Chain-of-Context Learning: Dynamic Constraint Understanding for Multi-Task VRPs
Authors: Shuangchun Gui
Suyu Liu
Xuehe Wang
Zhiguang Cao
Venue: ICLR 2026 Poster (2026)
Keywords: Multi-task learning
deep reinforcement learning
vehicle routing problems
Primary area: optimization

Abstract: Multi-task Vehicle Routing Problems (VRPs) aim to minimize routing costs while satisfying diverse constraints. Existing solvers typically adopt a unified reinforcement learning (RL) framework to learn generalizable patterns across tasks. However, they often overlook the constraint and node dynamics during the decision process, making the model fail to accurately react to the current context. To address this limitation, we propose Chain-of-Context Learning (CCL), a novel framework that progressively captures the evolving context to guide fine-grained node adaptation. Specifically, CCL constructs step-wise contextual information via a Relevance-Guided Context Reformulation (RGCR) module, which adaptively prioritizes salient constraints. This context then guides node updates through a Trajectory-Shared Node Re-embedding (TSNR) module, which aggregates shared node features from all trajectories' contexts and uses them to update inputs for the next step. By modeling evolving preferences of the RL agent, CCL captures step-by-step dependencies in sequential decision-making. We evaluate CCL on 48 diverse VRP variants, including 16 in-distribution and 32 out-of-distribution (with unseen constraints) tasks. Experimental results show that CCL performs favorably against the state-of-the-art baselines, achieving the best performance on all in-distribution tasks and the majority of out-of-distribution tasks.

Published as a conference paper at ICLR 2026
CHAIN-OF-CONTEX LEARNING: DYNAMIC CON- STRAINT UNDERSTANDING FOR MULTI-TASK VRPs
Shuangchun Gui', Suyu Liu?, Xuehe Wang2, Zhiguang Cao
1 School of Computing and Information Systems, Singapore Management University, Singapore 2 "Gunngdong Laboratory of AI and Digital Hconomy, Shenzhen, Guangdong, China
School of Artificial Intelligence, Sun Yat-sen University, Zhuhai, Guangdong, China gistuangchunoout.ooo com, liusuyu976 70 gmail com, wangxuehe@mail sysu. edu. cn, zgcac edu.sg
ABSTRACT
Multi-task Vehicle Routing Problems (VRPs) aim to minimize routing costs while satisfying diverse constraints. Existing solvers typically adopt a unified rein- forcement learning (RL) framework to learn generalizable patterns across tasks. However, they often overlook the constraint and node dynamics during the deci- sion process, making the model fail accurately react to the current context. To address this limitation, we propose Sluain-of-Contes Learning (CCL), a novel framework that progressively captures the evolving context to guide fine-grained node adaptation. Specifically, CCL constructs step-wise contextual information via Relevance- Guided Context Reformulation (RGCR) module, which adaptively prioritizes salient constraints. This context then guides node updates through a Trajectory-Sharee Node Re-embedding (TSNR) module, which aggregates shared node features from all trajectories' contexts and uses them to update inputs for the next step. By modeling evolving preferences of the RL agent, CCL captures step-by-step dependencies in sequential decision making. We evaluate CCL on 48 diverse VRP variants, including 16 in-distribution and 32 out-of-distribution (with unseen constraints) tasks. Experimental results show that CCL performs favorably against the state-of-the-art baselines, achieving the best performance on all in-distribution tasks and the majority of -uttof-distribution tasks.
INTRODUCTION
The vehicle routing problem (VRP) seeks to determine optimal routes for fleet of vehicles serve set of customers while satisfying operational constraints such as vehicle capacity. Efficiently solving VRPs can significantly reduce transportation costs and improve service quality, making a critical task in logistics and supply chain management (Toth and Vigo, 2014; Konstantakopoulos et al., 2022; Garaix et al., 2010; Dondo et al., 201 I). Traditional approaches (Perron and Furnon; Lin and Kernighan, 1973; Vidal et al., 2020) often rely on heuristic-based solvers, such as LKH (Lin and Kernighan, 1973) and HGS (Vidal et al., 2020). While effective in certain settings, these methods computationally intensive and typically require extensive hand-crafted rules to adapt different problem variants. Recently, neural networks have emerged as a promising alternative due to their flexibility and ability learn generalizable policies (Joshi al., 2019; Kool et al., 2019; Kwon et al., 2020; Wu et al., 2021; Ma et al., 2023; Sun and Yang. 2023; Bengio et al., 2021; Bogyrbayeva al., 2024; Hottung and Tierney, 2020; Hottung al., 2022: Xin et al., 2021: Chalumeau al., 2023; Ma et al.. 2023: Chen et al., 2023a). These neural solvers are trained offline using historical or synthetically generated instances, enabling fast inference at test time for a given VRP variant. However, real-world VRPs often involve more complex and diverse constraints beyond vehicle capacity, leading to multi-task VRPs, where each task involves different combination of constraints. This makes the neural VRP solvers for specific single task less effective due to the massive yet necessary re-training or fine-tuning.
"Corresponding author.

---

Published as a conference paper at ICLR 2026
In multi-task VRPs, the commonly studied constraints include backhaul demands (B) (Zong et al., 2022; Kong et al., 2024), open routes (O) (Tyasnurita et al., 2024; Bezerra et al., 2023), route duration limits (L) (Oliveira et al., 2025), customer time windows (TW) (Zhang et al., 2022; Lin et al., 2021), mixed backhaul (MB) (Wang et al., 2024), and multi-depot settings (MD) (Karakatić and Podgorelec, 2015). To tackle the multi-task scenario, number of neural models (Liu al., 2024; Zhou et al., 2024a; Berto et al., 2024; Li et al., 2025) have been developed using a unified reinforcement learning (RL) framework, which encodes both constraint information and node attributes into static embeddings. The decoding stage follows Markov Decision Process (MDP). For given VRP task, the model combines a global context, such as current time or remaining vehicle capacity, with these static node embeddings to select the next node. Since node priorities change across decoding, static node embeddings, which remain fixed across decoding steps, cannot reflect this dynamic property. While the context is updated, such a misaligned context-node pair may lead to inaccurate state estimation, thereby misjudging the next decision.
To overcome this limitation, argue that constraint requirements should be explicitly integrated into the step-wise context and used to adaptively refine node-level representations. In single-task VRPs, dynamic decoding mechanisms, such the removal of visited nodes (Xin et al., 2020), have been used to reflect evolving routing decisions. While conceptually related, extending such a mechanism multi-task settings introduces three unique challenges: (1) The importance of each constraint may vary across decoding steps, e.g., the open route constraint becomes more critical as vehicle's sub-route nears completion. Applying uniform attention across all constraints at each step, such as the one in (Li et al., 2025), limits the model's ability to focus on the most important ones. Moreover, performing RL-based node refinement into VRPs poses issues with efficiency and sequential dependencies. On the one hand, (2) tuati-trajectoriis involve different contexts at each step, and re-embedding the nodes for each context (e.g., (Xin et al., 2020)) causes a heavy computational burden. On the other hand, (3) multi-task VRP solvers (Li et al., 2025; Berto et al., 2024; Zhou et al., 2024a) typically refine the node representations at step-í using only the initial (step-0) embeddings and the current context. A misaligned state may fail to capture the status of the current decoding step, thereby limiting the model's 's ability to accurately represent the Markov property, which is essential for coherent sequential decision-making.
To address these challenges, we propose Chain-of-Conin Context Learning (CCL), a novel framework for constraint-aware, step-wise reasoning in multi-task VRPs. Specifically, to tackle Challenge (1), CCL constructs sep-wise contextual information using a Relevance- Guided Context Reformulation (RGCR) module. RGCR combines constraint-specific attributes (e.g., remaining capacity for B and current time for TW), and adaptively emphasizes each constraint according to its similarity to the current node embedding. To address Challenge (2), we design a Trajectory-Siie -Shared Node Re-embedding (TSNR) module, which enables efficient refinement of node features. TSNR employs shared node embeddings as queries and uses multi-trajectory contexts as keys and values in a multi-head attention mechanism, avoiding redundant re- rebeddnng for each trajectory. To resolve Challenge (3). TSNR updates node embeddings in the environment and feeds them as queries to the next decoding step. This design allows CCL to capture sequential dependencies and model the evolution of node importance over time.
We evaluate CCL. on the combinations of six core constraints (B, O, I., TW, MB, MD), resulting in 16 in-distribution and 32 out-of-distributio multi-task VRP variants. Our contributions are summarized as follows: (1) Conceptually, we correct misalignment in prior VRP formulation, by learning step-wise context and node status for a more accurate state. (2) Methodologically, we propose RGCR to integrate constraint requirements into the step context, along with TSNR to facilitate effective refinement and capture sequential dependencies. (3) Experimentally, our method achieves superior tasks.
results on all seen (in-distribution) tasks and the majority of unseen
2 PRELIMINARIES
Problem Definition. The classical vehicle routing problem (VRP) aims to determine set of sub- routes that minimize total travel cost while satisfying customer demands. In each sub-route, a vehicle departs from the depot, delivers goods to subset of customers, and returns to the depot, subject to the following standard constraints: (1) each sub-route starts and ends at the depot: (2) each customer is visited exactly once; and (3) the total demand on each sub-route does not exceed the vehicle's
2

---

Published as a conference paper at ICLR 2026
capacity. Formally, the problem is defined on graph where the set of nodes V {ep, D1 ........ UN represents the depot (ui) and N customer locations. Each customer node v; is associated with a demand value 6; E [0, Q), where denotes the vehicle's 's capacity.
Following (Berto et al., 2025), we extend this classical setting by considering six additional constraints commonly studied in multi-task VRPs: (1) Open Routes (O): In problems like OVRP, this constraint denoted by a binary flag o E {0, 1}, which defines whether a route must return to the depot. When 1, vehieles are not required to return the depot after completing their route. (2) Duration Limits (L): In problems like VRPL, this constraint enforces a maximum route length / to promote workload balancing across sub-routes. (3) Backhaul Demands (B): In problems like VRPB, customer nodes are classified into linehaul and backhaul types. The vehicle must first complete all linehaul deliveries (goods from depot to customers) before collecting backhaul items (goods from customers to depot). Each customer has two types of demand: 8 for linehaul and 67 for backhaul, with 6%, 8 E Q). (4) Mixed Backhaul (MB): In problems like VRPMB, this constraint relaxes the rrchaul-beforee backhaul requirement, allowing both types of customers to appear in any order along the route, while still respecting the capacity constraint. (5) Time Windows (TW): In problems like VRPTW, each customer v is associated with an early time !;, a late time 14, and a service duration 47. Vehicles must arrive before E and wait if they arrive earlier than t5. ensuring service occurs within the specified window. (6) Multi-Depot (MD): In problems like MDVRP, this constraint allows multiple depot nodes instead of a single depot. Vehicles may begin their routes from any depot in the set, introducing additional complexity in depot assignment.
Markov Decision Process for Multi-Task VRPs. The multi-task VRP solver acts as a single agent, using the encoder-decoder architecture its policy network. The policy generates a node sequence autoregressively, using a Markov Decision Process (MDP) environment
M = AS,A.,)
(I) State (S) consists of node embeddings and context embeddings. During decoding. following (Liu et al., 2024; Zhou et al., 2024a; Berto et al., 2024; 2025; Li et al., 2025), the model explores from diverse starting points, forming multiple trajectories in parallel. Each trajectory maintains its own context (e.g.. current time and used capacity), while all trajectories share the same set of node embeddings.
(2) Action (AA) corresponds to selecting the next node to visit. The policy network takes the current state as input and generates trajectory-speciff probability distribution over feasible nodes, allowing each trajectory to independently select its next action based on the predicted probabilities.
(3) Transition (P) updates the environment after a node selected. This modifies the environmental routing information, such as the vehicle's current position and remaining capacity. The updated environment then defines the next context embedding and continues the decision process.
(4) Reward (R) is defined as the negative total route length. After all nodes are visited, each trajectory computes its own negative route length as the reward. These rewards, together with the action log- pog--abbabilitiss produced by the policy network, are aggregated to form single training objective. The policy network parameters are then updated using the REINFORCE gradient (Williams, 1992):
VoJ Σ (R. - 6) log (a st),
(2)
where i is the index of the trajectory and (a s;) denotes the probability assigned to action a, conditioned on state 8;. is a shared baseline used to reduce gradient variance, computed as the average reward over all trajectories.
METHODOL.OGY
Existing works only update the context embeddings while keeping node embeddings fixed. As described in Section 2, the current state should include both candidate node embeddings and context embeddings. In method, treat context and node status pair, ensuring that both reflect the status of the current decoding step. During environment updates, we update both simultancously to maintain alignment between context and node information.
3

---

Published as a conference paper at ICLR 2026
EEncodingg
Ieput Altributss
Constraint Attribures
Current node Embedding
Node Embeddings
Context Embedking
Encoder
Linear Lnyers
TW
ooce
Decoder
4 CCL
Traj.
Traj. 2
Traj. N
n
Probubility
Action
RGCR
TSNR
Relevance
Relevance- Guided Context
Measurement
Reformulation (RGCR)
Scores
Feature Weighting
Concat
Decoding
1k
Concal
Norm
Trajectory-
Node Re-embedding (TSNR)
Add
Linear
Update
Update Xodes
Add & Norm
Add
Multi-Layer Perceptron
nall ngll nall
Update Context
& Lincar
V (
(a) Chain-of-Context Learning (CCL)
(b) Relevance- Gulded Context
(e) Trajectory-Shann Re-embedding
Figure 1: (a) CCL enables fine-gr fine-grained constraint understanding by integrating RGCR and TSNR into the decoding stage. (b) and (c) illustrate the internal architectures of RGCR and TSNR, respectively.
3.1 OVERVIEW OF CHAIN-OF-CONTEXT LEARNING (CCL)
Fig. (a) illustrates the training workflow of our proposed Chain-of-Conter Learning (CCL). It adopts the classic encoder-decoder paradigm, with Relevance- Guided Context Reformulation (RGCR) and Trajectory-Sharee Node Re-embedding (TSNR) integrated into the decoding stage. During encoding, each VRP instance-comprrsing constraints, depot, and node features-is embedded using a transformer encoder. Instances from 16 tasks, derived from the four constraints (B, L, O, and TW), combined into single batch for multi-task learning. In the RL-based decoding stage, CCL employs lightweight architecture to make decisions, with multiple trajectories explored in parallel from diverse starting points. At each decision step. RGCR aggregates the constraint-specific attributes and current node embedding to generate a context embedding. After collecting the context embeddings from all trajectories, TSNR refines the historical node embeddings by jointly processing them with the multi-trajectory contexts. These refined node embeddings are passed to the next step, progressively influencing context construction and forming a Chain-of- of-Context across decoding steps. The constructed context and refined node features are used together to make the decision, with all components jointly optimized using an RL objective. The inference procedure is similar to training setup, except is extended to evaluate generalization on two additional constraints, i.e., MB and MD, which are held out during training for zero-shot evaluation.
3.2 ENCODER
In the encoding stage, shown in Fig. (a), the inputs includes the contraint flag h and the node attributes h = [ho, hi,..., h.). These attributes are embeded through a transformer-based encoder E()), resulting in node embeddings H e RRN+1)xD
H = =[(h,h)
Following (Li et al., 2025), the constraint label h E R is a one-hot vector to indicate the presence of 4 constraints (i.e., B, O. L. TW). The depot attribute ho = (cT cy. 0,1} E R4 includes the depot Nx7 are customer features, with each node h, [[c] <3', [8). 62), [t; 14, (t]) specifing the coordinates, demands, and time windows. For simplicity, the encoder's input processing and architecture are provided in Appendix B.
coordinates, and labels of and L. he......... .hy E
3.3 RELEVANCE- GUUDDED CONTEXT REFORMULATION (RGCR)
4

---

Published as a conference paper at ICLR 2026
In multi-constraint scenarios, RGCR automatically lears the relative importance of constraints each step, enabling the model to focus on the most critical ones. In Fig. 1(b), RGCR undertakes three steps to formulate context embedding: (1) generating embedding for each constraint, (2) computing the correlation between each constraint embedding and the current node embedding, and (3) adaptively aggregating constraint embeddings based correlation scores.
In the constraint embedding formulation, we first extract the corresponding attributes and then project them through separate linear layers. For the i-th trajectory decoding step j. the current node index is denoted as Ti.j. The attributes for each constraint are summarized as follows:
c = 16 P Cash c - C .igh. ci = -17,,, " Gugg C P.gg 4.5):
= = G..: desh,
where 8', 86 denote the linehaul and backhaul demands, and qi2 is the remaining vehicle capacity. The coordinates c² e² specify node locations in the two-dimensional space, d is the remaining distance of the current sub-route, and d' is the total distance traveled. Moreover, 15, 19,"', ,,,,',, represent the earliest, latest, service times, and current time, respectively. These attributes are separately fed to linear layers for producing constraint embeddings, denoted as:
C = H(c),))
(5)
where C E ER., { O, for projection. In correlation computing, these constraint embeddings interact with the current node embedding to produce correlation scores, denoted as:
O, TW]ir is the constraint type, and H(--) denotes the linear layer used
H
(6)
where H.,, E RDD is the current node embedding, and denotes the dot product used for calculating the correlation scores (or similarities). In constraint aggregating, the unified constraint embedding is obtained by adding the original and enhanced constraint embedding, denoted as S., = S + S The original part is defined the concatenation of the four constraint embeddings from Eq. (5):
$.j= H(Concat ( (Ci. Cluy C..3 C )),
where (-ncat(-) denotes concatenation along the feature dimension, resulting in a concatenated embedding of size N 4D. H(-) is linear layer that projects the 4D input back to D, resulting in St I2 For the enhanced part, we apply a weighted sum over the constraint embeddings:
B L.. C + st. Ct, + C. + st.
TW C.
8
The final context embedding is aggregated from the unified constraint and current node embeddings:
Crj H(Concat (S.,,, Hr,,)).
3.4 TRAJECOORY-SARRD NODE RE-EMBEDDING (TSNR)
To capture spee-spe states influenced by the current context, we aggregate contextual semantics from other nodes and multi-trajectory contexts into the node embeddings. As illustrated in Fig. 1 (c), this is achieved via multi-head attention mechanism, where node embeddings serve as queries, and the unified node-context information acts as keys and values. Formally, step j. we denote the context embedding for N trajectories as C Concat (CI,j; C2.j.... ČN.s), where C₁ E NxD By using the last step node H R (N+1)) the query, keys, and values are represented as
qi H(Norm(HH--))
kj, V
(HooormmConcca(HI--1.. Ü,)),
(10)
where q NXD and ky. (N++)xX Norm(-) denotes the Root Mean Square (RMS) normalization layer (Zhang and Sennrich, 2019). For simplicity, we use the same notation H(()) to denote the module that produces k; and v,. To calculate attention weights, we further incorporate a distance-based bias to prevent the model from overfitting to TW. This bias term, denoted as consists of two parts: the node-node and node-context distance:
B, Concat (d .d.
n- {dm,n mM, n E {0, (0,1,...,N1) NJ).
(11)
d.- {dm, Im E (T. (T.,) :..)), n E (0, [,...,,,,] NN),
5

---

Published as a conference paper at ICLR 2026
where d=-= R RV++1)(N++1) d d,- RVN+++1) and each element of it takes the form dm,x ||cm c, with с {e² e denoting Fuclidean coordinates. For the node-context part, extract the coordinates of the current node for each trajectory (indexed by {Tı.j, 72.j..... -N.j }) and compute their distances to all candidate nodes. The attention weights are subsequently computed as
A = So Sofftax(q.kk) //DVV +D+ Bj),
(12)
where A,, B, E ( and Softmax()) is the softmax operation. The re-embedded node representations are computed as follows:
H, - Ayys, H H, + MLP MLP (Norm(H,))
(13)
We preserve the updated node embeddings Hj from the current step and use them as input queries for the next step, with update frequency controlled by probabilities Ptr (training) and Pt. (testing).
3.5 STEP-WISE DECISION AND TRAINING OBJECTIVE
Once the context embedding C NxD and current node embeddings H3 RNN+I)xD obtained, we use them to predict the selection of the next node, and then compute the RL objective function to optimize model parameters. In the step-wise decision stage, we employ classic decoder (shown in Appendix B.2) to acquire the probability of selecting the next node. This procedure is represented as follows:
P, H,,
(14)
where Pj E (Nx(+11), D(-) denotes the decoder, while M2 is a mask that prevents revisiting previously selected nodes. If all constraints are satisfied, the node with the highest probability is selected as the next node to visit. Otherwise, the depot is selected. After one interaction. the model generates N solution trajectories, each denoted as т₃ [Tr,1. T.. T }. where
{1, NJ and N' is the total number of decision steps. The RL objective is then computed using the reward of each trajectory and the log- prg-obbbbiliiies of selected nodes, as illustrated in Eq. 2.
EXPERIMENTS
4.1 DATASETS AND EVALUATION METRICS
We evaluate CCL on 48 VRP variants. Following (Berto al., 2025), node locations sampled uniformly from the 2D Fuclidean space 1). Fach vehicle starts at the depot with a capacity Q=I and maximum route duration (=3. Linehaul and backhaul demands are sampled as integers from [1, 10) and scaled by a factor of 30 + N/5, where N is the number of customers. In backhaul settings, 20% of the customers are designated as backhaul, and the remaining 80% as linehaul. For time window tasks, early arrival times, service durations, and time window lengths are independently sampled from [0.0126, 4.25), [0, 0.15), and [1.8, 2.0), respectively. Late times are computed as the sum of early times and window lengths. The training set consists of 100,000 instances uniformly distributed across 16 variants. The best model checkpoint selected based on validation performance on CVRP (Capacitated VRP), using a held-out set of 128 instances. The test set comprises 48 variants, each containing 1,000 instances. We benchmark CCL against state-of-the-at (SOTA) basclines under two settings: 55 and N=100. We evaluate performance using three standard VRP metries: total routing length (" ("Obj.") "), performance gap ("Gap") ) to the strong baseline HGS-PyVRP (Wouda et al., 2024), and inference time. All metrics computed over 1,000 test instances, with "Obj." and "Gap" reported as averages and inference time as total runtime.
4.2 IMPLEMENTATION DETAILS
Our method is implemented in PyTorch (Paszke et al., 2019). All experiments are conducted on machine with an AMD EPYC 7702P 24-core CPU and single NVIDIA RTX L40S GPU. We use a batch size of 256 during training. The model adopts a 6-layer Transformer encoder, with both encoder and decoder sharing the same architecture: embedding dimension D=128, 8 attention heads, and a hidden dimension of 512. During decoding. node refinement is applied probabilistically. For instances with N=50, the refinement probability is 0.75 during training and 1.0

---

Published as a conference paper at ICLR 2026
Table 1: Performance on 16 seen in-distribution tasks. * denotes the strong baseline used compute the gap. Best neural approach is highlighted in bold; best existing SOTA is underlined.
Methods
100
Mathods
50
Okg. 1
Cup
Time-
0kg.
Cup
Time
Otg.
Cup J
Time
Okg...
Cup L
Time
BIGS PyVRP
10,372
10.4m
15.628
20.8m
IIGS-Py PRPP
16.031
10.4m
25. 25.425
20.8mm
MIPOMO
10,520
1.23%
2x
15.94]
2,030%
MITPMM
16.419
2/43%
2x
26.433
3.962%
CYRP
MVMoFF
10.499
1.220%
15.888
1.601%
MVMoFF
2.208%
76.390
3.780%
11s
10,502
1.257%
15,800
1.534%
RE-E
16.341
1.933%
3.154%
CaDA
10.505
1.257%
26
15,843
1.42%
CaDA
16.312
1.745%
1s
26.169
2.925%
9
CaDA!
10:471
0.9495
36
15,740
100%
13%
CaDA!
16.299
1.6%%
36
26.105
2.608%
14s
CCL
10.473
0.977%
13.823
1.287%
19%
CCL
16,190
0.979%
25.913
1.908%
21s
CCL
10.463
0.881%
15.787
1.08%%
CCL
24s
HGGS-PyVR
6.507
10.4m
9.725
20.8m
HIGS-PyRRP
10,510
10
16.926
20.8m
MTPPMO
6.717
3. 1949
26
10.216
5.028%
$6
MTPOMO
10.676
1.558%
26
17. 17,442
3.0225
96.
OVRP
MVMoF
6,705
3.0039
36
10.177
4,617%
11s
MVMoF
10.674
1.541%
36
17.416
2,870/%
12%
2.65%%
10.115
3,9966
RF-TE
10.645
1.2649
2.352%
CaDA
2.585%
1s
10:095
3.78015
CaDA
10.630
1.122%
12.283
2,000%
CaDA!
6,652
2.212%
36
10,050
3.425%
12s
CaDA!
10.621
1,00%
26
12,246
1,868%
14s
CCL
6,635
1,957%6
56
10.068
3.511%
20c
CCL
10,569
0,543%
66
17.125
1.142%
216
CCL
6.610
1.566%
10,012
2.936%
25%
CCL
10.564
0.506%
1.033%
20s
HGS-PYVRP
10.335
HGS-PyVRP
MTPOМO
7,105
2.97355
2%
10.882
5.364%
MIPPOM
11.823
1.307%
19.656
2,5935
100
MVMoE
7,089
2.744%
36
10.841
4.809%
11s
MVMoE
11.816
1.2455
46
19.637
2,499%
136
RE-TE
7,065
2.385%
2s
10.774
4,2339%
RF-TE
11,790
1,022%
26
19.555
2,00256
9%
CaDA
7,064
2.377%
10.739
3,890%
CaDA
11.775
0.898%
19.495
1.754%
CaDA!
1.916%
10.682
3.329%
13s
CaDA!
11.768
19.089
1.617%
15s
CCL
7,008
1.568%
56
10.655
3.1799%
19%
OCL
11.721
0.436%
19.348
0,983%
21%
CCL
6.992
1.344%
66
10.624
2.775%
25p
OCL
11.718
0.416%
7%
19.329
0.888
27
HOS-PyVRP
6.899
10
10.335
20
HGS-Py VRP
11.668
10
19.156
20.8rm
MTROмO
3.053%
10,858
5.318%
MIROMO
11,823
1,315%
2.602%
MVMoE
7.094
2.7995
36
10:847
4.929%
11s
MVMoE
11.815
1.349%
19.690
2.514%
12s.
RE-TE
7,068
2.417%%
25
10.778
4.200%
RP-TE
11.789
1.017%
2s
19.554
2.061%
9%
CaDA
7,062
2,339%6
Is
10.741
3.900%6
CaDA
11.777
0.914%
26
19.497
1.762%
9%
CaDA!
7,054
1.955%
10.086
3.308%
CaDA!
11.769
0.848%
1.602%
15%
CCL
1.309%
10.681
3.323%
20s
11.721
0.442%
19.345
0..77%%
22s
CCL
6.992
1.335%
66
10,609
2.611%
21p
CCL
11.718
0.144%
7s
19.334
0.915%
27%
BIG PyVRP
6.507
10.4m
9.724
20.8m
HIGS VRP
10,510
10.4m
16.926
20.8cm
МTPOМO
6,720
3.248%
28
10.224
5.112%%
MIPOMO
10.677
1.572%
26
17,442
3.000%
MVMoF
3.078%
10.184
6.00%%
MVMoE
10.677
1,564%
12s
6,683
2.6805
10:121
4.054%
RF-TE
10.645
1.367%
12.328
2,3525
CaDA
6,680
2.623%
16
10,095
5.773%
85
CaDA
10,631
1,133%
1s
37.280
2,073%
%.
CaDA!
6,052
2,200S
26
10,050
3.432%
136
CaDA!
10.621
1003%
36
17.264
1.861%
14s
CCL
6.637
1.988%
10.067
3,495%
20s
CCL
10,569
0.546%
17.125
1.142%
22s
CCL
1.569%
10,000
24s
OCL'
1.06%%
26
BIGS-IyVRP
9.887
10.4m
14.377
20.8m
IIGS-PyVRP
18.292
10:4m
29: 09:467
20.8m
MIPОМM
10.056
3 3.699
28
15.102
5.0555
36
MTPOMO
18,649
1.938%
28
30.478
3,426%6
9%.
MVMoF
10,007
3 3,2555
36
15.023
5055
10%
MVMoF
18.622
1.841%
36
30.437
3.284%
12x
RF-TE
3.000%
14,955
3.006%
RF-E
18.575
1.517%
2.641%
CaDA
3.010%
14.910
3.721%
CaDA
18.523
1.301%
30.174
2.390%
CaDA!
9,922
2.40%%
26
14,838
3,222%
12s
CaDA!
18:528
1.270%
26
20:113
2,183%
14s
CCL
9,916
2,352%
56
14,882
3.526%
19%
CCL
18,450
0.738
66
29.911
1.194%6
218
CCL
9,875
1.921%
14.780
CCL
18.419
0.678%
1.357%
206
HGS-IYVRP
10.185
14.779
20
HGP-PPVRP
20.8mm
MTPOMO
10.679
4.700%
2%
15.718
6.204%
MITPMO
19.001
1.199%
30.948
3.704%
MVMoE
10,639
4 38.5%
36
15.642
5.771%
11%
MVMoE
18,953
2.075%
36
30.892
3.609%
12%
RE-TE
10,569
3.713%
24
15.525
5,008%
RF-
18,900
1.713%
26
30.700
2.5778%
CaDA
10,576
3.716%
13.490
4.731%
CaDA
18.896
1.623%
30.620
CaDA!
10,503
3,064%
15,389
4.093%
12s
CaDA!
1.50%%
30.520
2.531%
15s
CCL
10,984
2.583%
56
15.407
4.219%
19%
CCL
18.775
0.976%
66
30.366
1.84276
211
CCL
10.440
2.40%%
66
15,297
3.472%
24s
CCL!
18,758
0.899%
7%
30.323
1.697%
25s
HGS-PyVRP
10.567
10.4m
15.766
20.8m
HGS-Py VRP
16.356
10:4m
25.757
20.8rm
MTROMO
10,775
1.733%%
26
2.483%%
MIROMO
16.822
2.877%
4.455%
MVMoE
10.753
1.525%
36
16.099
2.113%
11s
MVMME
16.817
2.783%
26.866
4,272%
12x
RE-E
10.747
1.435%
2s
16.057
1.838%
35
RF-TE
16.728
2.2485
2s
26.706
5.645%
5%
CaDA
10,749
1.508%
1s
16,036
1.725%
CaDA
16.709
2.130%
1s
26.631
3.358%
9s
CaDA!
10.207
1.112%
15.984
1.400%
CaDA!
2.054%
26.556
3.065%
14h
CCL
10.710
1.145%
36
16,009
1.501%
19%
CCL
16.579
1.333%
26.366
2.321%
206
CCL
10,698
1.027%
66
15,960
1.245%
23s
OCL
16.556
1.192%
3s
26.324
2.157%
24s
during testing. For N=100, the respective probabilities are 0.25 and 0.5. The model is optimized using Adam with a learning rate of 3 x 10-4 and a weight decay of 1 x 10-6, A multi-step learning rate scheduler is used with milestones at epochs 270 and 295, decay factor of 0.1, and gradient clipping set to 1. Training is conducted for a total of 300 epochs. Our code is available at git.
/ttpps:/giithue ccmm//ehuunggchu//CCI-WWWrr
4.3 COMPARISON WITH THE SFATE-OF
Baselines. We compare CCL with state-of-the-ar multi-task VRP solvers, including MTPOMO (Liu et al., 2024), MVMoE (Zhou et al., 2024a). RouteFinder (RF-TE) (Berto et al., 2025), and CaDA (Li et al., 2025). Among these, RF-TE and CaDA have reported the strongest performance, and
7

---

Published as a conference paper at ICLR 2026
Table 2: Generalization on 32 unseen out-of-distribution tasks.
MDOVRPB, MDOVRPL MDVREBL, MDOVRPBL MIXVRR,, MIOVER, MDVRPPR MDVRPI. VRPMB, OVRPMB, VRPMBL. OVRPMBL
MDDVRPBTW. BEOPHTT..DOORRPEE MDOVRPLTW. MDVRPBLTW. MDOVRPBLTW
MIXCRPPTW, MDOVRPIW, MOVEPHIW, MOVRPLIT
VRPMBTW, OVRPMBTW, VRPMBLTW. WR MDVRFMBTW, MDOVRPMBTW. MDVRPMBLTW. MDOVRPMBLTW
Methods MDVRPMB, MDOVRPMII, MDVRPMBL, MDOVRPMBL
N = 50
N 100
N 50
N 100
Obj.
Gap
Time
Obj.
Gap 1
Time
Obj +
Gap
Time 1
Obj.
Gap
RF-TE
9.651
40.472%
14
14.746
45.724%
91
14.557
32,586%
2s
24.217
36.564%
10s
CaDA
38.860%
2x
47.156%
14,410
23.523
32.512%%
CaDA'
9.285
34.169%
3s
14.395
41.419%
16s
14,830
34. 66.6655%
45
21.328
36.774%
17s
15.6
38.624%
13,923
23.375
31.159%
18s
CCL'
8.673
25.781%
5s
13.777
35.413%
22s
13.536
22.422%
5s
24.025
34.163%
25s
Table 3: Ablation on key modules within CCL.
Methods
CVRP
OVRP
VRPB
VRPL
OVRPB
OVRPL
VRPBL
OVRPBL
Avg.
CCL'
0.881% %
1.566%
1.921 %
1.027%
1.344%
1.569 %
2.450%
1.335%
1.512%
RGCR
0.874% %
1.710%
1. 1.969%
0.993%
1.396%
1.712%
2.486%
1.407%
1.568%
TSNR
0.961 S%:
2.284%
2.413%
1.131%
1.969%
2.311%
3.001
1.973%
2.005%
RGCR- TSNR
1.014%
2.395%
2.416%
1.160%
2.036%
2.411%
3.088%
2.041%
2.070%
Methods
VRPTW
OVRPTW
VRPBTW
VRPLTW OVRPBTW
OVRPLIW
VRPBLTW
OVRPBLTW
Avg.
CCL
0.907 %
0.506 %
0.678%
1.192%
0.416%
0,501 %
0.899%
0.414%
0.689%
RGCR
0.938%
0.521% %
0.720%
1.235%
0.419%
0,519%
0.926%
0.427%
0.713%
TSNR
1.539%
0.947%
1.204%
1.857%
0.795%
0,957%
1.409%
0.805%
1.189%
RGGCR- TSNR
1.615%
0.969%
1.266%
1.930%
0.813%
0.958%
1.492%
0.810%
1.232%
include them in both in-distribution (Table 1) and out-of-distribution (Table 2) evaluations. To ensure a fair comparison, we reimplement CaDA in the RouteFinder framework (Berto et al., 2025), which also serves as the basis for RF-TE and our CCL. As shown in Appendix C.I.1, our reproduction closely matches the performance reported in the original paper (Li et al., 2025). To further enhance performance, we integrate a context-aware module, ReLD (Huang et al., 2025), into both CaDA and CCL., denoted CaDA and CCL respectively.
In-Distribution Evaluation. In the Table 1, CCL outperforms CaDA across both N=50 and N=100 settings. Specifically, for N=50, both CCL and CCL achieve lower performance gaps than CaDA on all 16 evaluated tasks. For N=100, the gap relative to the HGS-Py VRP baseline narrows even further. We also observe complementary strengths between ReLD and CCL. ReLD performs particularly well on variants without time windows (TW), leveraging its ability to extract globally shared constraint signals through step-wise context. contrast, CCL's dynamic node refinement excels on TW tasks, offering finer-grained adaptation to local, node-specific constraints. By combining both, the resulting CCL achieves the best overall performance across all the in-distribution tasks.
Diit----Ditiitttionn Evaluation. We present the averaged performance for both tasks with or without TW in Table 2 (detailed results for each task are presented in Appendix C.1.3, where CCL outperformed CaDA on the majority). Table 2 shows that CCL consistently outperforms other methods under the N=50 setting. For N=100, while CCL$ maintains competitive performance, shows slightly higher gap TW tasks compared to standalone CCL. We hypothesize that this may be due to a low test-time update rate, which can cause the model to overfit to static constraint structures and under-adapt time-sensitive variations, thus increasing the gap. Nevertheless, either equipped with ReL.D or not, our CCL exhibits superior overall performance to CaDA (and its counterpart).
4.4 ABLATION STUDIES
Ablation on Key Modules within CCL. We conduct ablation experiments to validate the effective- ness of RGCR and TSNR in CCL Table 3 reports the results for 5500. Removing RGCR leads to a smaller gap increase than TSNR, and even slightly reduces the gap on CVRP and VRPL. It likely that the relatively simple constraints of the two VRP tasks make relevance weighting less effective. Moreover, removing both modules yields the highest gap, highlighting their complementary effective- ness. We also conduct these ablations CCL (the variant without ReLD), showing that the main performance gains come from CCL itself rather than ReLD. Details are presented in Appendix C.
8

---

Published as a conference paper at ICLR 2026
Gap (%)
Gap (%)
Complexity
TW
Methods
Avg.
wio
TW
Params
(M)
Time (5)
2.01
1.67
1.52
1.51
Atrributes
1.156
1,584
1.19
1.10
Concat Embeddings
L.441
1,568
0.713
4,05
5.7
0.88
+ Random Scores
1.092
1.477
0,707
4,05
6,5
0.69
+ Cosine Similarity
1.108
1.480
0.737
4.05
7.2
Attention
X
Dot Product (RGCR)
1.100
1.512
0.689
4.05
Updates
X
X
X
Figure 2: Ablation on key components within RGCR (Left) and TSNR (Right), respectively.
Ablation on Key Components within RGCR and TSNR. We further conduct ablation studies using 5= to evaluate the components within RGCR and TSNR.
Regarding RGCR, we first examine direct concatenation of attributes (as CaDA) and embeddings RCCL in Table 3). We then evaluate three correlation scores, namely random, cosine similarity, and dot product, as defined in Eq. 6. The left part of Fig. presents the averaged gap and the corresponding model complexity. Compared with direct concatenation, all three correlation scores reduce the gap in both settings. Notably, the dot product achieves the smallest gap on tasks with TW, demonstrating its superiority in handling complex constraints. More experimental setups and analyses are provided in Appendix C.3.2.
Regarding TSNR, the right part of Fig. 2 shows that combining node-level attention, embedding updates, and the distance bias in Eq. 12 achieves the lowest gaps, indicating that all three elements are essential for improving the overall performance. Moreover, a detailed analysis that node-level atten- tion reduces model complexity compared to the vanilla Transformer is provided in Appendix C.3.3,
DISCUSSION
We discuss the strengths and limitations of CCL. Its main drawback is the longer inference time required for better performance. However, flexible parameter settings can mitigate this issue, enabling CCL to perform well on large-scale real-world instances.
Table Complexity analysis.
Complexity Analysis. Table compares the model com- plexity of SOTA methods and our CCL. We further com- pare CCL with a heavy-decoder variant of the SOTA model, denoted as CaDA' -HD. Detailed configurations of this variant are presented in Appendix C.4. All models are trained and evaluated the 16 VRP variants with N=50 using an L40S GPU. "Time" denotes the total inference time over 1,000 test instances, "# Params" refers to the
Gapl
Memory. (GiB)
# Purams]
Time.
Methods
(%)
(M)
(S)
CaDA
1.90
6.01
3,37 0.1
1.5
CaDA
0.3
2.7
CaDA
HDA HHD
1.53
7.55
3,37+10
5.1
CCL
1.28
8.13
3.39 0.45
5.4
CCL
1.10
8.76
3.39
3.39+0..6
6.5
ASW-TAM
29.93
15.47
3.39 0.66
96.1
total number of parameters in the encoder and decoder, and "Memory" indicates the peak memory usage during testing across all 16 variants. Compared to CaDA and CaDA", our CCL and CCL intro- duce only moderate increase in memory usage and parameter count, while achieving a substantial performance improvement. The inference time longer due to additional computation, but the gain in solution quality justifies the cost. In addition, we also apply the step-wise refinement strategy, iee, ASW-TAM (Xin al., 2020) to the multi-task setting, where each route is re-embedded individually. However, due to memory constraints, original one. Results show that the naive refinement strategy leads to significantly higher gaps, longer inference time, and larger memory consumption, which further validates the effectiveness of CCL. Moreover, we observe that CCL achieves a comparable model cost while reducing the gap by 0.25% compared with CaDA HDD These findings indicate that the effectiveness of CCL stems from its design rather than from an increased network scale.
adopt much smaller batch size that is only 1/16 of the
Performance-Cost Trade-off. In Section 3.4, Ptr and Pts denote the probabilities of updating node embeddings during training and testing, and we assess their impact on model performance and inference efficiency, which also leads to lightweight version of CCL. We first conduct sensi- tivity studies using 55 and evaluating all 20 combinations of Ptr E [0.25, 0.5, 0.75, 1} and Pts E (0.0.25,0.. 0.5, 0.75, 1). Fig. shows the corresponding gap values and inference time.

---

Published as a conference paper at ICLR 2026
Gap (%)
Time (s)
We observe that, for fixed probabil- ity Per during training, the gap tends to be smaller when the probability Pts during testing is slightly higher. For example, when 0,,=0.25 0.5, the best performance is achieved at Pt.00.5 and 0.75, respectively. This work adopts Per = 0.75 and Pts as this setting yields the lowest gap. Meanwhile, reducing Pt. leads to shorter inference time, as fewer re- finement steps are involved. While this leads to a higher gap. it provides trade-off between solution quality and inference efficiency. Motivated by this, we design a lightweight version of CCL' using Per = 0.25 and Pt. 0.25. It achieves average gap of 1.38% across 16 VRPs with an average inference time of 4.6s, while the existing SOTA CaDA attains a gap of 1.63% in 2.7s (see Table 4). This enables users adjust Pt. based the requirements of practical deployment scenarios, and more detailed comparisons between CCL and CaDA are provided in Appendix C.4.
6.5
5.49
4.05
3:08
2.01 1.14
5.0
4.1
4.8
3.4
6.0
6.6
4.5
6.0
5.11
2.00
1.34
1.14 1.10
4.2
4.7
5.3
5.9
6.5
3.5
5.5
3.0
3.25
1.48
1.23
1.16 1.17
4.1
4.6
5.2
5.9
6.6
5.0
2.5
0.25
2.34
1.34
1.39
1.50
0.3
4.0
4.6
5.3
4.5
4.0
6
0.25
0.5
0.75
1
0
0.25
0.5
0.75
P
Figure 3: Gap (Left) and inference time (Right) under dif- ferent training and testing update rates.
Large-Scale Real- Practicality. We evaluate zero-shot generalization on 60 real-world VRPTW instances, each with N=600 (Homberger and Gehring, 1999). The model is trained on 16 tasks with =100 During inference, we apply an up- date probability of 1. I to reduce computational cost. Ta- ble 5 reports the averaged results across these instances (per- instance results in Appendix C.7), showing that CCL achieves the lowest average gap while maintaining comparable infer-
Table 5: Results on large-scale real- world VRPTW instances (N=600).
Methods
Obj.
Gap.
Time.
RF-TE
29558
145.593%
1.4s
CaDA
22917
18.188%
1.4s
CCL
20633
70.961 %
2.0s
ence time. Appendix C.7 also reports results on VRPTW instances with N=100 (Solomon, 1987), where CCL outperforms SOTAs 24 out of 27. It is further evaluated CVRP instances with N e [100, 251] (Uchoa et al., 2017), where CCL achieves the best performance on 16 out 27. These findings indicate that method is well-suited for deployment in real-world scenarios, particularly for problems with complex constraints such as time windows.
6 CONCLUSIONS
Existing neural multi-task VRP methods often neglect the evolving nature of node states during decoding, limiting their ability to respond accurately constraint requirements. To overcome this, proposed Chain-of-Conted Learning (CCL), a step-wise framework that updates node embeddings based on the current decision context. Through relevance guided constraint reformulation and trajectory-shared re-embedding, CCL captures the agent's 's evolving preferences and improves solution quality. Experiments on 48 VRP variants show that CCL achieves SOTA performances on all in- distribution and most cut-of-diisribution tasks. One limitation of CCL lies in its slightly longer inference time. Although flexible parameter settings can mitigate such issue, trade-off between computation cost and solution quality still remains. In future, we plan to explore more advanced techniques to further improve the inference efficiency while preserving superior solution quality.
THE USE OF LARGE LANGUAGE MODELS (LLMS)
We employed LLMs for polishing the paper and assisting with simple coding tasks.
ACKNOWLEDGMENTS
This research supported by the National Research Foundation, Singapore under its Al Singapore Programme (AISG Award No: AISG3-RP-2022-031) This research is also supported by the Lee Kong Chian Fellowship awarded to CAO Zhiguang by Singapore Management University.
10

---

Published as a conference paper at ICLR 2026
REFERENCES
Paolo Toth and Danicle Vigo. Vehicle routing: problems, methods, and applications. SIAM, 2014.
Grigorios D Konstantakopoulos, Sotiris P Gayialis, and Evripidis P Kechagias. Vehicle routing problem and related algorithms for logisties distribution: literature review and classification. Operational research, pages 1-30, 2022.
Thierry Garaix, Christian Artigues, Dominique Feillet, and Didier Josselin. Vehicle routing problems with alternative paths: An application to on-demand transportation. European Journal of Operational Research, 204(1):62-P5 75, 2010.
Rodolfo Dondo, Carlos A Méndez, and Jaime Cerdá. The multi-echelon vehicle routing problem with cross docking in supply chain management. Computers & Chemical Engineering, (0022)))0022-2244 2011.
Laurent Perron and Vincent Furno.. optimi zatien/.
Or-tools.
URL httpps://aavvlloprrs..ooo// www.:opperrr.oog..o..ooo
Shen Lin and Brian W Kernighan. An effective heuristic algorithm for the traveling-salesman problem. Opera- rions research, 21(2):498-511 1973.
Thibaut Vidal, Gilbert Laporte, and Piotr Mail. A concise guide to existing and emerging vehicle routing problem variants. European Journal of Operational Research, (6662):401-466 2020.
Chaitanya K Joshi, Thomas Laurent, and Xavier Bresson. An efficient graph convolutional network technique for the travelling salesman problem. arXiv preprint arXiv: 01227, 2019.
Wouter Kool, Herke van Hoof, and Max Welling. Attention, learn to solve routing problems! In International Conference on Learning Representations, 2019.
Yeong-Dac Kwon, Jinho Choo, Byoungjip Kim, Iljoo Yoon, Youngjune Gwon, and Seungjai Min. Pomo: Policy optimization with multiple optima for reinforcement learning. Advances in Neural Information Processing Systems, 33:21 188-21198, 2020.
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, 33(9):007--506, 2021.
Yining Ma, Zhiguang Cao, and Yeow Meng Chee. Learning to search feasible and infeasible regions of routing prohlems with flexible neural k-opt. Advances in Neural Information Processing Systems, 36:49555-49578, 2023.
Zhiqing Sun and Yiming Yang. Difusco: Graph-based diffusion solvers for combinatorial optimizttoonin.......i Advances in Neural Information Processing Systems, 36:3706-3731, 2023.
Yoshua Bengio, Andrea Lodi, and Antoine Prouvost. Machine learning for combinatorial optimization: methodological tour d'horizon. European Journal of Operational Research, 290(2):405-421, 2021.
Aigerim Bogyrbayeva, Meraryslan Meraliyev, Taukekhan Mustakhov, and Bissenbay Dauletbayev. Machine learning solve vehicle routing problems: A survey. IEEE Transactions on Intelligent Transportation Systems, 4575) 4772, 2024.
André Hottung and Kevin Tiemey. Neural large neighborhood search for the capacitated vehicle routing problem. In ECAI 2020, pages 443-450. IOS Press, 2020.
André Hottung, Yeong-Dae Kwon, and Kevin Tierney. Efficient active search for combinatorial optimization problems. International Conference on Learning Representations, 2022.
Liang Xin, Wen Song, Zhiguang Cao, and Jie Zhang. Neurolkh: Combining deep learning model with lin- kemnighnn-helsguun heuristic for solving the traveling salesman problem. Advances Neural Information Processing Systems, 34:7472-7483, 2021.
Felix Chalumeau, Shikha Surana, Clément Bonnet, Nathan Grinsztajn, Arnu Pretorius, Alexandre Laterre, and Tom Barrett. Combinatorial optimization with policy adaptation using latent space search. Advances in Newrud Information Processing Systems, 36:7947-7959, 2023.
Jinbiao Chen, Jiahai Wang. Zizhen Zhang, Zhiguang Cao, Te Ye, and Siyuan Chen. Efficient meta neural heuristic for multi-objective combinatorial optimization. Advonces in Neural Information Processing Systems, 36:56825-5683,, 2023a.
11

---

Published as a conference paper at ICLR 2026
Zefang Zong, Meng Zheng, Yong Li, and Depeng Jin. Mapdp: Cooperative multi-agent reinforcement learning solve pickup and delivery problems. Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pages 9980-9988, 2022.
Detian Kong, Yining Ma, Zhiguang Cao, Tianshu Yu, and Jianhua Xiao. Efficient neural collaborative search for pickup and delivery problems. IEEE Transactions Pattern Analysis and Machine Intelligence, 46( 12): 11019-11034 2024.
Raras Tyasnurita. Ender Özcan, John H Drake, and Shahriar Asta. Constructing selection hyper-heuristicr for open vehicle routing with time delay neural networks using multiple experts. Knowledge-Bord Systems, 295: 111731, 2024.
Sinaide Nunes Bezerra, Sérgio Ricardo de Souza, and Marcone Jamilson Freitas Souza. A general vns for the multi-depot open vehicle routing problem with time windows. Optimization Letters, 17(9):203--200 2063, 2023.
Bruno Oliveira, Artur Pessoa, and Marcos Roboredo. Hybrid iterated local search algorithm for the vehicle routing problem with lockers. Journal of Heuristics, 31(2):22, 2025.
Rongkai Zhang, Cong Zhang. Zhiguang Cao, Wen Song. Puay Siew Tan, Jie Zhang, Bihan Wen, and Justin Dauwels. Learning to solve multiple-tsp with time window and rejections deep reinforcement learning. IEEE Transactions on Intelligent Transportation Systems, 24(1):1325-133) 2022.
Bo Lin, Bissan Ghaddar, and Jatin Nathwani. Deep reinforcement learning for the electric vehicle routing problem with time windows. IEEE Transactions Intelligent Transportation Systems, 23(8):11528111 11538, 2021.
Conghui Wang. Zhiguang Cao, Yaoxin Wu, Long Teng. and Guohua Wu. Deep reinforcement learning for solving vehicle routing problems with backhauls. IEEE Transactions on Neural Networks and Learning Systems, 36(3):4779-4777 4773, 2024.
Saso Karakatič and Vili Podgorelec. A survey of genetic algorithms for solving multi depot vehicle routing problem. Applied Soft Computing, 27:519-532, 2015.
Liu, Xi Lin, Zhenkun Wang, Qingfu Zhang, Tong Xialiang, and Mingxuan Yuan. Multi-task learning for routing problem with cross- pross-rrobemm zero-shot generalization. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pages
2024.
Jianan Zhou, Zhiguang Cao, Yaoxin Wa, Wen Song, Yining Ma, Jie Zhang, and Xu Chi. MVMoF: Multi-task vehicle routing solver with mixture-of-experts In Proceedings of the 41st International Conference on Machine Learning, volume 235, pages 1804-618244 2024a.
Federico Berto, Chuanbo Hua, Nayeli Gast Zepeda, André Hottung, Niels Wouda, Leon Lan, Kevin Tierney, and Jinkyoo Park. Routefinder: Towards foundation models vehicle routing problems. ICML 2024 Workshop on Foundation Models in the Wild, 2024, URL https: / /ttp///////////oeeeo//e/o// net/forumt id= @Ciail6c4C.
Han Li, Fei Liu, Zhi Zheng, Yu Zhang, and Zhenkun Wang. CaDA: Cross-problem routing solver with constraint-aware dual--ttention. In Forty-second International Conference on Machine Learning, 2025. URL
net. / forum? d=CSARyQUTis
Liang Xin, Wen Song, Zhiguang Cao. and Jie Zhang. Step-wise deep learning models for solving routing problems. IEEE Transactions on Industrial Informatics, 17(7):4861 4871, 2020.
Federico Berto, Chuanbo Hua, Nayeli Gast Zepeda, André Hottung, Niels Wouda, Leon Lan, Junyoung Park, Kevin Tiemey, and Jinkyoo Park. Routefinder: Towards foundation models for vehicle routing problems. Transactions Machine Learning Research, 2025. ISSN 2835-8856.
Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8:229-256, 1992.
Biao Zhang and Rico Sennrich. Root mean square layer normalization. Advances in Neural Information Processing Systems, 32, 2019.
Niels A Wouda, Leon Lan, and Wouter Kool. Pyvrp: A high-performance vrp solver package. INFORMS Journal Computing, 36(4):943-955, 2024.
12

---

Published as a conference paper at ICLR 2026
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. Pytorch: an imperative style, high-performance deep learning library. In Advances in Newral Information Processing Systems, pages 8024-8035. Curran Associates, Inc., 2019.
Ziwei Huang, Jianan Zhou, Zhiguang Cao, and Yixin XU. Rethinking light decoder- based solvers for vehicle routing problems. In The Thirteenth International Conference on Learning Representations, 2025, URL https: / http//ppeeeeeiieeeeeeeeeeeeeeeee net/ forum? 4wwkkyaaau
Jörg Homberger and Hermann Gehring. Two evolutionary metaheuristics for the vehicle routing problem with time windows. INFOR: Information Systems and Operational Research, 37(3):297-33 1999.
Marius Solomon. Algorithms for the vehicle routing and scheduling problems with time window constraints. Operations research, 35(2):254 265, 1987.
Eduardo Uchoa, Diego Pecin, Artur Pessoa, Mareus Poggi. Thibaut Vidal, and Anand Subramanian. New benchmark instances for the capacitated vehicle routing problem. European Journal of Operational Research, 257(():845-858, 2017.
Mohammadreza Nazari, Afshin Oroojlooy, Lawrence Snyder, and Martin Takác. Reinforcement learning for solving the vehicle routing problem. Advances in Neural Information Processing Systems, 31, 2018.
Sirui Li, Zhongxia Yan, and Cathy Wu. Learning to delegate for large-scalc vehicle routing. In Advances in Neural Information Processing Systems, volume 34, pages 26198-26211, 2021a.
Qingchun Hou, Jingwei Yang. Yiqiang Su, Xiaoqing Wang, and Yuming Deng. Generalize learned heuris- tics to solve large-scale vehicle routing problems in real-time. In International Conference on Learning Representations. 2023.
Haoran Ye, Jiarui Wang, Helan Liang, Zhiguang Cao, Yong Li, and Fanzhang Li. GLOP: Learning global partition and local construction solving large-scale routing problems real-time, Proceedings of the AAAI Conference on Artificial Intelligence, 2024.
Chaitanya Joshi, Quentin Cappart, Louis-Martin Rousseau, and Thomas Laurent. Learning tsp requires rethinking generalization. In International Conference on Principles and Practice of Constraint Progranuming 2021.
Jicyi Bi, Yining Ma, Jiahai Wang, Zhiguang Cao, Jinbiao Chen, Yuan Sun, and Ycow Meng Chee. Learning generalizable models for vehiele routing problems via knowledge distillation. In Advances in Neural Information Processing Systems, volume 35, pages 3 31226-31238, 2022.
Simon Geisler, Johanna Sommer, Jan Schuchardt, Aleksandar Bojchevski. and Stephan Günnemann. Generaliza- tion of neural combinatorial solvers through the lens adversarial robustness. In International Conference on Learning Representations, 2022.
Jianan Zhou, Yaoxin Wu, Wen Song. Zhiguang Cao, and Jie Zhang. Towards omni-generalizable neural methods for vehicle routing problems. In International Conference on Machine Learning, pages 222769-47788 PMLR, 2023.
Ni Zhang and Zhiguang Cao. byyrid-balance GFlownet for solving vehicle routing problems. In The Thirty- ninth Annual Conference on Neural Information Processing Systems, 2025. URL https:t////att //pprr/veee........../// net / forum? didmmiiiGiGeGGeDDD
Ni Zhang. Jingfeng Yang. Zhiguang Cao, and Xu Chi. Adversarial generative flow network for solving vehicle routing problems. In The Thirteenth International Conference on Learning Representations, 2025. URL
https: : htttps://cpeeee http//ppeeeeeeeeeeeeeeeeeeeeeeee ew. net/ / forum? i.dtttott 4
Yeong-Dat Dae Kwon, Jinho Choo, Iljoo Yoon, Minah Park, Duwon Park, and Youngjune Gwon. Matrix encoding networks for neural combinatorial optimization. In Advances in Neural Information Processing Systems, volume 34, 2021.
Jingwen Li. Liang Xin, Zhiguang Cao, Andrew Lim, Wen Song, and Jie Zhang. Heterogeneous attentions for solving pickup and delivery problem via deep reinforcement learning. IEEE Transactions on Inielligent Transportation Systems, 23(3):200--215, 202
Minsu Kim, Junyoung Park, and Jinkyoo Park. Sym-NCO: Leveraging symmetricity for neural combinatorial optimization. In Advances in Neural Information Processing Systems, volume 35, pages 1936-1949, 2022.
13

---

Published as a conference paper at ICLR 2026
Nathan Grinsztajn. Daniel Mtrells--Blance Shikha Surana, Clément Bonnet, and Thomas D Barrett. Winner takes all: Training performant RL populations for combinatorial optimization. Advances in Neural Information Processing Systems, 2023.
Jinbiao Chen, Zizhen Zhang, Zhiguang Cao, Yaoxin Wu, Yining Ma, Te Ye, and Jiahai Wang. Neural multi- objective combinatorial optimization with diversity enhancement. In Advances in Neural Information Processing Systems, volume 36, pages 39176-39188, 2023b.
Chengrui Gao, Haopu Shang, Xue, Dong Li, and Chao Qian. Towards generalizable neural solvers for vehicle routing problems via ensemble with transferrable local policy. In Proceedings of the Thirty-Thine International Joint Conference on Artificial Intelligence, pages 6914-6922, 2024.
André Hottung, Mridul Mahajan, and Kevin Tiemney PolyNet: Learning diverse solution strategies for neural combinatorial optimization. In International Conference Learning Representations, 2025.
Chuanbo Hua, Federico Berto, Jiwoo Son, Scunghyun Kang, Changhyun Kwon, and Jinkyoo Park. CAMP: Collaborative Attention Model with Profiles for Vehicle Routing Problems. In Proceedings of the 2025
International Conference on Autonomous Agents and Multiagent Systems, 2025a.
Darko Drakulic, Sofia Michel, Florian Mai, Arnaud Sors, and Jean-Mare Andreoli. BQ-NCO: Bisimulation quotienting for efficient neural combinatorial optimization. In Advances in Neural Information Processing Systems, volume 36, 2023.
Fu Luo, Xi Lin, Fei Liu, Qingfu Zhang, and Zhenkun Wang. Neural combinatorial optimization with heavy decoder: Toward large scale generalization. In Advances in Neural Information Processing Systems, volume 36, pages 8845 8864,886622 2023.
Fu Luo, Xi Lin, Yaoxin Wu, Zhenkun Wang. Tong Xialiang, Mingxuan Yuan, and Qingfu Zhang. Boosting neural combinatorial optimization for large-scale vehicle routing problems. In International Conference on Learning Representations, 2025.
Jonathan Pirnay and Dominik G. Grimm. Self intromepoveeentt for neural combinatorial optimization: Sample without replacement, but improvement. Transactions on Machine Learning Research, 2024. ISSN 2835-883 9935-8856.
Darko Drakulic, Sofia Michel, and Jean-Marc Andreoli. GOAL: A generalist combinatorial optimization agent learner. International Conference on Learning Representations, 2025.
Zhuoyi Lin, Yaoxin Wu, Bangjian Zhou, Zhiguang Cao, Wen Song, Yingqian Zhang, and Senthilnath Jayavelu. Cross-prote pross-problem learning for solving vehicle routing problems. In Kate Larson, editor, Proceedings of the Thirty-Thire International Joint Conference on Artificial Intelligence, pages Conferences Artificial Intelligence Organization, 2024.
International Joint
Zefang Zong, Xiaochen Wei, Guozhen Zhang, Chen Gao, Huandong Wang, and Yong L.i. Unico: Towards unified model for combinatorial optimization problems. arXiv preprint wwXiv:2505.06244 2025.
Xia Jiang. Yaoxin Wu, Yuan Wang. and Yingqian Zhang. Unco: Towards unifying neural combinatorial optimization through large language model. arXiv preprint arXiv:2408. 12214, 2024.
Jianan Zhou, Yaoxin Wu, Zhiguang Cao, Wen Song, Jie Zhang, and Zhiqi Shen. Collaboration! towards robust neural methods for routing problems. Advances in Neural Information Processing Systems, 37: 121731-121764. 2024b.
Chuanbo Hua, Federico Berto, Zhikai Zhao, Jiwon Son, Changhyun Kwon, and Jinkyoo Park. Uspr: Learning a unified solver for profiled routing. arXiv preprint 2555::250..05 09, 2025b.
Chuanbo Hua, Federico Berto, Jiwoo Son, Seunghyun Kang, Changhyun Kwon, and Jinkyoo Park. Camp: Col- laborative attention model with profiles for vehicle routing problems. In Proceedings of the 24th International Conference on Autonomous Agents and Multiagent Systems. pages 1015-1024, 2025c.
Jiwoo Son, Zhikai Zhao, Federico Berto, Chuanbo Hua, Changhyun Kwon, and Jinkyon Park. Neural combina- torial optimization for real-world routing. arXiv preprint arXiv: 16159, 2025.
Zhenwei Wang, Ruibin Bai, and Tiehua Zhang. Towards constraint-base adaptive hypergraph learning for solving vehicle routing: An end-to-end solution. arXiv preprint arXiv:2503. 10421, 2025.
Suyu Liu, Zhiguang Cao, Nan Yin, and Yew-Snon Ong. Scale-net: A hierarchical u-net framework for cross-scale generalization in multi-task vehicle routing problems. In Proceedings of the AAAI Conference on Artificial Intelligence, 2026.
Suyu Liu, Zhiguang Cao, Shanshan Feng, and Yew-Soon Ong. A mixed-curvature based pre-training paradigm for multi-task vehicle routing solver. In Fecond International Conference Machine Learning, 2025.
LLC Gurobi Optimization. Gurobi optimizer reference manual, 2024. URL https: htww./wwwwb......o//////oo com.
14

---

Published as a conference paper at ICLR 2026
A RELATED WORK
Neural Solvers for Single-Task VRPs. A common paradigm in neural solvers for single-task VRPs to construct solutions in an autoregressive manner. These methods typically employ an encoder to embed the VRP instance into node representations, followed by a decoder that sequentially predicts the probability of selecting the next node. To reduce the computational overhead of reinforcement learning (RL), most approaches adopt static node embeddings during decoding (Joshi et al., 2019; Nazari et al., 2018; Kool et al., 2019; Kwon et al., 2020; Huang et al., 2025; Li et al., 2021a; Hou al., 2023; Ye et al., 2024; Joshi et al., 2021; Bi et al., 2022; Geisler et al., 2022; Zhou al., 2023; Zhang and Cao, 2025; Zhang et al., 2025). One influential method in this line is the Attention Model (AM) (Kool et al., 2019), which uses Transformer- (Kwon et al., 2020) enhances AM by introducing Policy Optimization with Multiple Optima (POMO), which leverages multiple solution trajectories and data augmentation to achieve strong performance on TSP and CVRP. POMO has since become widely adopted baseline (K Woon et al.. 2021; Li et al., 2021b; Kim et al., 2022; Grinsztajn et al., 2023; Chen et al., 2023b; Gao al., 2024; Hottung et al., 2025: Hua et al., 2025a: Li et al., 2021a; Hou et al., 2023; Ye et al., 2024; Joshi et al., 2021; Bi et al.,
policy network to guide node selection.
2022: Geisler et al., 2022; Zhou et al., 2023). To improve context modeling, ReLD (Huang et al.,
2025) proposes an enhanced decoder architecture incorporating identity mapping and feed-forward layer to better capture local and global dependencies. In terms of dynamic node re-embedding, (Xin et al., 2020) introduces step-wise RL framework that removes visited nodes each decision step, enabling the model to represent distinct node states as the context evolves. An alternative direction involves building heavy decoder-based solvers trained with supervised learning (SL) (Drakulic et al., 2023; Luo et al., 2023; 2025; Pirnay and Grimm, 2024; Drakulic et al., 2025). While these methods demonstrate strong empirical performance, their reliance on multi-layered decoder architectures results in high computational cost, making them unsuitable for RL-based training, which does not need optimal solutions as the labels.
Neural Solvers for Multi- Task VRPs. Multi-task VRPs generalize single-task VRPs by involving varied combinations of constraints, resulting in multiple task variants within a shared framework. Recent works train a single model to capture transferable patterns across tasks (Lin et al., 2024; Liu et al., 2024; Zhou et al., 2024a; Berto et al., 2024; Zong et al., 2025; Drakulic et al., 2025; Jiang et al., 2024; Zhou et al., 2024b; Hua et al., 2025b;c; Li et al., 2025; Berto et al., 2025; Son et al., 2025; Wang et al., 2025; Liu et al., 2026; 2025). (Lin et al., 2024) shows that a pre-trained TSP model can be fine-tuned to handle other VRP variants. To expand constraint coverage, (Liu et al., 2024) and (Zhou et al., 2024a) introduce models that handle B, L, O, and TW constraints, training on single-constraint tasks with the goal of generalizing to tasks with mixed attributes. (Berto et al., 2024) presents a foundation model trained on 16 variants and fine-tuned on an unseen constraint (MB) across 8 new variants. A subsequent extension (Berto et al., 2025) adds MD to the task space, culminating in a benchmark of 48 variants. Most recently, (Li et al., 2025) proposes Constraint-Awan Dual-Attention (CaDA), which incorporates constraint prompts and global-sparse attention to enhance encoder performance in capturing both broad and localized constraint-relevan node information. Despite their progress, these methods generally struggle to capture the dynamic, fine-grained impact of constraints during decision-makee making particularly when certain nodes become increasingly urgent due to time-sensitive or context-dependent requirements. In contrast, our work introduces a step-wise, context-aware refinement mechanism to better model these evolving constraint-driven priorities.
B DETAILS OF MODEL ARCHITECTURE
B. ENCODER ARCHITECTURE
As illustrated in Section 4.2, the input attributes includes three parts: the constraint label ho e R the depot attribute hn E TR: and the customer features (h,, h2,..., h,) E Nx77 The depot and customer attributes are separately processed by two linear layers, and the outputs are concatenated to generate the input node embedding as follows:
I Concat(H((b)) H((h, be....,,yyl).
(15)
where IE ()) Concat(-) denotes the concatenate operation, and H(-)) represents the linear layer. Similarly, we project the constraint labels into the prompt embedding space and expand them
15

---

Published as a conference paper at ICLR 2026
to match the shape of the node embeddings:
L = Exxannd(((hh))
(16)
where E R(N+1)xD, H(() is linear layer to project the feature dimension from D, and Expand()) is the duplication and expansion operation to reshape the feature map from RIxD to R(N+1)xD Subsequently, unified input embedding is formed by concatenating the projected constraint embedding and the node embedding:
I H(Concat(I.L))
(17)
The dual-attention encoder processes the original node embeddings and the unified input embeddings I through the sparse and global branches, respectively. Each branch contains Transformer layer and a linear layer for fusion, with the overall computation defined as:
A 10 = 7 (H(
H (0) = T (H(-))) HO H + H (H))),
(18)
H = (0) + (H
where H HH H (N++)xxD and denotes the index of the encoder layer. In the first layer, initialize H and =I Tg¹) .10)) denote the Transformer and linear layers of the global branch, respectively, while T 71 correspond to those of the sparse branch. The Transformer layer employs the pre-norm design from (Berto al., 2024). and integrates sparse attention based on (Li et al., 2025). The emhedding output by the final global layer is passed through a normalization layer, and the normalized embeddings are used as the initial node embeddings for decoding:
H = No Norm (errm(((k)) (H()),
(19)
where E [(++))x and K=6 denotes the number of encoder layers.
B.2 CLASSIC DECODER
Following RGCR and TSNR, a classic decoder is employed to calculate the action probability distribution using multi-head head attention mechanism. At step j. the context embedding generated by RGCR denoted as C, E R.N where is the number of trajectories, equal to the number of customers. We directly use the context embeddings as the query, i.e., = Cj. In the multi-head attention mechanism, the key and value embeddings are derived from the node embeddings produced by TSNR, i.e., k,. v, *HHH)). where k,. Vj (NN+1)x Since each node is visited only once, we apply a mask M, to the visited nodes when computing the attention weights:
A = Softmax O:k
OM,),
(20)
where Aj. M, E Nx++1), Softmax(-) denotes the Softmax operation, and @ ensures that multiplication values for visited nodes are set to -0. The context query is computed as q H(A)v,)) and the candidate node representations are obtained as k, H(H)). Based on these, the action probability distribution is derived as follows:
k
D;
(21)
DD
where â RND k, E ((N+1)xD and Dj NxN+1 To generate solutions, the unnormalized
log-probability (logit) is calculated as
n = Tanh (D,) 0 M,,
(22)
where Tanh(-) is the Hyperbolic Tangent operation, and {=10 is predefined clipping hyperparameter. The final selection probabilities for each node are computed by applying the Softmax operation: Pj So Softmax(uu)
16

---

Published as a conference paper at ICLR 2026
Table 5: Performance on 16 seen in-distribution tasks. * denotes the strong baseline used compute the gap. Best neural approach is highlighted in bold; second underlined.
2V 50
N 2001
50
N - 1001
Methods
Methods
Cop
Obj
Cap
Time
Okg
Gap
Ohjj
Cap |
Time
HGS PyVRP
10,372
10.4m
15.628
20.8m
HGS PyVRP
16.031
10:4m
25.423
20.8m
CVRP
CaDA!
10.814
1,825
25
15,870
1.578%
16,278
1.5355
26.070
2.53095
CIDA
10,505
1.287%
25
15,843
1412%
CaDW
16.312
1.745%
26.169
2.925%
CCL
10,473
0.977%
3s
15:523
1.287% %
19%
CCL
16.190
0.979%
56
25.913
1.908%
216
HGS-PyVRP
9.725
8m
HGS-P%VRP
10,510
10
16.906
OVRR
CaDA!
6.670
2.468%
24
10.121
4,045%
86
CaDA!
10.613
0.9579
26
17.226
1.751%
9%
6.677
2,5859
10.095
1.786%
CADA
10,630
1.122%
17.283
7.08899
CCL
6.636
1.957%
10,068
CCL
10.569
0.543
1.142%
216
OVRPB
HGS--y HGS-PyVRR
6.896
10,4m
10.335
20.8m
TIGS-PyVRP
11,689
10,
19.136
20.8m
CaDA
2.159%
2s
10.762
4.099%
11.261
0.779%
19:426
1.41%%
CíDA
7,064
2.377%
10.730
1.800%
CaDA
11:775
8.808%
26
19.495
1,754/
CCL
7,008
1.568%
$s
10,666
3.179%
19%
CCL
11.721
0.436%
6s
19.348
0.983
21s
OVRPBL
HGS-PPVRR
10:335
20.8a
HGS-PyVRP
11,668
19.156
20.80s
CADA
2,051
2.100%
2s
10.762
4.102%
$6
CaDAF
11,260
0.721%
2s
19.425
1.429%
9
7,862
2.539%
1s
10.741
1,900%
CaLAA
11.777
0.914%
19.497
1.76256
CCL
7,009
1.569%
56
10
3.323%
20s
CCL
11,721
0.442%
6x.
19.346
0.977 %
234
JIGS PyVRP
6.507
10,4m
9.724
20 20.8m
JIGS HCPS-PYVRP'
10.510
10,4m
16.926
20.8m
OVRN
CADA!
2.475%
10.122
4.053%
10,613
0.961%
17:236
1.752%
CaDA
6,680
2,623%
1s
10.093
3.773%
$6
CaDA
10.631
1.133%
Is
17.280
2.073%
9%
OCL
1.96%%
10.067
20c
10.569
0.546%
1.143%
27A
HGS PyVRP
9.687
10.4m
14.377
20.8m
IIGS PyVRP
18.292
10.4m
29.467
20.8m 00.8m
VRPB
CaIM?
2,960
2,8008
25
14.960
4.038%
CaDA?
14,500
1.12%
26
301089
1.9997
%
3.010%
14,910
3.721%
C:DA
18,543
1.361%
2.390%
CCL
9,916
2.352%
55
14.882
3.526%
19%
CCL
18,430
0.738
65
29.911
1.494%
21s
HGS-P.VRP
10.186
10:4m
14:779
HCS.P/VRP
18.361
20
VRPBL
CIDA?
10,543
3.461%
15.525
5.001%
CADA
18.848
1.376%
30 521520
2.3595%
CADA
10.576
3.776%
14
15.400
4.771%%
CADA
18.894
1.623%
1s
30.600
2.7009
9%
10.484
2888%
15,407
4.319%
18,773
0.976%
21s
HGS-PsVRP
10,567
10.4m
15.766
20 2m
HCS-PPVRP
16.356
10
25.757
20.8m
VRPL
CADA
10.731
1.333%
2s
16.057
1.847%
CaDA
16,689
1.8995
2s
26 36.540
2.995%
96
CADA
10.749
1.505%
16.036
L.725%
CADA
16:709
2.130%
26 26.631
1,358%
CCL
10,710
1.145%
5s
16.009
1.561 %
19%
CCL
16,579
1.333%
6s
26.366
2.321 %
20s
HGS
10:4m
12.584
20.8m
HGP-PPVRPP
14.175
22.730
Avg.
CADAF
8,646
2.2564%
24
13.022
3,595%
86
CADAF
14.380
1,12%
24
23.314
2.033%
9%
CaDA
3.662
2.437%
1s
12.993
1,372%
CaLA
14.409
1.300%
1s
23.394
2.381%
9%
CCL
1.802%
12.950
14.319
0.749%
1,476%
21s
Gap (5%
CaDA
CalkA
CCI-mean
OCL-std
3.4
39
4.5
1.4
2.9
2.9
24
21
2.1
1.9
Tooks
8.9
CVRP
OVRP
OVRPB
OVRPSL
OVRPL.
VRPB 0052
VRPBL 0.015
VRPIL 0.085
OVRP BLTWW
OVRP STW
OVRF LTW
OVRP TW
VRP BLTW
VRP RTW 0,02
LTW
VRF
VRP
TW
Std
0.052
0.905
0,627
0,000
0A82
0,006
0013
0038
DU07
0013
0,856
0,017
Figure 4: Error-bar analysis of CCL under 1=100.
C ADDITIONAL ANALYSES AND DISCUSSIONS
C.I COMPARISON WITH NEURAL SOTA METHODS
C.I.1 COMPARISON BETWEEN RE-IMPLEMENTED SOTA AND REPORTED SOTA
Main Results. We compare CCL against the reported SOTA method (CaDA (Li et al., 2025)) and our re- re-implemented one (CaDA (Li et al.. 2025)), with the strong heuristic baseline HGS-PyVRP (Wouda et al., 2024) included for reference. Table 5 presents the results for all 16 in-distribution tasks, and the overall average scores across these tasks both with and without the TW constraint. Under the N=100
17

---

Published as a conference paper at ICLR 2026
Table 6: Improvement of CCL' L over CaDA?,
Tasks
P-yalue
Tasks
P-salue
Tasks
A
P-value
Tasks
P-value
CVRP
8.16%
2.7e-05
OVRP
29.21%
1.2e-76
VRPELTW
41.6%
2.2c 68
OVRPBLTW
51.19%
6.2c-68
VRPB
20.14%
5.0e-42
OVRPA
29.86%
1.4e-22
VRPRTW
46.84%
4.0e-81
OVRPATW
50.63%
2.8c-68
VRPBL
20.03%
2.0e-39
OVRPBL
31.05%
4.7e-68
VRPLIW
41.39%
2.2e-97
OVRPLIW
51.50%
2.9e-79
VRPL
7.66%
0.4
OVRPI.
28.69%
3.5e:74
VRPTW
45.72%
2.9..99
OVRPTW
50.85%
3.2e-75
Table 7: Per-task results on 32 unseen
tasks (N=50).
Tasks
RP-TE
CaDA
CaDA
CCL
CCL
[Rtj
Clap
Time
Okg_ J
Cap 1
Time
Obj 1
Gap
Time
Cljj 0
Cap 1
Time
04g
Gep L Time
VRFMB
9.579
8,561%
1s
9,781
7.749%
2s
9.722
7.097%
35
9,943
9.538%
3s
9,940
9
4s
MECCVR
56.957%
25
12.335
54,08336
13.595
70.775%
41
10.829
54.446
10.554
31.361%
MDOVRP
6,876
1s
28.36915
6.260
36
20-155%
6.2%6
17.71916
56
MDVRPB
12.725
60.956%
25
12,651
60.054%
2s
12.190
54.066%
3s
11,579
47.450%
4s
11,001
38.750%
Ss
MDVREL
12.618
$7.370%%
2s
12,426
51.965%
26
13:433
67.697
4s
11.278
40.248%
4s
10,912
35.667%
OVRPMB
6.949
13,607%
1s
6.872
12.440%
6,819
11.570%
36
6.917
13.149%
36
6.854
12. 1244%
46
VRPMBI.
8,059%
10.163
10,061
6.130%
10.376
9.456%
7,898%
MDOVRPD
7.556 31
1s
7,622
33.009%
6,999
22,0325
34
6,876
19.787%
4s
6.738
17.332%
Sc
MDOVRPL
6.871
28 9469
16
6.807
27. 746466
26
6.270
17.504%
36
6.442
20.7166
46
6.305
18 07.07
So
MDVRPBL.
12.831
61. 1755%
26
12,585
58.011%
24
12.077
50.770%
34
11,453
43, 8469%
46
11.208
40.4133%
64
MEVRRMBB
544%
12.493
71,55066
12,066
68.185%
11,540
58.118%
11,032
51,055%
66
MEVREIW
17,818
48.941%
25
16.971
41.739%
17.581
46.365%
16.107
34.402%
15.475
29.004%
OVRPMBL
6.949
13.686%
1s
6.871
12.423%6
6.818
11.563%
3s
6,932
13.401%
35
6.819
11.555%
4s
VRPMETW
17,298
8.074%
1s
17.198
7,43456
26
17.282
7.954%%
36
16,988
6.099%
36
17.158
7. 7.142%6
56
MEDOVRFBL
7,500
31.772%
7.606
32.71396
1.6%
36
19.463%
6.827
18.939%
MDOVREMB
7.617 47.411%
1s 7,541
45.953%
32.344%%
36
526%
MDOVRPTW
10.818
34.976%
Is
10,204
29.610%
10.407
32.287%%
46
9,783
1507
4s
9,632
22.146%
MOVRPETW
18.591
37 3649
26
19,025
40.629%
26
19.977
47.721%
44
18,455
36.000%
46
17.645
30.121
So
MDVRPLIW
18.127
51 2766
24
17,125
42.765%
2A
17.965
49 2515
46
16,709
59, 7287228
16,126
34.252%
64
MDVRIMBL
2
12.434
56.969%
36
11,74
40.9
OVRPMETW
11.132
6 2666
Is
11.0K7
849%
11.121
1,00%
34
10,866
4.65%%
10,855
3.617%
VRPMBLTT
17.597
7.0625
Is 17.495
7.337%
2A
17,559
7.728%
36
17.514
7.433%9
36
17.402
6.710%
ADDVRPBTV
11,399
32.332%
24
11,190)
29.807%
26
11.425
52. 52.600%
4:
10.774
24. K305
46
10.313
19.359%
56
MEOVVPPITY
34.731%%
25
10.196
29.501%
10,408
52.2866
25.331%
9,704
23,040%
MDOVREMBL
7,602
47.108%
7,540 1s
6,848
32.401%
3s
30.088%
6,545
56
MOVRPBLIW
19,045
40.553%%
25
19.243
42,01156
26
20.076
48.201%
46
18:544
36.702%
56
17,612
29.641%
74
MDVRFMETW
17,830
48.435%
25
18,394
53.345%
25
19.327
6110839
4s
17.103
42.3144%
5s
16.729
39.084%
56
OVRFMBLTW
11,138
6.317%
L1,090
5.875%
11.116
6.109%
3s
10,993
4.912%
10,835
3.427%
S6
MDOVRPOLTW
32.207%
2>
11.155
29.743%
32.2166
74.%
19.783%
MIDVRPMBTW
11.055
153%
24
10,833
37.252%
11.150
41.3366
44
10.171
658%
10.110
27.872%
MDVRFMBLIW
18,222
$1.55 11.5544%
2s
18.511
51.0266
2s
19,362
61.119%
4s
17,940
19.174%%
5s
16.825
39, 63.6335%
fis
MDOVRPMBLTW
11.054
40.136/6
2s
10.818
37,044%
11.135
41.125%
10,200
29.114%
9,804
23.943%
6x
Ang.
36.529%
34,866%
34.417%
27.588%
24.102%
Bost (Best/oual)
0/32
0/32
5/32
1/32
26/32
without TW, CaDA outperforms the reported CaDA+, while in all other settings it is slightly inferior to CaDAt. In contrast, CCL consistently surpasses both CaDA and CaDA across all 16 tasks.
Error Bar Analysis. Since the testing update probability Pts is set to 0.5 for N=100, we further analyze the error bars of CCL under different random seeds. Fig. 4 plots the mean gap and its standard deviation over three independent test runs of CCL. Across all 16 tasks, the standard deviation of CCL's gap remains tightly bound between 0.005% and 0.060%. Visually, the error bars in Figure 5 are negligible compared to the performance difference between CCL and CaDA/CaDA indicating that the choice of seed has minimal impact on test-time results.
C.1.2 STATISTICAL SIGNIFICANCE
We continue to include t-tests to assess statistical significance. We first collected the gap values of 1,000 test instances from both CCL' and the strongest SOTA CaDA from Table 1, then we report the improvement percentages (A) along with the corresponding p-values (shown in Table 6). Here, the improvements are computed as the average gap reductions of CCL$ over CaDAt, i.e. - -Gaap(CCL)) Gap (ap(CaDA))) /Gap(CaDAT) x 100%. Across all 16 tasks, CCL achieves 7-51 % improvement. In particular, OVRPBLTW, OVRPBTW, OVRPLTW, and OVRPTW exceed 50%. All p-values are below 0.001, indicating that these gains are statistically significant.
C.1.3 DETAILED RESULTS ON UNSEEN OUT-OF-DISTRIBUTION TASKS
We provide per-task results on 32 unseen out-of- tasks. Each method is evaluated in zero-shot setting (i.e., directly tested without fine tne-tuning). For the N=50 setting, the test-time update probability is set to 1,,50.15, and for N=100, it is set to =.020.02. Table 7 and Table 8 present the full results under the 55 and --100 settings, respectively. For each task, we report the objective (Obj.), performance gap (Gap), and inference time (Time). Additionally, the bottom rows summarize
18

---

Published as a conference paper at ICLR 2026
Table 8: Per-task results on 32 unseen out-of-distriutiion tasks (N=1000.
Tasks
TE
CaDA
CaDA!
CCL
CCL
Ohgy
Gipp
Time
Olg.
Giop L
Time
Ohi
Giep 1 Time
Ohi
Glop L Time
Ohj
Gop
Time
VRPMI
14.888
10. 189%
56
14.710
8.822%
86
14,652
8.399
15%
15.322
13. 13.438%
14c
14.936
10.530%
18c
MICVVR
67.10%%
20.628
75.392%
115
20.964
78.117%
16.769
41.675%
178
16,834
MIDVERP
34,308%
96
10.605
33.38716
10%
9.849
10,095
26.715%
15%
9.753
22.409%
MDVRPB
18.721
61.761%
9
19.894
08.606%
11s
18.780
62.22116
17%
18.185
56.977%
196
13.625
50.9319
25s
MDVREL
20,100
70.198%
10s
20.867
77.336%
12s
21.001
78. 78.27716
18s
17.494
47.699%
24s
18.407
55 $5.828%%
29c
OVREMB
16.429%
10.468
16.205%
19. 19.642%
10.744
19.23516
VREMBL
15:198
10.375%
15.015
9.011%
14,949
8.536%
15.761
14.464%
14s
15.536
12,833%6
MDOVRPB
11.752
35 65.6
11.790
16.137%
10s
11.104
28.078%6
166
10,959
26.27.%
166
10.533
21.340%
215
MDOVRPL
10.703
34.620%
9%
10.574
32. 18.9
10s
9.854
23.749%
15%
10.272
29.03355
16x
9.71K
21.967%
20c
MDVRPRI
19,606
68. 8915
11%
19.827
70.875%
136
18.693
60 60.860%
18k
18.446
58. 5205
214
17.588
51.069%
284
MEVRPMB
76. 36.325%
83.817%
23995
17.544
MIVREIWW
29.468
53.28346
25.955
34.783%
10%
28.751
49.517%
26-901
39.557%
186
29.016
5082916
28%
OVRPMBL
10.709
18.877%
7%
10.485
16.399%
86
10.470
16.225%
136
10.883
20.748%
14s
10.749
19.28-16
196
VREMBTW
28.256
10.840%
S6
28.317
11.074%%
9%
35.310
11.038%
14s
28, 100
10.2391%
15s
27.971
9.722%
20s
MDOVRPBL
11.761
35.771%
9%
11.768
36, 16.106%
10s
11.105
28. 08.077%
16
10.963
26. 545%%
16x
10.511
21.090%
21s
MIDVRPMMR
53.010%
45,077
44.044%
MDDVEPTW
41 35/55
17.205
22.8045
10
17.636
36,20155
17A
17:002
31, 21.20095
17%
16.733
29.0765
MOVRPETW
30.681
39. 99.926
10k
30.176
37,589%
10s
31.716
44.719%
196
30U627
39.759%
19%
$1.269
42,473%
506
MDVRPLIW
29.640
53.9 17699
1Is
26. 200
35,862%
10s
29.327
52. 28416
17%
27.977
45.0555%
256
29.593
53.413%6
316
MDVRPMBLL
19,460
18.479
73.647%
70.333%
OVRPMETW
15.449
9.724%
18.478
8.9011%
18,430
8.00%%
18-427
5.590%
166
18.211
7.321%
216
VRFMBLTW
28.004
10.80556
35
28.658
11.00216
9%
23.641
10.2511
14s
28.374
9.907%
15%
25.582
10.09816
20s
MDOVRPOTW
19.590
36 897%
10s
19,305
34.918%
10s
19.341
35.15
18s
18.761
30 00.960
17%
18.265
27.473%
266
MDOVRPLTW
18.232
40,817%
10s
17.221
32,9235%
10s
67.665
36.128%
17%
16838
29 857%
18x
16.707
28.872%
236
MIOVRPMRII
52.726%
41.416%
35.973%
MDVRPBEIW
31.044 41 41.49
36.537
39.033%
10k
$2,239
46.9239
50.667
39.535%
24h
350%
MOVRPMBTW
29,651
54.3959E
10k
29.385
53.0015 5%
10s
30,722
15%
28.388
47.661%
186
50.84
50.555%
266
OVRPMBLTW
18.452
8.739%
18,476
8.88716
9%
18.415
8.51616
15s
18,404
8.443%
166
18.476
8.877%
206
MDOVRPBLTW
19:553
36.6.2%
19,341
35.156%
10s
19.349
35.212%6
18.553
31. 399%
17s
18.194
26.963%
MDOVRPMBTW
16.2166%
18.735
45.019%
10s
45.232%
37.8 17.823
MOVRPMBBLY
205355
132%
29.611
53,9895
11%
157
67.147%
18k
29.000
50.703%
23
057
4114
MIOVRPMRITW
18,846
45.873%
10k
18,770
45.2866%
10s
18.787
45. 45.427%
17%
17,803
57.665%
17%
17.667
36.646%
24h
Aug. Clep
41.144%
39.83496
39.09999
34,8925
34,788 %
(Res/Total)
(V32
4/32
7/32
17/32
each method's average gap and the number of tasks where it achieves the best performance, reported in the format "# Best (best/total)". Across both the 5= and N=100 settings, CCL consistently achieves the lowest average gap, demonstrating strong generalization unseen out-of-distribution tasks. In terms of per-task performance, CCL and CCL together outperform all baselines on 27 out of 32 tasks for N=50, and on 24 out of 32 tasks for N=100, further highlighting the robustness and effectiveness of our method across different problem scales.
C.2 COMPARISON WITH TRADITIONAL SOLVER
We follow RouteFinder (Berto et al., 2024; 2025) and CaDA (Li al., 2025) in using HGS- PRP (Wouda et al., 2024) as a strong tradi- tional solver. Moreover, we compare our method against additional traditional solvers, includ- ing Gurobi (Gurobi Optimization, 2024) and LKH (Lin and Kernighan, 1973) on CVRP in- stances with 50 customers. The total times, de-
Table 9: Comparison with traditional solvers
Methods
Obj. +
Time
Avg. Time +
HGGS-Py
10.372
10m
10.0s
Gurobi-15m
10.568
120m
15.0m
LKII
10.392
63s
1.1s
CCL
10.463
6s
0.2s
noted as "Time", are accumulated over 1000 instances, which are exactly the same as the ones used in (Berto et al., 2024; 2025; Li et al., 2025). We also provide the average per-instance time for refer- ence, denoted "Avg. Time". As shown in Table 9, the results of HGS-PyVRP are taken from (Li et al., 2025), while the results of Gurobi and LKH are obtained using a 32-core CPU. Based on this, Gurobi further uses threads per CPU core, enabling 4x32 instances be solved in parallel. LKH is executed for 10 runs with a 10-second time limit per instance. We set a 15-minute limit on Gurobi and report its generated (approximate) solutions. These results show that CCL achieves performance comparable to traditional solvers, while its total inference time is approximately 10x, 1,200x, and 100x faster than LKH, Gurobi, and VIGS-PyVRE respectively (corresponding per-instance speedups of 5x, 75x, and 50x). This demonstrates that learning- based models are practical for real-time appli- cations, especially when solving multiple VRP instances simultaneously. Moreover, in multi-task scenarios, CCL can learn generalizable patterns across different VRP variants, without requiring experts to manually design heuristics. Once trained, the model can solve 48 VRP variants without re- re-training, which will broaden its practical deployment. These findings show that both traditional algorithms and our learning- based method have their own merits and demerits. Research in either direction provides important insights for the VRP community.
19

---

Published as a conference paper at ICLR 2026
C.3 ABLATION RESULTS
C.3.1 ABLATION RESULTS WITHOUT RELD
We conducted ablation studies to examine whether the effectiveness of our method depends on ReLD (Huang et al., 2025). Starting from CCL', we remove ReLD, RGCR. TSNR, and both RGCR and TSNR. The corresponding av-
Table 10: Ablation on CCL (variant w/o ReLD).
Methods
Obj-4
Gapl
Time!
CCL
11,447
1.10%
6.5s
ReLD (CCL)
11.464
1.28%
5.4s
erage results across 16 in-distribution tasks are presented in Table 10. The results show that both RGCR and TSNR remain effective even without ReLD. Moreover, CCL reduces the average gap by 0.6% compared with CEL-TSNR-RGCK whereas CCL provides only 0.18% improvement
RGCR
11.477
1.40%
4.6s
TSNR
11.518
1.76%
2.8s
TSNR- RGCR
11.529
1.88%
2.0s
CCL. This indicates that the main performance improvement comes from CCL rather than ReLD.
C.3.2 ABLATION DETAILS WITHIN RGCR
To validate the effectiveness of RGCR, we test 16 in-domain VRP variants, each with 1,000 instances, and compare RGCR with four alternatives: (1) "Concat Attributes", which directly concatenates the constraint attributes; (2) "Concat Embeddings", which embeds each constraint into a high- dimensional space and concatenates them; (3) "+ Random Scores" employ random importance weights as the correlation scores; and (4) "+ Cosine Similarity"
cosine similarity measure the correlation scores. We present both the model complexity and performance results on the left of Fig. Specifically. "# Params" denotes the total number of parameters in the encoder and decoder, and "Time" is the accumulated inference time over 1,000 instances. "Avg. Gap" denotes the average gap across all 16 tasks, while "w/ TW" and "w/o TW" refer the subsets tasks with and without time-window constraints, respectively. Compared with concatenating attributes, RGCR achieves strong performance while increasing the model size by only 0. parameters and adding 1.3s to inference time. Compared to random weights, RGCR shows modest performance in average gap across the 16 tasks, but demonstrates clear superiority on tasks with time windows. These results demonstrate that RGCR benefits more complex tasks than simpler ones. This may be attributed to the fact that tasks without time windows often include a lot of padding information, which may introduce some noise during model training. Moreover, RGCR introduces no additional parameters compared to the "Concat Embeddings" setting, yet still reduces the average gap by 0.04 I %. This indicates that the gains arise from improved constraint prioritization rather than model capacity.
C.3.3 DESIGN CHOICE OF ATTENTION STRATEGY IN TSNR
In TSNR, we adopt a cross-attention mechanism, where the node embedding serves as the query and the unified node-constraint embedding as the key and value. The following theoretical analysis and empirical results show that this approach reduces computational complexity compared to the vanilla Transformer, which applies self-attention on the unified embedding. Specifically, the unified embedding has dimension (N (NN+N)1)) x while the context and node embeddings have dimensions N x and (N + 1) D, respectively. Consequently, self-attention computes (2NV + 1) x (2N + 1) attention weights, whereas cross-attention computes only (N + 1) x (2N + 1). As shown in Table 11, cross-atie cross-attention achieves comparable performance while reducing inference time. For example, across tasks with TW (i.e., the right half of Table 11), it narrows the gap from 0.727% to 0.689% and reduces inference time by Is.
C.4 PERFORMANCE COMPARISON UNDER MATCHED INFERENCE TIME
To further validate that the effectiveness of CCL is not merely due to increased inference time, we introduce a heavy decoder variant of CaDA, denoted as CaDA-HD. This version deepens the original 1-layer Transformer decoder to 4 layers, resulting in an inference time that is comparable to CCL. Table 12 presents the performance comparison between CaDA-HD and CCL under similar inference budgets. These results indicate that CCL and CaDA-HD perform similarly windows (TW), with CaDA-HD sometimes showing slightly better results. In contrast, on tasks with TW, CCL consistently outperforms CaDA-HD by significant margin. This suggests that CCL's
tasks without time
20

---

Published as a conference paper at ICLR 2026
Table 1I: Performance comparison under different attention mechanisms.
-ttss-Attettion
Self-Attention
Cross-Atlention
Self Self-Attention
Tasks
Tasks
Obj
Gap
Time
Ohj.
Gap
Time
Ohj 4
Gap
Time
Obj
Gap J Time
CVRP
10.463
0.881%
6s
10.461
0.867%
7s VRPTW
16.177
0.907 %
7s
16,
0.955%
8s
OVRP
6.610
1.566%
6s
6.611
1.582%
7s OVRPTW
10.564
0.506%
78
10,568
0.540%
8s
VRPB
9.875
1.921%
6s
9.873
1.896%
VRPBTW
18.419
0.678%
18.427 0.719%
VRPL
10.698
1.027%
6s
10,694
0.993%
VRPLIW
16.556
1.192%
16,564 1.246%
OVRPB
6.992
1.344%
6s 6.992
1.337%
7s OVRPBTW
11.718
0.416%
7s
11.721
0.444%
8s
OVRPL
6.610
1.569%
6s 6.611
1,582%
7s OVRPLTW
10,564
0.501 %
7s
10,565
0,517%
85
VRPBL
10.440
2,450%
6s
10.445
2.489%
VRPBLTW
18.758
0.899%
18.769
0.952%
OVRPBL
6,992
1.335%
6s
6.992
1.330%
OVRPBLTW
11.718
0.414%
11.721
0.446%
Avg.
8.585
1.512%
6s
8.585
1.510%
Avg.
14.309 0.689%
14.315
0.7279
8
Table 12: Performance comparison under matched inference time.
CaDA-HD
CCL
CaDA-HD
CCL
Tasks
Tasks
Obj.
Gap
Time
Obj. L
Gap J Time J|
Obj L
Gap
Time
Obj. L
Gup J Time J
CVRP
10.468
0.926%
6s
10.473
0.977%
VRPTW
16,293
1631%
0.979%
OVRP
6.635
1.937%
5s
6.636
1.957%
5s OVRPTW
10.615
0,986%
5s
10.569
0.543%
VRPB
9.908
2.267%
5s
9.916
2.352%
Ss VRPBTW
18,529
1.280%
5s
18.430
0.738%
66
VRPL
10,704
1.091
5s
10.710
1.145%
VRPLTW
16.676
1.900%
5s
16.579
1.333%
OVRPB
7,019
1.725%
5s
7,008
1.568%
OVRPBTW
11.772
0.874%
6s
11.721
0.436%
OVRPL
6.633
1.908%
5s
6.637
1.968%
5s OVRPLTW
10.615
0,986%
5s
10.569
0.546%
6s
VRPBL
10.480
2.842%
5s
10.484
2.883%
5s VRPBLTW
18.872
1.498%
5s
18.773
0.976%
6is
OVRPBL
7.020
1.721%
5x
1.569%
OVRPELTW
11.771
0.864%
0.442%
Avg.
8.608
1.802%
5s
8.609
1. 1.802%
Avg.
14.393
1.256%
Ss 14.319 0.749%
design is particularly effective in handling temporally constrained problems, and its advantage is not merely& a result of longer inference time.
C.5 CONVERGENCE ANALYSIS
train/loss
-0.01
Fig. 5 shows the training loss of CCL and CaDA. CCL achieves faster convergence reaches lower final loss compared to CaDA. For instance, at epoch 50, the loss of CCL is 0.0129, while CaDA is 0.0198. By the end of training, CCL attains 0.0090 versus 0.0129 for CaDA. These results indicate that CCL con- tributes more efficient and effective training, yielding both faster convergence and improved final performance.
-0.02
the early epochs and
0.03
0.05
-006
0.07
0
50
100
150
200
250
Figure 5: Training loss convergence.
C.6 GENERALIZATION OF CCL ON ADDITIONAL SCENARIOS
Same as RouteFinder (Berto et al., 2024; 2025), CaDA (Li et al., 2025), MTPOMO (Liu al., 2024), and MvMoE (Zhou et al., 2024a), our work also focuses on solving routing problems solely. Moreover, the design of CCL can be useful for other decision-making problems. We conduct preliminary experiment on the Flexible Flow Shop Problem (FFSP). When assigning an operation to machine, we incorporated TSNR to allow operation embeddings integrate information from the current machine, thereby updating the operation's state. Results in Fig. 6 (a) show that the method converges, but training can become slightly unstable from the middle late stages. This suggests that certain modifications to TSNR may be needed for the best performance, for example, to filter out irrelevant information in the machine embeddings that does not contribute to subsequent decision-making.
We also conduct an experiment using graph-structured inputs instead of coordinates. Specifically, each node's coordinates were replaced with vector of distances to all other nodes, which is then concatenated with demand and other attributes to form the node inputs. We retrain CCL across 16 tasks, each with 50 customers. During the 300 training epochs, we report the training loss and the validation average objective length across 128 CVRP instances (also with 50 customers). Fig. 6 (b) shows that both the training loss and the validation scores of CCL converge quickly within the first 50 epochs, suggesting the potential of CCL for graph-structured VRPs. To further investigate the effectiveness of RGCR and TSNR, we apply this setting retrain the corresponding baseline model
21

---

Published as a conference paper at ICLR 2026
Loss ObJ(FFFPP)
Graph-Structured CCL
CCL-RGCR- - TSNNR
Obj: (FVSP)
VRPBL
VRPBLTW
36
VRPL.
VRPLTW
21
41
61
VRPB
VRPBTW
Epoch
(a) Model Convergence of CCL FESP.
CYRP
VRETW
Loss
(bj.(CVPP)
OVRPBL
OVRPBLTW
Obj. (CVRP))
0.16
12
OVRPL
OVRPLTW
OYRPS
OVRPHTW
51
101
151 Epoch
201
251
OVRP
OVRPTW
(b) Model Convergence on Graph-Structured CCI.
(c) Ablation of Ohjective Values on Graph-Structured CCL,
Figure 6: Generalizing CCL on the flexible flow shop problem (FFSP) and graph-structured input.
Table 13: Per-instance results on small-scale real- wol-d instances. In CVRP, "X-n**** denotes the customer number of each instance, while each VRPTW instance has 100 customers.
CVRP
RP
CaDA
CCL
CCL-Bis
VRFTW
RP-TE
CaDA
Instmnors
0gt
Ohg
Gap
Otj
Cup L Ghjj |
Gop 1
OPg.
Cap
Testamnoss
Ope
[1b
Cap
Ohj ]
Gop
Ohg
Gop
Km101-225
17591
25087
5.422%
28765
4.255%
28765
4.255%
28717
4.117%
R100
1638
160
-2.068%6
1612
-1 1.569%
1619
-1.142%
X-n10-214
26362
27162
3.035%
27069
2.6825
26066
2.2911
26864
1.904%
R102
1467
1567
6. 6.8469)
1572
7.1875
1583
5.891 %
Xm110 k13
14971
15314
2.291%
15425
3.033%
15386
2.772%
15185
1.429%
R100
1209
1493
23.521%
1458
20.625%
1444
19.467
%
X-115-k10
12547
13338
4.638%
13143
3.107%
13334
4 4.65%
13162
3. 1.255%
R104
972
1315
35. 35.358
1325
36. 36.387%
1308
346637 %
X-1120-66
13392
13765
3.245%
3.900%
1355
7.430%
1463
7.947%
1443
6.471
X
55539
56525
5.378%
57943
4.328%
57671
3.33%
57442
3.426% %
R106
1235
1455
17.852%
1420
15.017%
1405
13.80%
Xan12-a18
28940
20598
2.274%
20517
1,948-220
29499
2.277
29458
1.790%
R107
10K5
1388
30.3785
1378
23.4
1327
24.645%
X-n134a13
19916
13585
6.29%
11468
5.057
1146
50029
11464
5,0208
R108
932
1310
411543%
1285
37.8611
1266
35.822% %
X-n159-210
13590
13812
1.645%
11661
2.008
13002
2.206%
18852
1926
R100
1147
1801
19 50494
1994
21 545%
1.899
18.40%%
%
X-n13427
15700
16257
3.548%
16231
3.305
13985
1815%
15985
18155%
R110
1068
1527
42. 42.978%
1984
29. 5885
1282
20
%
X 4:6
43448
45036
3.655%
45995
1.681%
68124
4 4318%
44053
3.661%
R11E
1049
40 460%
1424
35. 35.787%
1366
36,25 %
X
21220
23478
10.641%
7
25245
23172
91
R112
1276
34.725%
1210
X
16876
17339
2.744%
17225
2,009
17184
1825%
17131
1.5115%
RC191
1620
2.852%
1663
26675
1661
2.544%
Xa162-211
14138
14664
3,720%
14584
3.155%
14702
3 0999
14072
5777%
RC102
1457
1731
18.773%
1717
17.813
1646
12.941%
X-n167-100
20557
21435
4.715
21305
3,629%
20967
2.002
201154
1-884%
RC133
125%
1760)
39 39.300%
1656
31 6388
1624
29.044 1
X-n177-551
45607
45120
5.530%
47727
4.64%%
46152
S 800%
47816
16 1887%
RC104
1150
1610
42, 188%
1407
32 2005
1534
34 5054
X-1176-266 47812
51400
7.504%
52117
9.1 0.190%
51485
7 T662
51164
7.011%
RC100
1514
1967
23 540%
1755
1555415
1751
15.77% %
X
25569
26097
2 1065%
26228
2.577%
26180
2.390%
Z60TS
1.999%
RC1
1373
1964
21 21.2211
1634
190355
1611
18.088
X-n166-115
25140
4.11%
5.164%
55040
17.2%
1500
1601
24.077
X
16980
17892
5,371%
17726
4.292%
17547
3,330
37547
1,23%
RC198
1114
58.679% 1577
1564
40.200%
1504
34.98%
X 451 44225
47390
7.157%
46585
5,306
5/48%
46121
4.25%%
RG29)
1262
24 24.980%
1606
27.278%
1931
21-494%
X-n200-336
58578
61199
4.474%
61048
4.17%
4.797%
61385
4.797%
RC232
1082
1553
42.177%
1480
35 35,494%
1433
31.191 8
X 30656
31876
3.990%
32005
4,000
32131
5.474%
32216
1.089%
BC203
924
1465
58.601%
1490
61 11.30
1439
55.787 %
X-n28-223
25742
28798
11.72%
18318
10.046%
27641
7,377%
2764]
7. 1.377%
RC204
754
1372
75. 1121
1278
63. 114%
1225
56.350%
X k14
27042
29595
9.441%
29R30
10.310%
29816
10
256%
29616
10. 258%
RC206
1051
1573
19,653%
1447
37.663%
1456
38.522%
X-n2474500
8.537%
10.710%
10.710%
RC207
56.0915
1433
48.821
X k26
3868-
40099
4.433%
40360
4.333%
40125
5.276%
40505
4.70%%
RC208
89.76416
1433
84.641%
1335
72.014%
Avg. Cap
5/0865
4.707%
4.261%
Avg. Cup Bost (Ross/iosal)
36,5796
30.828%6
27.114%
Best (Bes//Tol))
3/27
8/27
4/27
13/27
1/27
2/27
24/27
(i.e., the version without RGCR and TSNR). Results in Fig. 6(c) show that, except for CVRP, CCL consistently reduces the average length compared to the baseline. This demonstrates that CCL is a aou--nnd-play strategy, which can be integrated into VRP solvers with various input structures.
C.7 EVALUATION ON REAL- WORLD BENCHMARK INSTANCES
Table 13 and Table 14 present per-instance results on small and large-scale real-world benchmarks, respectively. The small-scale set consists of 27 CVRP and 27 VRPTW instances, while the large-scale set contains 60 VRPTW instances. Following (Zhou et al., 2024a), the model is trained on 16 synthetic tasks with N = 100 and directly applied to all instances in a zero-shot manner. For CVRP, we set the test-time update probability Pr. to 0. denoted as CCL.. We also design an ensemble variant, CCL-Ens, which applies the trained model with four update probabilities Pte E {0.1, 0.25, 0.3) and selects the best solution. For scall-scale VRPTW, Pt. is fixed at 0.25.
On the CVRP benchmark, CCL yields a slightly higher average gap compared to CaDA. However, its ensemble variant CCL-Ens achieves the best overall performance, demonstrating the benefit of test-time adaptation via varying update probabilities. In total, CCL and CCL-Ens together outperform baselines on 16 out of 27 instances. On the VRPTW benchmark, CCL attains the lowest average gap and surpasses baselines on 24 out of 27 scall-scale instances and 35 out of 60 large-scale instances.
22

---

Published as a conference paper at ICLR 2026
These results indicate that our method can be effectively deployed in real-world settings, especially on complex constraints such as time windows.
Table 14: Per-instance results on large-scale real-world instances. The customer number is 600.
VRPTW
RF-TE
CaDA
CCL
Instances
Opt.
Obj. 4
Gap 4
Time
Obj.
Gap 4
Time
Obj. 4
Gap J
Time
C1-6-1
14077
17537
24.583%
1.4s
17355
23.290%
16418
16.633%
C1-6-10
13618
35201
158.498%
1.2s
24365
78.924%
1.3s
17861
31.162%
1.5s
C1-6-2
13948
20505
47.007%
1.1s
21571
54.650%
1.1s
20980
50.413%
1.6s
C1-6-3
13757
22648
64.635% %
1.1s
23142
68, 68..226%
1.18
24090
71.117%
1.6s
C1-6-4
13539
22743
67. 98.9
1.1s
22601
66.937%
1.1s
23787
75.698%
1.7s
C1-6-5
14067
18265
89.8%
1.2s
17643
25, 22.42
1.1s
16368
16.359% %
1.65
C1-6-6
14071
22405
59.229%
3.3s
19998
42.123%
3.3s
19481
38.449% %
4.9%
CI-6-7
14067
26369
87.456%
1.3s
23920
70.046%
1.2s
16848
19.771 %
1.6s
C1-6-8
13991
26618
90.248%
1.2s
20504
46.54 46.549%
1.1s
16844
20.390 %
1.6s
C1-6-9
13665
60014
339.196%
1.55
36753
168.967%
1.3s
18274
33.733%
1.65
C2-6-1
7752
15193
98.9%
3.1s
12018
55.027%
3.18
12410
60.084%
4.9%
C2-6-10
7124
29321
31 111.586%
1.38
17609
187.1
1.18
12161
70.707 %
1.68
C2-6-2
7472
14789
97. 939%
3.1s
14420
93.0005%
3.18
13642
82.587% %
4.7%
C2-6-3
7215
15033
308.358%
3.1s
16259
325.
3.18
18351
154.445%
4.7s
C2-64
6877
15039
118.685% %
3.18
16236
136.091% %
3.18
18952
175.585%
4.8%
C2-6-5
7554
22865
2012.695%
1.28
13343
76. 66.64
1.18
13040
72.628 %
1.5s
C2-6-6
7450
22171
197 605%
1.2s
13312
78 6899
1.1s
12032
61.508% %
1.5s
C2-6-7
7491
25219
236.644%
3.1s
14632
320%
3.1s
12908
72.307% %
4.8s
C2-6-8
7304
22619
209.6924
1.2s
13469
84.413%
12266
67.942%
1.5s
C2-6-9
7303
23663
224.009%
3.1s
15017
105.622%
3.18
12569
72.103%
4.76
RI-6-1
21274
29154
37.039%
1.2s
25041
17.706%
1.1s
26556
24.827%
1.66
RI-6-10
17584
30508
73.502%
1.2s
26126
48.581%
1.Is
26121
48.552%
1.6s
R1-6-2
18520
26017
40.482%
L.Is
26262
41.80%%
27011
45.849%
1.7s
R1-633
16875
26105
54.697%
1.18
26601
$7.636%
1.18
28093
66.478%
1.68
R1-64
15721
24450
55.526%
1.1s
24521
55.978%
1.18
26939
71.359%
1.9s
R1-6-5
19295
31715
64.370%
1.2s
24975
29.438%
1.1s
25367
31.470%
1 1.8s
R1-6-6
17764
25692
44.632%
1.1s
25559
43.8K3%
26890
51.376%
R1-6-7
16496
25749
56.099%
1.1s
26401
60 64.04
26693
61.813%
1.6s
R1-6-8
15584
23857
53.084%
1.1s
24533
57, 57.421%
1.18
24796
10.%
1.5s
R1-6-9
18474
30700
179%
1.2s
25067
35.687%
1.Is
25931
40.364%
1.76
R2-6-1
15145
31072
105, 159%
1.2s
22482
44.4
1.Is
21855
44.302% %
1.6s
R2-6-10
11837
35862
202.965%
1.2s
23395
97.643%
19894
68.066 %
R2-6-2
12976
22676
74.749%
1.0s
21124
62.789%
1.0s
24214
86.602%
1. 6.6
R2-6-3
10455
20072
91.97%%
1.0s
19274
84.347%
1.0s
24258
132.016%
1.8s
R2-6-4
7915
16925
113.848%
1.0s
16589
109.603%
1.0s
22399
183.012%
1.7%
R2.6-5
13790
33895
145.790%
1.2s
22311
61.789%
21,811
56.495% %
1.7%
R2-6-6
11848
22695
91.555%
1.1s
20914
76.522%
1 1.0s
221125
86.744%
1.5s
R2-6-7
9770
19723
101 867%
1.0s
18900
93.443%
1.0s
23231
137.772%
1 8.8
R2-6-8
7512
16596
120.918%
1.0g
16716
122.515%
1.0s
23249
209 479%
2 2.0k
R2-6-9
12737
35883
181.727%
1.3s
22963
80,2896
1.1s
20863
63,801 %
1.7s
RC1-6-1
16944
44173
160.697%
1.3s
28659
69.138%
22506
32.824% %
1.7s
RC1-6-10
15651
47967
206.473%
1.3s
33429
113.586%
1.2s
23311
48.940 %
1.6s
RCI-6-2
15891
24480
$4.053%
1.1s
25282
59.100%%
1.1s
23526
48.050% %
1.6s
RCI-6-3
15181
23667
55. 896
1.Is
23640
$5.718%
1.1s
23809
56.831%
1.6%
RC1-6-4
14753
23076
56.414%
1.Is
22451
52.177%
22178
50.327% %
1.6s
RC1-6-5
16536
45720
176.483%
1.3s
29036
75. 58.88
1.28
22505
36.095% %
1.7s
RC1-666
16473
47520
188.467%
1.3s
30741
86.611% 11
1.2s
22586
37.107% %
1.7%
RC1-6-7
16055
45359
182.517%
1.36
31323
95.0945%
1.28
21936
36.628% %
1 1.68
RC1-6-8
15892
42548
167.736%
1.3s
32086
101.903%
1.2s
22792
43.420% %
1.6s
RC1-6-9
15804
47394
199, 896%
1.3s
33968
114,940%
24490
54,966 %
1.7s
RC2-6-1
11966
46194
286.041 9%
1.35
26852
124.401 9
1.28
20004
67.172%
1 1.6s
RC2-6-10
8973
44372
394, 489%
1.3s
28880
221 8844%
1.2s
17107
90.643%
1 1.6s
RC2-6-2
10337
21343
106.474%
1.1s
20623
99,509%
1.1s
21034
103 485%
1.6s
RC2-6-3
8895
17942
101.711%
1.0s
17270
94.156%
22858
156.979%
1.65
RC2-6-4
6968
14864
113.333%
1.0s
14459
107.521% %
1.0s
18521
165. 820%
1.6s
RC2-6-5
11081
45546
31 311.039%
1.3s
27674
149.750%
1.2s
19270
73.906 %
1 1.68
RC2-6-6
10831
46487
329. 329.223%
1.36
27920
157.790%
1.28
18374
69.651 %
1 1.6x
RC2-6-7
10289
46929
356.091%
1.3s
28173
173.806%
18364
78.475% %
RC2-6-8
9779
45416
364.424%
1.38
29578
202.464%
1.24
18548
89.672%
1.7%
RC2-6-9
9436
44922
376.070%
1.38
29079
208, 208.171 96
1.28
16951
79.642 %
1.7%
Avg.
12694
29558
145.593%
1.4s
22917
88.188% 188%
1.4s
20634
70.961 %
2.0s
Best (Best/Tatal)
10V60
15/600
35/60
23


Reviews:
[{"review_id": "v31Lpt2258", "reviewer": "Reviewer_A3oC", "summary": "The paper presents Chain-of-Context Learning (CCL), a novel framework designed to enhance decision-making in multi-task Vehicle Routing Problems (VRPs) by dynamically adapting to evolving constraints. Traditional heuristic methods for VRPs are computationally intensive and inflexible, while existing neural approaches rely on static embeddings that fail to capture changing constraint priorities. CCL overcomes these challenges through two key modules: the Relevance-Guided Context Reformulation (RGCR) module, which dynamically prioritizes salient constraints, and the Trajectory-Shared Node Re-embedding (TSNR) module, which refines node representations by aggregating contextual information from multiple trajectories. Using a transformer-based encoder and reinforcement learning, CCL jointly models node and constraint dynamics to make adaptive routing decisions. Extensive experiments on 48 VRP variants show that CCL consistently outperforms state-of-the-art baselines, including CaDA, achieving superior routing efficiency and generalization to out-of-distribution tasks. Ablation studies confirm the complementary effectiveness of RGCR and TSNR, while complexity analysis shows that the slight increase in inference time is justified by significant performance gains. Furthermore, CCL demonstrates robust performance on real-world VRPTW instances, achieving the lowest average performance gap among all tested methods.", "questions": "1. In equation 1, shouldn't the dimensions of H be (N+2) x D?\n\t\n2. What is N&L, and C&L in figure 1?\n\n3. What does \"Multiple Trajectories\" mean in figure 1?\n\n4. What does \"multi-trajectory context\" mean?\n\n5. What is the dimension of the concatenated vector in equation 5?\n\n6. What does \"trajectory\" refers to in the context of the manuscript? Is it the same as a sub-route?\nThe linehaul and backhaul demands are a node-specific quantity. It is not clear why they are expressed using the notations for trajectory an decoding step (i and j).", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The problem addressed in this work is of high significance in the field of operations research. It deals with a complex and practically relevant challenge that aligns with current trends in optimization and decision-making under uncertainty. \n\t\n2. The authors have provided a thorough and well-balanced review of the existing literature related to the problem under study. The cited works effectively capture both the foundational research and recent advancements in the area, demonstrating a clear understanding of the current state of the art. The discussion successfully positions the proposed work within the broader research landscape, highlighting how it builds upon and differentiates itself from prior studies.\n\t\n3. The manuscript is well organized and presented in a logical, easy-to-follow manner. The problem statement, methodology, and experimental results are clearly delineated, making it straightforward for the reader to follow the development of ideas. The figures, tables, and explanations are used effectively to support the narrative and to communicate key insights.\n\t\n4. The authors have conducted a comprehensive experimental analysis that strengthens the credibility of their claims. The inclusion of evaluations on out-of-distribution scenarios demonstrates the robustness and generalizability of the proposed approach. Furthermore, the trade-off analysis and ablation studies provide valuable insights into the contribution of individual components and the practical implications of design choices.", "weaknesses": "1. The problem considered in this work is not a notoriously difficult problem to solve with existing learning and non-learning-based methods. Considering the complexities and the size of the problem, optimal solutions can be obtained by formulating the problem as Integer Linear Programming, and using commercial solvers given enough computing time. Since the problem is deterministic, computing time is not a limitation unless the problem size is significantly large. Therefore,  I would encourage the authors to provide more justification on why a learning-based method should be used for a small-scale deterministic combinatorial optimization problem.\n\n2. The baselines used for comparison are not strong. Provided that the problem along with the six constraints do not invoke any dynamically generated tasks or uncertainties, there are many traditional methods including Integer Linear Programing (ILP) based methods to compare. I would also encourage the authors to use the LKH solver to compare the performance of their proposed method. These centralized traditional methods usually provide optimal solutions compared to learning-based method. For deterministic problems like this, where optimal solution can be pre-computed, the use of learning-based approaches is not justified. I would also encourage the authors to provide a short (1-2 sentence) description about the baselines used, including any parameter/configuration settings.\n\n3. The manuscript lacks a formal problem formulation. In order to apply Reinforcement Learning, the problem has to be formulate as a Markov Decision Process, explaining the state space, action space, and the reward. Having the problem formulation gives the reader a better understanding of  how/why the proposed decision-making architecture works.\n\n4. I would encourage the authors to use statistical tests such as T-test or ANOVA to compare their method against other baselines. The statistical tests can provide evidence if the difference in mean is significant.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "P6yr7JlikS", "reviewer": "Reviewer_jrJZ", "summary": "This paper tackles the niche problem of multi-task vehicle routing optimisation, where a vehicle must deliver goods from a depot to customers arranged variably in 2D space as a graph of nodes. The authors introduce a method to aggregate the context (encoding information from multiple constraints and node features) while incorporating step-wise relevance of each constraint using its similarity to node features. They further add a module to incorporate sequential/temporal information from multiple trajectories and other nodes using multi-head attention. Contexts/features are encoded and actions are decoded using an encoder-decoder architecture. Results over a range of VRP tasks show slight improvements over state-of-the-art baselines in terms of performance, but with increased inference time.", "questions": "* I don’t understand how the action selection is RL? It seems to be a form of offline return (sum of future rewards) maximisation but there is no value iteration or policy gradient", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "* The methodology and diagram are clear\n* Fairly many tasks (comparative and ablative) are tested on\n* The results discussion and analysis are detailed", "weaknesses": "* The motivation for this architecture is not very clear to me. There have already been works that incorporate context from other nodes (Kool et al 2019) or use sequential information (Nazari et al 2018)\n* The architecture consists of several components which are not novel in this space. At the same time, they are combined in a complicated way and I don’t understand the need for this complication (e.g. combining constraint embeddings with node features many times over in different ways in RGCR)\n* The results show extremely incremental improvements in performance, to the point where it’s hard to tell if they came from the architecture or just the size of the networks etc.\n* The degradations in inference time, by contrast, are significant\n* The method only applies to a single agent (let me know if I’m wrong here as this wasn’t completely clear), unlike the SOTA baseline MTPOMO\n* Minor: ‘bias’ is misspelled ‘bais’ in the architecture diagram and the bar chart", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "4fxdmuHEY0", "reviewer": "Reviewer_yziX", "summary": "This paper proposes Chain-of-Context Learning (CCL), a novel reinforcement learning framework for solving multi-task Vehicle Routing Problems (VRPs). CCL introduces two key modules:  Relevance-Guided Context Reformulation (RGCR), which dynamically prioritizes constraints at each decoding step. Trajectory-Shared Node Re-embedding (TSNR), which efficiently refines node embeddings across multiple trajectories using shared context. The method is evaluated on 48 VRP variants (16 in-distribution, 32 out-of-distribution), demonstrating state-of-the-art performance on both seen and unseen tasks, particularly those involving complex constraints like time windows.", "questions": "1. The update probabilities $P_{tr}$ and $P_{ts}$ are set empirically. Have the authors considered a learned or adaptive scheduling mechanism for these parameters?\n2. How does CCL perform on highly constrained or infeasible instances? Is there a mechanism to handle or recover from constraint violations during decoding?\n3. The paper focuses on Euclidean VRPs. Can CCL be extended to non-Euclidean or graph-structured routing problems? What modifications would be required?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "**Originality**: The idea of step-wise, context-aware node re-embedding is novel in the multi-task VRP setting. Unlike prior methods that use static embeddings, CCL captures evolving constraint priorities and node states, addressing a clear gap in the literature.\n\n**Quality**: The paper is technically sound, with well-designed modules (RGCR and TSNR) and thorough experiments. Ablation studies and complexity analyses validate the design choices.", "weaknesses": "**Weaknesses**\n\n1. **Methodological Complexity and Limited Generalizability**: The proposed CCL framework introduces significant architectural complexity through its two specialized modules (RGCR and TSNR). While effective for multi-task VRPs, the approach appears highly tailored to this specific problem domain. The paper would benefit from discussing how these components might generalize to other combinatorial optimization problems beyond VRPs, or what adaptations would be necessary for broader applicability.\n2. **Narrow Technical Contribution Focus**: The primary innovation appears concentrated on neural architecture design rather than broader methodological contributions. The work introduces novel network components but doesn't substantially advance the underlying reinforcement learning paradigm, problem formulations, or theoretical foundations. A more balanced contribution spanning architectural, algorithmic, and theoretical dimensions would strengthen the paper's impact.\n3. **Insufficient Formalization of RL Framework**: The reinforcement learning formulation lacks rigorous mathematical specification. The paper would be significantly strengthened by explicitly defining the core Markov Decision Process (MDP) components.\n4. **Presentation Issues in Figures**:\n    - **Figure 1**: The loss term contains unclear symbols or potential encoding issues.\n    - **Figure 2**: There is a typo in the label \"Bais\" which should be corrected to \"Bias\".", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "fIXeIP729N", "reviewer": "Reviewer_Uhy6", "summary": "This paper addresses the challenge of solving multi-task VRPs, where existing neural solvers often fail to react to dynamic, step-wise changes in constraints and node states. The authors propose Chain-of-Context Learning (CCL), a novel framework that enhances the decoding process. CCL introduces two key modules: (1) a Relevance-Guided Context Reformulation (RGCR) module, which constructs a step-wise context by adaptively prioritizing salient constraint information, and (2) a Trajectory-Shared Node Re-embedding (TSNR) module, which efficiently updates node representations based on this dynamic context, capturing sequential dependencies without full re-computation. The authors evaluate CCL on a comprehensive benchmark of 48 VRP variants, demonstrating SOTA performance on all 16 in-distribution tasks and 32 out-of-distribution (OOD) tasks.", "questions": "In Table 3, the ablation studies are conducted based on ReLD (CCL$^\\dagger$}). I would like to know whether the effectiveness of the proposed model depends on the ReLD module.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The core contribution, explicitly modeling step-wise constraint dynamics within the decoder, is a novel and intuitive approach.\n2. The paper is well-written, and its structure is logical.\n3. The paper's empirical evaluation is thorough and rigorous. The use of a large benchmark (48 variants)  provides strong evidence for the method's generalization capabilities. The ablation studies are comprehensive.", "weaknesses": "1. In the Introduction (lines 56–57), the paper claims that existing methods fail to consider node priority. Could the authors provide illustrative examples or visualizations to demonstrate this point and further clarify how the proposed method captures node priority more effectively?\n\n2. Works such as Jieyi Bi’s PIP and others have also addressed constraint-aware learning, though often in single-task settings like TSPTW. It would be valuable to distinguish the proposed method from these studies and discuss whether such single-task methods could be adapted to multi-task problems.\n\n3. The rationale for the superiority of RGCR in integrating constraints into the embedding space is unclear. Incorporating constraints to improve performance is intuitively reasonable, but the paper does not explain why the proposed approach achieves better integration. A more intuitive explanation or analysis would be helpful.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "qZtFVExmZa", "reviewer": "Reviewer_xK8d", "summary": "The paper proposes Chain-of-Context Learning (CCL), a new neural framework for multi-task Vehicle Routing Problems (VRPs) that aims to improve constraint awareness and dynamic adaptation during solution construction. CCL combines two novel modules: Relevance-Guided Context Reformulation (RGCR), which adaptively highlights salient constraints for the current state, and Trajectory-Shared Node Re-embedding (TSNR), which stepwise updates node embeddings based on historical context and multi-trajectory exploration. The model achieves better results on multi-task VRP benchmarks of 48 variants.", "questions": "Given that many of your settings are flawed yet yield effective results, I would appreciate it if you could provide a reasonable explanation. If you can articulate the strengths in a reasonable manner, I am willing to increase the rating.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. This paper is clearly written.\n2. CCL achieves state-of-the-art results on all in-distribution tasks and most out-of-distribution tasks, with performance improvements that are robust to hyperparameter and random seed variations, as shown in standard deviation analyses.\n3. The ablation study is thorough.", "weaknesses": "1. **My main concern with this paper is that there is no theoretical basis for your design.**\n\n\nSome designs are not reasonable, violating some common recognition. You mentioned in the Introduction that \"They fail to leverage information accumulated in previous decoding steps, thereby limiting the model’s ability to capture and preserve historical preferences, which is an essential factor for coherent sequential decision-making.\" However, usually we define the RL process on the Markov Process, but your statement violates the Markov property. If you intend to include all past information in the state space, this would undoubtedly significantly increase the learning complexity, but this phenomenon is not observed in the training curves. \n\nSpecifically, I do not get the importance of providing the whole change trace of constraints as you mentioned. Your example \"For example, if a customer’s time window is about to be violated, the model should prioritize handling that node immediately.\", why can the customer feel the emergency with only existing scaler-based methods? They will probably make a wise decision based on the low number. \n\nWe know that, according to the Markov property, we can simplify this problem by partially solving it. According to you, such simplification (losing information about historical constraints) is actually detrimental to decision-making, which clearly contradicts the facts.\n\nIn conclusion, I believe the motivation in this paper is somehow flawed, and the authors should rethink their interpretation of their empirical findings.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)