). The number of augmentations is set to 8 for the CVRP and VRPTW, and 128 for the PCVRP. Note that this configuration was manually selected rather than derived from automated hyperparameter tuning. In Appendix we analyze the impact of the hyperparameters on search performance.
Results Table presents the performance of all compared methods on the test data. The gap is reported relative HGS for the CVRP, and to PyVRP-HGS for the VRPTW and PCVRP. By choosing HGS for the gap caleulation, we follow recent publications and ensure easy comparability of results with these earlier works. Across the 12 test settings, NDS delivers the best performance in 11 cases, with HGS being the only approach able to outperform it the CVRP with 100 customers. Compared to other learning-basee methods, NDS shows significant performance improvements across all CVRP sizes. On the CVRP with 2000 customers, NDS achieves a 6.83% improvement over the best-performing learning-based method, LEHD (when both are given the same runtime), and an 11.13% improvement over GLOP. Against the state-of-the art HGS and extension, PyVRP-HGS, NDS performs especially well on larger instances, achicving an improvement of more than 2% across all problems for instances with 2000 customers. For the PCVRP, NDS also attains substantial improvements relative PyVRP-HGS, exceeding 4% instances with 500
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CVRP5500
VRPTW500
PCVRP500
36.025
NDS 5258a
48.0
44.1
36.600
36.577
Costs
48.9
Closss
$1.0
96.050
48.8
48.7
43.8
36.500
48.6
36.470
43.7
50
100
150
200
250
50
100
150
200
260
30
100
160
200
230
Runtime
Runtime
Runtime
Figure 3: Comparison of NDS and SISRs performance across varying runtime limits.
nodes. When compared to SISRs, NDS maintains small advantage on larger instances and demonstrates significantly better performance on small instances.
5.3 Performance Across Different Runtime Limits
To better understand the application scenarios of NDS, we evaluate its performance under varying runtime limits. In this experiment, compare NDS only to SISRs as it is the overall best performing baseline in the previous experiment (see Section 5.2) We use the same trained models and evaluation setup as in the previous experiment, but with test sets reduced to 100 instances each to manage computational costs.
Figure 8 presents the results for instances with 500 customers. For each problem, five different runtime limits ranging from 15 to 250 seconds are considered. The hyperparameters of STSRs and NDS remain consistent with those used in previous experiments across all runtime limits. Notably, NDS outperforms SISRs across all runtime limits, except for the PCVRP, where SISRs perform better under a 15-second runtime limit. The relative performance gap is particularly striking for the CVRP, where running NDS for just 30 seconds surpasses the performance of SISRs at 250 seconds. Similar trends are observed for instances with 100, 1000, and 2000 customers. Additional details and results for other problem sizes are provided in Appendix D
5.4 Ablation Studies
We perform a series of ablation experiments to assess the importance of different components of NDS. These experiments are conducted on separate validation instances with N=500 customers. The parameter configuration remains identical to the previous section, except the training is reduced to 1,000 epochs and the ASA search is limited by the number of iterations. For the CVRP and VRPTW, we run 1,000 iterations using 8 augmentations, while for the PCVRP, perform 50 iterations with 128 augmentations. In addition to the ablation results reported here, present an ablation study the impact of the initial improvement step during training in Appendix
Network Architecture We assess the impact of the message passing layer (MPL.) and tour encoding layer (TEL) overall performance by training separate models without these components. Table 2a sunmarizes the resulting scarch performance. Excluding both layers Icads to significant performance drop, with a 1.5% reduction on the PCVRP. Even the removal of a single layer causes a notable performance decline, particularly for the VRPTW and PCVRP. The VRPTW in particular benefits from both layers, likely due to the MPL's ability to better interpret and handle time windows.
Insertion Order The inscrtion algorithm reinserts removed customers in a specified order. During testing, we reconstruct deconstructed solution five times using different customer orders and retain the best solution. For the first reconstruction iteration, we use the customer order provided by the DNN, while for the remaining four itcrations we use random order. We compare this standard setting to using only random orders across all five insertion iterations assess whether the DNN has learned to select an order that improves the overall search performance. The results in Table 2b show that using only random orderings leads to significantly
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Table 2: Ablation experiments. All tables report objective function values.
(b) Inscrtion order
(a) Impact of the message passing layer (MPL) and the tour encoding layer (TEL) performance.
Order
CVRP
VRPTW
PCVRP
DNN Random
36.81
47.68
42.96
MPL
TEL
CVRP
VRPTW
PCVRP
Random
36.86
47.76
43.05
36.81
47.68
42.96
(c) Deconstruction policy
X
36.82
47.75
43.13
36.81
47.74
42.98
Policy
CVRP
VRPTW
PCVRP
36.87
47.87
43.62
DNN
36.81
47.68
42.96
Heuristic
37.03
48.16
43.61
Table 3: Out-of-distribution (OOD) vs. in-distribution (ID) performance on the CVRP500.
Uniform Locations
Clustered Locations
Method
Low Capacity
High Capacity
Low Capacity
High Capacity
Obj.
Gap
Time
Obj.
Gap
Time
Obj.
Gap
Time
Obj.
Gap
Time
HGS SISRs
91.73
47.89
60
88.20
60
44.53
91.34
-0.38%
47.79
-0.17%
60
87.78
-0.48%
60
44.31
-0.49%
NDS (OOD)
91.15
-0.59%
60
47.70
-0.36%
60
87.75
-0.33%
60
44.29
-0.54%
60
NDS (ID)
91.14
-0.59%
60
47.69
-0.38%
60
87.70
-0.58%
60
44.26
-0.60%
60
performance across all three problems, indicating that the learned policy not only plays crucial role in deconstruction, but also siguificantly influences reconstruction.
Learned Policy We assess the relevance and effectiveness of the learned deconstruction policy by replacing it with a handcrafted heuristic based on the destroy procedure outlined in Christiaens & Vanden Berghe (2020). The resulting approach eliminates any learned components, but is otherwise identical to NDS. The performance comparison, shown in Figure 2c reveals that the heuristic deconstruction policy performs significantly worse than the learned counterpart, with performance gaps of up to 1.5% on the PCVRP. This demonstrates that the DNN is capable of learning highly efficient policy that surpasses handorafted methods.
5.5
Generalization
One major advantage of learning-based solution approaches is their ability to adapt precisely to the specific type of instances at hand. However, in real-world scenarios, concept drift in the instance distributions cannot always be avoided. In this experiment, we evaluate whether the learned policies of NDS can handle instances sampled from a slightly different distribution. For the CVRP with N=500, we train a policy on instances with medium-capacit, vehicles and customer locations that follow a mix of uniform and clustered distributions. We then evaluated the learned policy on instances with low- and high-capacity vehicles, and customer locations following either uniform or clustered distributions. Additionally, we train distribution- specific models for each test setting for comparison. As baseline, we compare against HGS and SISRs, giving all approaches the same runtime. The results are shown in Table 3 where NDS (OOD) represents the model's performance when the training and test distributions differ, and NDS (ID) represents the setting where the training and test distributions are identical. Overall, the performance difference between the two set settings is minimal, indicating that NDS generalizes well across different distributions. Interestingly, the distribution of customer locations has a larger impact on performance than vehicle capacity. In Appendix we also evaluate NDS's ability to generalize both smaller and significantly larger instances.
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5.6 Scalability Analysis
Relative Resouroe Usage
We assess the scalability of NDS by analyzing its runtime and GPU memory consumption CVRP instances of varying sizes. Figure presents the relative resource usage func- tion of problem size. Overall, NDS demonstrates strong scal-
2.4
Time
2.2
Memory
2.0
1.8-
ability to larger instances. Notably, solving instances with
1.6-
1,000 customers requires only 61% more runtime and 23% more memory compared to instances with 100 customers, despite the problem size increasing by an order of magni- tude.
1.4
1.2
1.0
100
500
1000
2000
Problem Size
6 Conclusion
Figure 4: Scalability
In this work, we introduced novel search method, NDS, which leverages learned policy to doconstruct solutions for routing problems. NDS presents several key advantages. First, delivers excellent performance, consistently matching outperforming state-of-the-art OR methods under equal runtime. Second, NDS scales effectively to larger problem instances, handling up to - customers, due to the fact that the mumber of customers selected by the policy is independent of the problem size. Third, it demonstrates strong generalization across different data distributions. Finally, NDS easily adaptable to new vehicle routing problems, requiring only small adjustments to the greedy insertion hcuristic and the model input.
A notable limitation is the reliance on a GPU for executing the policy network. Future research could explore model distillation techniques to lower the computational requirements or investigate whether the underlying principles of the learned policies Another limitation of NDS is its reliance on training data that closely resembles the instances encountered during testing. Future work could investigate whether fine-tuning techniques enable fast adaptation of a pretrained foundation model to new problem instances. This could significantly reduce training time and minimize the amount of required training data.
be approximated using faster, more efficient algorithms.
Acknowledgments
André Hottung was supported by the Deutsche Forschungsyemiinschall (DFG, German Research Founda- tion) under Grant No. 521243122. Paula Austauschdienst (DAAD) RISE Germany program for this rescarch. Additionally, we gratefully acknowledge the funding of this project by computing time provided by the Paderborn Center for Parallel Computing (PC2). Furthermore, some computational experiments this work have been performed using the Bielefeld GPU Cluster. We thank the HPC.NRW team for their support.
received funding from the Deutscher Akademischer
References
Irwan Bello, Hien Pham, Quoe V Le, Mohammad Norouzi, and Samy Bengio. Neural combinatorial opti- mization with reinforcement learning, arXie preprint or XxX::1611.09940, 2016.
Federico Berto, Chuanbo Hua, Laurin Luttmann, Jiwoo Son, Junyoung Park, Kyuree Ahn, Changhyun Kwon, Lin Xie, and Jinkyoo Park. Parco: Learning parallel autoregressive policies for efficient multi-agent combinatorial optimization. arXiv preprint arXiv:2409. 009.011,,,081,, 2024a.
Federico Berto, Chuanbo Hua, Junyoung Park, Laurin Luttmann, Yining Ma, Fanchen Bu, Jiarui Wang. Haoran Ye, Minsu Kim, Sanghyeok Choi, Nayeli Gast Zepeda, André Hottung, Jianan Zhou, Jieyi Bi, Yu Hu, Fei Liu, Hyeonah Kim, Jiwoo Son, Haeyeon Kim, Davide Angioni, Wouter Kool, Zhiguang Cao, Qingfu Zhang, Joungho Kim, Jie Zhang, Kijung Shin, Cathy Wu, Sungsoo Ahn, Guojie Song, Changhyun Kwon, Kevin Tierney, Lin Xie, and Jinkyoo Park. RL4CO: an extensive reinforcement learning for com- binatorial optimization benchmark. Xiu preprint ar arXio:2306. 1 17100, 2024b.
13

---

Published in Transactions on Machine Learning Research (05/2025)
Federico Berto, Chuanbo Hua, Nayeli Gast Zepeda, André Hottung, Niels Wouda, Leon Lan, Kevin Tierney, and Jinkyoo Park. Routefinder: Towards foundation models for vehicle routing problems. arviv preprinl arXiv:2406.15000 15007, 2024c.
Xinyun Chen and Yuandong Tian. Learning to perform local rewriting for combinatorial optimization. Advances in Neural Information Processing Systems, pp. 6278-6289. 2019.
Kyunghyun Cho. Learning phrase representations using rmn encoder-docoder for statistical machine trans- lation. Xiv preprint orXiv: 106. 1078, 2014.
Jinho Choo, Yeong-Dae Kwon, Jihoon Kim, Jeongwoo Jae, André Hottung, Kevin Tierney, and Youngjune Gwon, Simulation-guideer beam search for neural combinatorial optimization. Advances in Neural Infor- mation Processing Systems, 35:8760-8772, 2022.
Jan Christiaens and Greet Vanden Berghe. Slack induction by string removals for vehicle routing problems. Transportation Science, 54(2):417-433, 2020.
Paulo da Costa, Jason Rhuggenaath, Yinggian Zhang, and Alp Eren Akçay. Learning 2-opt Heuristics for the Traveling Salesman Problem via Deep Reinforcement Learning. Asian Conference on Machine Learning, 2020.
William Dees and Robert J Smith. Performance of interconnection rip-up and reroute strategies. 18th Design Automation Conference, pp. 382-390, 1981.
Darko Drakulic, Sofia Michel, Florian Mai, Arnaud Sors, and Jean-Mare Andreoli. BQ-NCO: Bisimulation Quotienting for Generalizable Neural Combinatorial Optimization. arXiv preprint 0rXi:22301003111 2023.
Jonas K Falkner and Lars Sehmidt-Thicmn Learning to Solve Vehicle Routing Problems with Time Windows
through Joint Attention. arXiv preprint 20i::2006.09100, 2020.
Jonas K Falkner and Lars Schnidt-Thieme.. Too big, fail? enabling neural construction methods to solve large-scale routing problems. arXiv preprint arXiv:2309.1108 2023.
Ana M Ferreiro, JA García, José G López-Salas, Salas, and Carlos Vázquez. An efficient implementation parallel simulated annealing algorithm in gpus. Journal of global optimization, 57:863-890, 2013.
Zhang-Hua Fu, Kai-Bin Qiu, and Hongyuan Zha. Generalize a small pre-trained model to arbitrarily large TSP instances. Proceedings of the AAAI conference on artificial intelligence, 35 35(87:7474 7482, 2021.
Michel Gendreau, Jean-Yves Potvin, et al. Handbook of metaheuristics, volume 2. Springer, 2010.
Nathan Grinszlajn, Daniel Furelos- Blanco, Shikha Surana, Clément Bonnet, and Tom Barrett. Winner takes it all: Training performant Il populations for combinatorial optimization. Advances in Neural Information Processing Systems, 36:48485-48509, 2023.
Keld Helsgaun. An effective implementation of the Lin Kernngnnnn traveling salesman heuristic. European Journal of Operational Research, 126:106-130, 2000.
Keld Helsgaun. An extension the Eexernighan-Hesgauee TSP solver for constrained traveling salesman and vehicle routing problems. Roskilde: Roskilde Universily, 2017.
André Hottung and Kevin Tierney. Neural Large Neighborhood Search for the Capacitated Vehicle Routing Problem. European Conference Artificial Intelligence, pp. 443 450, 2020.
André Hottung, Mridul Mahajan, and Kevin Tierney. PolyNet: Learning diverse solution strategies for neural combinatorial optimization. International Conference on Learning Representations, 2025.
André Hottung, Yeong-Dae Kwon, and Kevin Tierney. Efficient Active Search for Combinatorial Optimiza- tion Problems. International Conference on Learning Representations, 2022.
14

---

Published in Transactions on Machine Learning Research (05/2025)
Chang-Sung Jeong and Myung-Ho Kim. Parallel algorithm for traveling salesman problem SIMD machines using simulated annealing. Proceedings of the International Conference on Application Specifie Army Processors, pp. 712 721, 1990.
Chaitanya K Joshi, Thomas Laurent, and Xavier Bresson. An Efficcintt Graph Convolutional Network Technique for the Travelling Salesman Problem. arXiv preprint arxi 1906.99002229 2019.
Minsu Kim, Jinkyoo Park, and Joungho Kim. Learning Collaborative Policies to Solve NP-hard Routing Problems. Advances in Newral Information Processing Systems, 2021.
Minsu Kim, Junyoung Park, and Jinkyoo Park. Sym-NCO: Leveraging Symmetricity for Neural Combina- torial Optimization. Advances in Neural Information Processing Systems, 2022.
Minsu Kim, Sanghyeok Choi, Jiwoo Son, Hyeonah Kim, Jinkyoo Park, and Yoshua Bengio. Ant colony sampling with gflownets for combinatorial optimization. arXiv preprint SF8::2408.00444 2024.
Wouter Kool, Herke van Hoof, and Max Welling. Attention, Learn Solve Routing Problems! International Conference Learning Representations, 2019.
Wouter Kool, Laurens Blick, Danilo Numeroso, Yingqian Zhang, Tom Catshock, Kevin Tierney, Thibaut Vidal, and Joaquim Gromicho. The EURO Meets NeurlPS 2022 Vehicle Routing Competition. Proceedings of the NeurIPS 2022 Competitions Track, 2022a.
Wouter Kool, Herke van Hoof, Joaquim Gromicho, and Max Welling. Deep Policy Dynamic Programming for Vehicle Routing Problems. Inlegration of Construind Programming, Artificial Intelligence, and Opera- tions Research, 2022b.
Yeong-Dac Dae Kwon, Jinho Choo, Byoungiip Kim, Iijoo Yoon, Youngjune Gwon, and Seungjai Min. POMO: Policy Optimization with Multiple Optima for Reinforcement Learning. Advances in Neural Information Processing Systems, 33:21188 21198, 2020.
Sirui Li, Zhongxia Yan, and Cathy Wu. Learning to delegate for large-scale vehicle routing. Advances in Newral Information Processing Systems, 1126198-262111 2021.
Fu Luo, Xi Lin, Fei Liu, Qingfu Zhang, and Zhenkun Wang. Neural Combinatorial Optimization with Heavy Decoder: Toward Large Scale Generalization. Newral Information Processing Systems, 2023.
Yining Ma, Jingwen Li, Zhiguang Cao, Wen Song, Le Zhang, Zhenghua Chen, and Jing Tang. Learning Iteratively Solve Routing Problems with Dupe-Aspect Collaborative Transformer. Neural Information Processing Systems, 2021.
Yining Ma, Zhiguang Cao, and Yeow Meng Choe. Learning to Search Feasible and Infeasible Regions of Routing Problems with Flexible Neural k-Opt. Neural Information Processing Systems, 2023.
Yimeng Min, Yiwei Bai, and Carla P Gomes. Unsupervised Learning for Solving the Travelling Salesman Problem. Neurol Information Processing Systems, 2023.
Mohammadreza Nazari, Afshin Oroojlooy, Lawrence Snyder, and Martin Takác. Reinforcement learning for solving the vehicle routing problem. Advonces in Neural Information Processing Systems, pp. 2018.
Esin Onbaşoglu and Linet Ozdamar. Parallel simulated annealing algorithms in global optimization. Journal of global optimization, 19:27 50, 2001.
Eduardo Queiroga, Ruslan Sadykov, Eduardo Uchoa, and Thibaut Vidal. 10,000 optimal CVRP solutions for testing machine lcarning based heuristics. AAAI-22 Workshop on Machine Learning for Operations Research (ML4OR), 2022.
Gerhard Schrimpf, Johannes Schncider, Hermann optimization results using the ruin and recreate principle. Journal of Computational Physics, 159(2): 139-171, 2000.
and Gunter Ducck. Record breaking
15

---

Published in Transactions on Machine Learning Research (05/2025)
Paul Shaw. Using constraint programming and local search methods to solve vehicle routing problems.
International conference on principles and praclice of constraint programming. pp. 417-431, 1998.
Marius M Solomon. Algorithms for the vehicle routing and scheduling problems with time window con-
straints. Operations research, 35(2):254-266 1987.
Paolo Toth and Daniele Vigo. Vehicle routing: problems, melhods, and applications. SIAM, 2014.
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is All you Need. Aduances in Neural Information Processing Systems, 30, 2017.
Thibaut Vidal. Hybrid genetic search for the CVRP: Open-source implementation and SWAP* Neighbor- hood. Computers & Operations Research, 140:105643, 2022.
Thibaut Vidal, Teodor Gabriel Crainic, Michel Gendreau, Nadia Lahrichi, and Walter Rei. Hybrid Genetic Algorithm for Multidepot and Periodic Vehicle Routing Problems. Operations Reseurch, 60(3):611 624, 2012.
Oriol Vinyals, Meire Fortunato, and Navdeep Jaitly. Pointer networks. Processing Systems, pp. 2692 22007000 2015.
Advances in Newral Information
Niels A Wouda, Leon Lan, and Wouter Kool. PyVRP: A high-performance VRP solver package. INFORMS Journal on Computing, 2024.
Yaoxin Wu, Wen Song, Zhiguang Cao, Jie Zhang, and Andrew Lim. Learning Improvement Heuristics for Solving Routing Problems. IEEE Transactions on Neural Networks and Learning Systems, 2019.
Haoran Ye, Jiarui Wang, Zhiguang Cao, Helan Liang, and Yong Li. DeepACO: neural-enhanced ant systems for combinatorial optimization. Advances in Neural Information Processing Systems, 36, 2024a.
Haoran Ye, Jiarui Wang, Helan Liang, Zhiguang Cao, Yong Li, and Fanzhang Li. GLOP: Learning global partition and local construction for solving large-scale routing problems in real-time. Proceedings of the AAAT Conference on Artificial Intelligence, 38(18):20284 20224-2222-- 2024b.
Zhi Zheng, Changliang Zhou, Tong Xialiang, Mingxuan Yuan, and Zhenkun Wang. UDC: A unified neural divide-and-conquer framework for large-scale combinatorial optimization problems. Advances in Neural Information Processing Systems, 2024.
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A Baseline Configurations
HGS We conduct evaluation runs using the publicly available implementation from the authors that has been designed and calibrated for medium-scale CVRP instances with up to 1,000 customers. We use a
wall-time-based stopping criteria to limit the duration of each search run.
SISRs Since the code of SISRs is not publicly available, we reimplement SISRs in C++ based on the description of the authors. Our implementation matches the performance reported in the original paper. We use a time based stopping criteria to limit the duration of each search run.
SGBS-EAS We conduct evaluation runs using the code that was made publicly available by the authors. For the CVRP with 100 customers, we use the publicly available model which has been trained on instances from the same distribution our test instances. We run SGBS-EAS using the configuration proposed in the original paper in which search runs are limited based on the number of iterations. Additionally, we conduct experiments in which we limit the search to the same runtime as NDS. More precisely, we limit the total runtime to 13.9 hours, which corresponds to 5 seconds per instance. We increase the beam width to 7 in this experiment to increase exploration and to better use the available runtime. For the VRPTW with 100 customers, no pretrained models are publicly available. We hence train the models ourselves on instances sampled from the same distribution as our test instances to allow for fair comparison. We use the same training configuration in the original paper (Kwon et al. 2020) and train the model for 10,000 epochs with each epoch comprising 10,000 instances. At test time, use the same setup for the VRPTW for the CVRP.
NeuOpt We evaluate NeuOpt on the CVRP with 100 customers using the code and trained model made available by the authors. Note that the model has been trained on CVRP instances sampled from the same distribution as the test instances. Like SGBS-EAS NeuOpt is evaluated only on instances with 100 customers, as its autoregressive nature makes training larger instances impractical. We conduct two sets of experiments: one where the search is limited to 10,000 ilerations (as in the original paper) and another where the search is constrained by runtime, using the same runtime limits applied to NDS. For the former, we set the dynamic data augmentation (D2A) parameter to 1, and for the latter, we set D2A to 5 to enhance exploration during the search.
BQ We conduct evaluation runs using the code and trained model made publicly available by the authors. BQ is supervised learning approach that focuses on generalization and requires near-optimal solutions during training, which are only obtainable for smaller instances. Consequently, we use the publicly available model, trained on instances with 100 customers, to solve all four problem sizes considered in our experiments. Note that the model has been trained on the same distribution as our test instances with 100 customers. We perform two sets of experiments: one with a beam width of 16 and another with a beam width of 64. In both sets, we use a batch size of 1, as the beam search constructs multiple solutions in parallel, fully utilizing the parallel computing capabilities of our GPUs. For the CVRP with 2000 customers, results are only reported for beam width of 16, beam width of 64 leads to out-of-memozy issues.
We evaluate LEHD using the publicly available code provided by the authors. LEHD is a super- vised learning approach focused on generalization and requires near-optimal solutions during training, which are only obtainable for smaller instances. Consequently, we use the publicly available model, trained on instances with 100 customers, to solve all four problem sizes considered in our exporiments. Note that the model was trained on the same distribution using the random re-construct method proposed by the authors. We conduct two sets of experiments: one where the scarch is limited to 1,000 iterations (as in the original paper) and another where the search is constrained by runtime, using the same runtime limits applied to NDS.
LEHD
our test instances with 100 customers. We evaluate LEHD
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UDC We evaluate UDC using the code that made publicly available by the authors. For fair comparison, we use custom models that are trained specifically for each considered problem size, instead of using the provided one-size-fits-al trained model. We use the default training configuration, but change sample_st size to 30 for the CVRP with 1,000 customers to reduce GPU memory usage. For the CVRP with 2,000 customers, we are not able lo conduct training due to memory constraints. The training instances are sampled from the same distribution as our test instances.
We evaluate UDC's test-time performance in two sets of experiments: one with the search limited to 250 stages (as in the original paper) and another constrained by runtime, using the same limits applied to NDS, For the CVRP with 500 customers, batch size of 6 is used, while for the CVRP with 1,000 customers, batch size of 4 is applied. The batch size is chosen to fully leverage the available GPU memory.
GLOP We evaluate GLOP using the code and trained models provided by the authors 7 GLOP is a divide-and--onquer approach tailored for large-scale problems, and we evaluate it only on CVRP instances with 1,000 and 2,000 customers, as these are the instance sizes for which trained models are available. The training instances used by the original authors are sampled from the same distribution our test instances. We run GLOP with the configuration outlined in the original paper, using LKH3 subsolver and single CPU core. Notably, there is no straightforward way to extend GLOP to longer runtime settings for the CVRP.
http:///gttab..cm//mmmyyyy/////p
18

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Published in Transactions on Machine Learning Research (05/2025)
B
Training Curves
Figure presents the validation performance throughout training for all experiments conducted across the three problem lypes and four problem sizes. Note that the training of the models for 2-2000 is warm-started using the model weights from N=1000 after 1,500 epochs.
CVRP100
VRPTW100
PCVRP100
15.71
10.06
13.17
10.04
Costss
13.16
Coetes
13.15
10.02
13.14
10.00
13.13
WW
mmm.
13.12
500
Epoch
Epoch
Epoch
CVRP500
VRPTW500
PCVRP500
50.50
45.00
38.0
50.25
46.75
50,00
Costs
69.75
44.25
37.6
69.50
46,00
37.4
69.00
43.50
300
1500
300
Epoch
Epoch
Epoch
CVRP1000
VRPTW1000
PCVRP1000
43.75
94
86
43:25
Costs
Costs
8G
43.00
84
91
62.50
300
2000
500
1500
Epoch
Epoch
Epoch
CVRP2000
VRPTW2000
PCVRP2000
161.30
08.20
174.8
161.25
08:24
161.20
58.22
Cootas
174.6
Costs
161.15
161.10
58.20
174.4
161.05
58.18
161.00
174.2
Epoch
Epoch
Epoch
Figure 5: Validation performance throughout the training process.
19

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Published in Transactions on Machine Learning Research (05/2025)
C
Visualizations of Policy Rollouts
Figures 6 and show visualizations of different policy rollouts for the CVRP, PCVRP, and VRPTW, respectively. For cach problem, we display two different instances, and for each instance, six rollouts are shown. Customers selected for deconstruction are circled in red. We note that the nodes selected for each deconstruction differs, sometimes significantly, allowing NDS to try out a variety of options in each iteration.
(a) Rollouts instance 1.
Roilort
Rollout
Rollont 5
Roilout 6
Rollouts for instance 2.
Hollout
Rollout 2
Rollout
Rollont
Rollout 5
Rollsnt
Figure 6: Rollouts for two selected instances for the CVRP with - (best viewed in color).
20

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Published in Transactions on Machine Learning Research (05/2025)
(a) Rollouts for instance 1.
Rollout
Rollout 2
Rollout 3
Rollout.
Rollout
Roilout
(b) Rollouts for instance 2.
Rollout
Rollout 2
Rollout 3
Rolloat
Rollout
Rollost
Figure 7: Rollouts for two selected instances for the VRPTW with N=100 (best viewed in color).
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Published in Transactions on Machine Learning Research (05/2025)
(a) Rollouts for instance 1.
Rollout
Rollout 2
Rollout
Rollout
Rollout
Roilout
(b) Rollouts for instance 2.
Rollout
Rollout 2
Rollout 3
Rolloat
Rollout
Rollost
Figure 8: Rollouts for two selected instances for the PCVRP with N=100 (best viewed in color).
22

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Published in Transactions on Machine Learning Research (05/2025)
D Performance Across Different Runtime Limits
We evaluate the performance of NDS and SISRs under varying runtime limits. The results are based on the same trained models and evaluation setup described in Section [4.4] but with test sets reduced to 100 instances each to manage computational costs. The hyperparameters remain as described in Section 4.4 except for the number of augmentations for the PCVRP. Specifically, for the PCVRP with 1000 customers, 256 augmentations are used for runtime limits of 240 seconds or more. For the PCVRP with 2000 customers, 64 augmentations are applied for runtime limits of 240 seconds or less, and 256 augmentations for a runtime limit of 16 minutes. The runtime limits this experiment are selected based on instance size, ranging from seconds for instances with 100 customers to 16 minutes for instances with 2000 customers.
9 presents the results across all problems and problem sizes. Overall, NDS significantly outperforms SISRs all problems. SISRs only surpasses NDS in few cases under very short runtime limits. The performance gap between NDS and STSRs appears to depend more problem size than problem type. For example, on instances with 100 customers, SISRs struggles to achieve the solution quality that NDS attains in just seconds, even after 60 seconds of runtime.
Figure
CVRP100
VRPTW100
PCVRP100
12,00
16.57
Method
NDS
16.06
12.55
9.70
Chels
15.55
Clata
Costs
16.54
12.86
9.74
15.53
9.72
15.32
1284
20
60
50
Rantime
Rantime
Puraiuee
CVRP500
VRPTW500
PCVRP500
Method NDS
36.00
statll
Coels
Costs
44.0
06.55
48.8
43.9
68.7
36.50
48.6
43.8
4377
Rantine
Rantimm
Funnime
CVRP 1000
VRPTW1000
PCVRP1000
81.8
Methed SD8
41.00
81.0
Ststt
41.25
87.6
Coels
Coets
81.4
41.20
87.4
81.2
41.15
87.2
81.0
41.10
80.8
1011
2001
3001
4001
5001
100
200
3000
4101
500
100
200
3000
500
Rantine
Ramtime
Puntinne
CVRP2000
VRPTW2000
PCVRP2000
Mathee
XDS
169.5
159.3
Costs
56.2
Coells
169.0
Costs
56.1
159.0
168.3
56.0
168.0
55.9
2011
1001
stil
1000
200
600
1000
200
400
GI0
800
1600
Ramtine
Rantime
Rantine
Figure 9: Performance of NDS and SISRs across different runtime limits.
23

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Published in Transactions on Machine Learning Research (05/2025)
E Impact of Hyperparameters on Search Performance
We evaluate the impact of key hyperparameeeers in the ASA search procedure. To manage computational costs, this experiment is conducted solely on the CVRP with 500 nodes. We perform multiple evaluation runs the test set, varying the number of augmentations A, the number of rollouts K, and the threshold factor 6, while keeping all other parameters at their default values specified in Section 5.2
The results, presented in Figure I indicate that values of A equal or greater than 8 yield the best perfor- mance. The number of rollouts K has little impact on performance within the tested range of 100 to 300. For the threshold factor 6, values above 15 yield consistently strong results. Overall, the search procedure demonstrates robustness hyperparameter choices, suggesting that tuning not critical for achieving good performance.
36.06
36.61
Costs
36.62
36.60
36.58
36.56
100
250
Number Augmentations A
Number Rollouts K
Threshold Factor
Figure 10: Search performance on the CVRP500 for different hyperparameter setting.
F
Ablation Study: Initial Improvement Steps
We assess the impact of initial solution improvement through ablation study. To this end, we train models on problem instances with 500 nodes, omitting any solution improvement steps before the main training phase (lines 10 17 in Algorithm I]. As result, the model also learns to refine low-quality start solutions.
Figure compares validation performance during training with and without the initial improvement steps. The results reveal substantial performance differences, with training runs that include initial improvement significantly outperforming those without, highlighting the effectiveness of this techmique. We hypothesize that initial improvement is beneficial because learning directly from poor-quality start solutions provides limited value; improving them is trivial, whereas most of the search process focuses on refining higher-quality solutions.
CVRP500
VRPTW500
PCVRP500
46
Costs
38.5
Costs
Coets
45
w/ Init. Imprro. w/o Init. Improv.
50
38.0
44
37.5
500
1000
1500
2000
0
500
1000
1500
2000
D0K)
1000
1500
2000
Epoch
Epoch
Epoch
Figure 11: Validation performance for training runs with and without initial solution improvement steps.
24

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Published in Transactions on Machine Learning Research (05/2025)
G Generalization to Smaller and Larger Instances
We also evaluate the generalization ability of NDS to larger and smaller instances by using models trained on instances with 500 customers to solve instances with 100, 1000, and 2000 customers. In this experiment, all hyperparameters are set identically to those the main experiment.
The results, presented in Table f also include in-distribution performance from the main experiment, where instance size-specific models are used, to allow for an easier comparison. The findings indicate that models trained 500 customer instances perform well instances with 100 and 1000 customers, with significant degradation in performance. However, on instances with 2000 customers, slight drop in performance is observed, particularly for the CVRP.
Table 4: disttf--iittibutiinn (OOD) performance of models trained on instances with 500 nodes, evaluated both larger and smaller instances, compared to the in-distribution (ID) performance of size-specific models.
N=100
N=1000
N=2000
(Smaller)
(Larger)
(Much Larger)
Obj-
Gap
Time
Obj-
Gap
Time
Obj- Gap
Time
CVRP
HGS
15.57
41.51
121
57.38
241
NDS (ID)
15.57
0.04%
41.11
-0.90%
120
56.00
-2.34%
240
NDS (OOD)
15.59
0.13%
41.32
-0.45%
120
57.63
0.43%
240
VRPTW
PyVRP-HGS
12.98
90.35
120
173.46
240
NDS (ID)
12.95
-0.19%
87.54
-3.14%
120
167.48
-3.50%
240
NDS (OOD)
12.97 -0.00%
87.54
-3.11%
120
167.76
-3.28%
240
PCVRP
PyVRP-HGS
10.11
84.91
120
165.56
240
NDS (ID)
9.90
-2.07%
5
80.99
-4.71%
121
158.09
-4.60%
241
NDS (OOD)
9.96
-1.50%
5
81.05
-4.53%
120
160.20
-3.24%
240
25


Reviews:
[{"review_id": "7sPIJcTUOV", "reviewer": "Reviewer_tv5y", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n\n- The paper is well-written and follows a logical structure.\n\n### Weaknesses\n\n- **Generally speaking, the performance gap between the proposed method and SISRs (which seems to be the best algorithm for three types of problems) is very mild (almost negligible), especially when $N$ is large. The authors should have reported the comparison results of their method against the best one (i.e., SISRs).**\n\nMy other concerns revolve around the fairness of some comparisons:\n\n- The authors note that some approaches process instances in batches, but not address the potential impact of batch size on resource allocation when time limits are imposed. With a fixed time limit, increasing the batch size may lead to fewer resources being allocated to each individual instance, potentially resulting in lower average performance. Moreover, the batch sizes used in different approaches are inconsistent, e.g., 1 for BQ and 4/6 for UDC, as detailed in Appendix A. I suggest the authors provide further clarification on this matter.\n\n- If I understand correctly, the baselines BQ and LEHD are trained using problem instances of size 100 and evaluated across four different problem sizes. In contrast, NDS is trained and evaluated for each problem size separately, rather than being trained on a specific size and evaluated across all sizes. For a fair comparison, BQ and LEHD should also be trained and evaluated for different problem sizes. The authors justify not doing so by claiming that these methods require near-optimal solutions for supervised learning, which are not obtainable for large problems. However, this reasoning is not entirely convincing. Off-the-shelf solvers and tools, such as SISRs, could be used to search for high-quality solutions as training labels (\"near-optimal\" may not be necessary). Besides, I also recommend reporting an additional NDS experiment that evaluates on the four problem sizes using the model trained on problem size 100.\n\nAdditionally, a minor concern is the lack of analysis on the impact of certain hyper-parameters in the ASA algorithm, such as the number of augmentations and rollouts. This would help readers better understand the roles of these components within the overall algorithm.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper explores learning-based solution construction for VRPs and their variants, proposing a deconstruction-then-reconstruction approach. It leverages guidance from trained DNNs and takes advantage of parallel computation via GPUs. While the individual elements of the approach are simple and not new, their combination seems effective. However, the proposed method only demonstrates minor performance improvements over the baseline algorithm SISRs on every dataset.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Below are some suggested changes:\n\n- Add clarifications (if any), regarding the major weaknesses in the manuscript.\n- Specify the static instance information mentioned in Section 3.3.1 and explain how it is encoded.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "kUUWIdR9ng", "reviewer": "Reviewer_QeAu", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**: see summary. \n\n**Weaknesses**:\n\n- **Search efficiency.** The authors mention in the introduction that existing DRL methods fail to generate as many feasible solutions as state-of-the-art heuristics. While I appreciate this observation, I am not entirely sure why this is the case. For example, existing DRL agents can also explore different solutions, and this process can be accelerated with GPUs and further enhanced by more sophisticated search techniques, e.g., beam search. Since this is a key observation in the paper, I believe the limitations of existing methods warrant a more detailed discussion. Specifically, why can’t current DRL approaches efficiently explore the solution space? Could the proposed method be used to enhance their search efficiency? Moreover, why is the number of feasible solutions explored more critical than solution quality? The final point has been empirically validated, but the argument could be further strengthened by providing additional intuition on the underlying reasons. \n\n- **Greedy insertion.** On my first read, I was initially confused by the fact that the insertion algorithm only reinserts removed nodes into the same route. For instance, in a CVRP instance with two vehicles, nodes removed from the first vehicle’s route would not be inserted into the second vehicle’s route. However, I later realized that in RL-generated VRP solutions, the routes of multiple vehicles are typically concatenated into a single array, separated by the depot node. As a result, performing removal and insertion within this concatenated array effectively enables node swapping between different routes. Is this understanding correct? The authors may want to clarify this point in the paper.\n\n- **Initial improvement steps.** In Algorithm 2, the authors apply a few local search steps to improve the initial solution quality before calling/training the RL agent. I wonder how crucial this step is. It would be helpful if the authors provided additional ablation studies to assess its importance, especially since it adds to the computational complexity---particularly given that such steps are typically performed on CPUs, whereas other RL steps run on GPUs.\n    \n- **Generalization.** I appreciate the experimental results demonstrating the model's ability to generalize across different data distributions. Another generalization study the authors might consider is evaluating how a model trained on small instances performs on larger instances. This could be of greater practical interest, as customer distribution and vehicle capacities are typically stable, if not fixed, within a city, while the number of customers can vary significantly over time. Since logistics companies often lack the resources to train a separate model for each $n$, it would be valuable to understand how well the model generalizes in this scenario.\n\n- **Table 2(a)**. No number is bold in the ``CVRP'' column. What do the numbers represent? Are they objective values?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a novel learning-based approach to solving vehicle routing problems. Unlike existing deep reinforcement learning (DRL) methods that iteratively construct solutions, the proposed approach trains a DRL agent to deconstruct a given solution by removing certain customer nodes, followed by a greedy heuristic that reinserts the removed nodes. This represents a fundamentally different strategy, as the focus of this algorithm is on efficiently exploring the solution space (like many meta-heuristics) rather than generating high-quality solutions in a few iterations (like existing DRL methods).\n\nOverall, the paper is easy to follow. I reviewed it critically but did not identify any major flaws. I believe this paper is a strong fit for TMLR. The authors are encouraged to consider the following points to further improve the paper.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Clarify the questions listed above\n- Potentially add ablation and numerical studies as suggested above. This is not necessary tho as the current experiments are already pretty comprehensive.", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "wSiUpEPzTB", "reviewer": "Reviewer_LxMb", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Ref to \"summary\"", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces a deep learning-based simulated annealing approach for solving vehicle routing problems. The motivation and proposed concept are clear and promising: while pure deep learning methods excel in intuitive heuristics but lack precision, traditional OR methods offer accuracy but lack adaptability. The key challenge lies in effectively integrating these two approaches. Experimental results indicate certain advantages of the proposed method. In particular, the out-of-distribution results are impressive. However, while the overall idea is well-articulated, the lack of technical details makes the paper somewhat difficult to follow.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Methodology descriptions are not clear enough:\n\n1. Section 3.1: The vehicle routing problem is not explicitly defined—what are the input parameters, and what are the expected outcomes? While VRP is a well-known problem, providing a precise definition would improve clarity.\n\n2. Sections 3.1 and 3.3: What constitutes an instance $l$, and how is it encoded in the neural network? The statement that it \"involves customers $c_1, \\cdots, c_N$\" is vague and lacks clarity.\n\n3. Sections 3.1 and 3.3: How is the solution $s$ defined? What features are used to describe a solution in neural networks? This information should be included, at least in the appendix.\n\nExperimental evaluations:\n\n4. Configuration: The paper provides detailed configuration settings but seems not clearly state how these were chosen (correct me if I am wrong). Are all the configurations tuned to the best? \n\n5. Robustness to Configuration Changes: If configuration parameters are perturbed, how does the robustness of the traditional OR method compare to the proposed learning-based method?", "broader_impact_concerns": "No broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers agree that the latest version of the manuscript should be accepted. Congratulations!

One remaining concern we discussed is how the GAP is (and should be) calculated in Table 3. One reviewer found that computing it with respect to the baseline might be misleading to readers at first glimpse. While NDS does seem to dominate other methods, it appears to be fairly close to the results of some of them (e.g. SISRs). On the other hand, I surveyed a few related papers, and it seems to be a common practice in the literature. 

In any case, the Obj seems to be the critical metric. With that in mind, I suggest the others focus on that metric and, for example, bold that column instead of the Gap. It would also be helpful to be explicit about what bolding means (e.g. by specifying it in the caption). If it's "simply" the highest value, it might be worth also highlighting the second-best (ideally, if you have a measure of uncertainty, bold all methods that are statistically indistinguishable). 

In the caption, you mention a 7% gap, but as far as I understand, it's closer to 5-6% ( (59.45 - 56)/59.45 ), no? Maybe I am misunderstanding. 

Please update the caption to reflect the changes you end up making.
Decision: Accept (poster)