Combination-of-Experts with Knowledge Sharing for Cross-Task Vehicle Routing Problems
Authors: Zikang Yu
Jinbiao Chen
Jiahai Wang
Venue: ICLR 2026 Poster (2026)
Keywords: neural combinatorial optimization
vehicle routing problem
cross-task generalization
combination-of-experts
Primary area: other topics in machine learning (i.e., none of the above)

Abstract: Recent neural methods have shown promise in generalizing across various vehicle routing problems (VRPs). These methods adopt either a fully-shared dense model across all VRP tasks (i.e., variants) or a mixture-of-experts model that assigns node embeddings within each task instance to different experts. However, they both struggle to generalize from training tasks with basic constraints to out-of-distribution (OOD) tasks involving unseen constraint combinations and new basic constraints, as they overlook the fact that each VRP task is defined by a combination of multiple basic constraints. To address this, this paper proposes a novel model, combination-of-experts with knowledge sharing (CoEKS), which leverages the structural characteristic of VRP tasks. CoEKS enhances generalization to constraint combinations via two complementary components: a combination-of-experts architecture enabling flexible combinations via prior assignment of constraint-specific experts, and a knowledge sharing strategy strengthening generalization via automatic learning of transferable general knowledge across constraints. Moreover, CoEKS allows new experts to be plugged into the trained model for rapid adaptation to new constraints. Experiments demonstrate that CoEKS outperforms state-of-the-art methods on in-distribution tasks and delivers greater gains on OOD tasks, including unseen constraint combinations (relative improvement of 12\% over SOTA) and new constraints (25\% improvement).

Published as a conference paper at ICLR 2026
COMBINATION-OO--XPPRRTT WITH KNOWLEDGE SHARING FOR CROSS-TASK VEHICLE ROUTING PROB- LEMS
Zikang Jinbiao Chen², Jiahai Wang**
1 School of Computer Science and Engineering, Sun Yat-sen University, P.R. China Department of Industrial Systems Engineering and Management, National University of Singapore 6ozakmmaiiln wyw.uu.suuuuu................. edu. en, bill. bill.c.jbmna juuuuu eg, wangjiahhmail 1. sysu.edu.cn
edu.
ABSTRACT
Recent neural methods have shown promise in generalizing across various vehicle routing problems (VRPs). These methods adopt either a fully-shared dense model across all VRP tasks (i.e., variants) or a -exture-of-experts model that assigns node embeddings within each task instance to different experts. However, they both struggle generalize from training tasks with basic constraints out-of- distribution (OOD) tasks involving unseen constraint combinations and new basic constraints, as they overlook the fact that each VRP task defined by a combi- nation of multiple basic constraints. To address this, this paper proposes a novel model, combination-oofexperrs with knowledge sharing (CoEKS), which lever- ages the structural characteristic of VRP tasks. CoEKS enhances generalization to constraint combinations via two complementary components: a combination- of-experts architecture enabling flexible combinations via prior assignment of constraint-speciffe experts, and a knowledge sharing strategy strengthening gen- eralization via automatic learning of transferable general knowledge across con- straints. Moreover, CoEKS allows new experts to be plugged into the trained model for rapid adaptation new constraints. Experiments demonstrate that Co- EKS outperforms ofate-of-thh-art methods on in in-distribution tasks and delivers greater gains on OOD tasks, including unseen constraint combinations (relative improvement of 12% over SOTA) and new constraints (25% improvement).
INTRODUCTION
Combinatorial Optimization (CO) plays pivotal role in numerous real-world applications, such as logistics (Zong al. 2022), transportation (Fu et al. 2025), supply chain management (Tirkolace)
et al. 2020), and resource allocation (Heydaribeni et al. 2024). The vehicle routing problem (VRP) stands one of the most fundamental yet challenging CO problems, requiring the determination of optimal routes for vehicle fleet serving a set of customers while satisfying multiple operational con- straints. Despite decades of algorithmic progress, traditional methods face significant limitations: exact approaches are computationally infeasible for large-scale instances due to the NP-hard nature of VRP (Wu et al., 2024), while heuristic approaches heavily rely on handcrafted expert knowledge and time-consuming iterative search from scratch for each new instance (Bogyrbayeva et al. 2024).
Recent advances in deep learning have introduced neural methods for VRPs, which autonomously learn heuristic policies from massive data end-to-end. These methods not only circumvent the de- pendency on expert domain knowledge but also produce high-quality solutions within short solving
time (Chen et al. 2025at Li al. 2025b; Goh et al. 2024 Chen et al. 2023b Zhang et al., 2023
Liao et al. 2025 Fang et al. 2026 Xiao et al. 2024). However, despite the promising performance,
most existing methods adopt task specifie learning paradigm, necessitating separate neural model for each VRP task. This lack of cross-task generalization capability incurs costly retraining and de-
ployment overhead when adapting to new tasks.
"oorresoondigg author.

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Published as a conference paper at ICLR 2026
More recently, some efforts have focused on developing unified models for cross-task VRPs. De- spite demonstrating feasibility,, current methods still perform suboptimally. especially under out-of- distribution (OOD) generalization scenarios: 1) tasks with unseen combinations of basic constraints, and 2) tasks involving new basic constraints. According to their model architectures, they fall into two categories: task-shared dense model and node-level mixture-of cefsuurr-off-xpertss (MoE) model. The task-shared dense model (Liu et al.) 2202b; Berto et al.] [024] Li et al. 2025a). whose parameters are fully-shared across all tasks, overemphasizes coupled representations but neglects task-specific ones, resulting in negative transfer among tasks. This leads to particularly poor OOD generalization. As an alternative, the nove-level MoE model (Zhou et al. 2024a Huang et al. (2025) employs a gat- ing mechanism to assign each node embedding within a task instance to different experts, fostering node-specialized experts. However, this gating mechanism restricts expert vision to narrow node
subset, which weakens experts' cognition of task-level knowledge.
Unlike general multi-task learning, we observe that VRP tasks often involve combinations of mul- tiple basic constraints. This motivates us to develop an innovative model architecture, called combination-oofexpertts with knowledge sharing (CoEKS). On the one hand, CoEKS utilizes constraint-specife experts to facilitate learning dedicated knowledge for every basic constraint. This enables flexible combinations of experts to manage diverse VRP tasks with constraint combinations and allows further plugging in of new experts adapt even to unseen basic constraints. On the other hand, CoEKS transfers general knowledge across constraints to foster collaboration among experts and enhance OOD generalization.
The contributions of this paper can be summarized as follows: 1) We introduce novel CoEKS model, building on the recognition of the prior structural characteristies of VRPs. This model is designed to enhance OOD generalization for VRPs with constraint combinations via two comple- mentary components and adapts to unseen constraints by plugging in new experts. 2) We design (CoE) architecture to acquire specialized constraint-level knowledge, in
a combination-of-expeee which each expert specializes in a basic constraint, enabling the model to effectively solve diverse VRPs by flexibly combining corresponding experts. 3) We propose a multi-view knowledge sharing strategy to transfer general knowledge across constraints, utilizing mutual distillation and shared transformation layers to automatically learn coordination among experts. 4) We demonstrate that CoEKS can be deployed on both state-of-the-art (SOTA) and classic backbones to show its univer- sality. Extensive experiments indicate that CoEKS outperforms SOTA cross-task neural methods for in-distribution (ID) and particularly OOD generalization.
RELATED WORKS
Neural combinatorial optimization for VRPs. Existing neural methods for VRPs can be primarily categorized into two distinct groups: 1) Neural construction methods leverage deep neural networks to generate feasible solutions in an end-to-end manner. Some works (Vinyals et al. 2015 Bello et al.,
2017 Nazari et al. 2018) pioneer this direction to address the Traveling Salesman Problem (TSP) and VRP Attention Model (Kool et al.) 2019). which combines Transformer with reinforcement learning, is regarded as a milestone. Policy Optimization with Multiple Optima (POMO) (Kwon ct al. 2020) leverages solution symmetry to improve policy learning and has become a typical back- bone model for many subsequent extensions (Bi al. 2024 Chen et al. 2025b 2023a| Zhou al., 2024b Hou et al.. (223) Fang et al. 2024) due to its prominent performance and Hlexibility. Recently,
RELD (Huang al. (2025) has emerged as the SOTA backbone model by incorporating feedfor- ward neural network (FFN) and identity mapping (IDT) into the decoder. 2) Neural improvement methods employ neural networks to replace handcrafted rules, iteratively refining an initial solution to meet specified requirements (Wu et al.) 202 Ma et al. 2021 Luo et al., [025 Ma et al.) 2023). Although such methods often yield superior solutions, their computational overhead is considerable. Consequently, this paper focuses on neural construction methods.
Some subsequent works enhance the generalization of neural construction methods across problem sizes (Luo et al.) 2023 Pan et al. 2025). node distributions (Bi et al. 2022] Liu et al., (024a) and their interactions (Manchanda et al. 2022 Wang et al., 2024h. Our work targets a more challenging and underexplored scenario: generalization across diverse VRP tasks.
Cross-task generalization for VRPs, Recent studies (Drakulic et al. 2024 Wang & Yu, 2023
Jiang al. 2024b) investigate unified model for cross-task CO problems, but focus only on II)
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Published as a conference paper at ICLR 2026
+6.
+0,
4.1
8.3
02
0.5
(C)
+0,4
+8.4
0044
Deput
Linchaul
Bockhaul
Time window
Feasible route
Figure
Illustrations of feasible solutions with various constraints.
generalization. In addition, Lin et al. (022) develop fine-tuning adapters on a pre-trained TSP model for new tasks. These methods fall short of zero-shot OOD generalization across diverse VRPs.
Existing approaches to zero-shot OOD generalization for diverse VRPs with constraint combina- tions can be divided into two categories. One type, such as POMO-MTL (Liu et al., 2024b), RouteFinnder (Berto et al., 2024), and CaDA (Li al.) 2025a), adopts task-shared dense model, which overemphasizes coupled representations at the expense of task-specife ones, leading to neg- ative transfer among tasks. The other type employs a node-level MoE model to route each node embedding to different experts, like MVMoE (Zhou et al. 2024a)- Rel.D-MoEL (Huang et al., 2025) integrates MVMoE with the ReLD backbone, achieving the SOTA performance on OOD generalization across VRP tasks. Nevertheless, these node-level MoE models suffer from narrow expert vision limited to a node subset, which deteriorates task-level generalization (Further compar- isons with the MoE models are provided in Appendix [..T) In summary, existing models overlook the structural characteristic of VRPs, which motivates us to propose a novel architecture, CoEKS. Along another orthogonal lines of research, [Liu et al. (2025) propose pre- pre-training paradigm for VRPs to improve generalization and Goh et al. (2025) study the multi-task multi-distribution VRPs using a mixture-of-depths architecture with clustering.
PRELIMINARIES
3.1 VEHICLE ROUTING PROBLEMS
Formally, VRP is modeled on complete graph G = E), where V Vi VV, includes depot node Vo and customer nodes V. {vi...., Ux). Each node i; E V is associated with 2D coordinate E [0, 1J.. Edge set E connects all node pairs, with each edge (i,j) E E associated with cost Gjj measured by the Euclidean distance. Each customer node v; E has a non-negative demand d, O, while the depot has zero demand. A VRP solution consists of a set of routes, each exccuted by a vehicle, that collectively visit all customers exactly once while satisfying task specific constraints. The objective is minimize total cost: min c(T), where c(T) Erer E(.n)r Cij. & denotes the set of feasible solutions, and represents route assigned vehicle.
VRP tasks are defined by applying different sets of practical constraints (see Figure I to reflect diverse real-world operational requirements. This paper focuses six basic constraints from recent studies (Berto et al. 2024 Zhou et al.) 2024a). 1) Capacity (C): Each vehicle has maximum capacity Q, i.c. the total demand of customers along any route must not exceed Q. Open Route (O): Vehicles are allowed to end their routes at the last customer instead of returning to the depot. 3) Backhaul (B): Customers are divided into linehaul nodes that require goods from the depot (delivery demand d,) and backhaul nodes that need goods to return to the depot (pickup demand P.). Vehicles serve both on single route, but all linehaul deliveries must precede backhaul pickups. 4) Duration I Limit (L.) Each route is subject to maximum duration or distance limit I., ensuring balanced workloads and operational feasibility. 5) Time Window (TW): Each customer n, has time window [e,, 4j] and service duration s;. Service must start within [e,, 4... Early arrivals must wait, and service is not allowed after 4, Additionally, all vehicles must return to the depot before a global time limit T.axx- 6) Mixed Backhauls (MB): Unlike backhaul, MB relaxes the strict linehaul- before-backhaul priority, allowing flexible sequences.
Each basic constraint exist individually in combination, resulting in rich set of VRP tasks. Their interplay introduces significant task diversity, requiring flexible and generalizable methods. More details on the VRP configurations and data generation process are provided in Appendix A
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3.2 NEURAL CONSTRUCTION METHODS
Neural construction methods represent cutting-edge approach to solving VRPs, using deep re- inforcement learning to construct solutions in an autoregressive manner, eliminating the need for precomputed labels or handcrafted heuristics. Typically, such methods adopt deep neural network with an encoder-decoder architecture to parameterize a stochastic policy. The encoder processes static VRP features (e.g., node coordinates and demands) to produce node embeddings. At each step, the decoder integrates these embeddings with dynamic context (e.g., remaining vehicle capac- ity, current route length) to output a probability distribution over unvisited nodes, from which the next node is sampled. This process repeats until all customers have been visited, forming a com- plete solution. The solution construction is modeled a Markov Decision Process, where state consists of the instance and current partial solution, and the action comprises the set of selectable nodes. Given a graph G. the policy network specifies the probability of a solution т, expressed autoregressively as po (r(G) = II Pa pa(ar[s)), where at and s, are the action and state at step t, respectively. T is the total number of decoding steps. The reward is defined as the negative cost of tour T, i.e., r(r) = - c(T). The task loss Cp is defined as the expected total cost. The policy is op- timized via REINFORCE with shared baseline b(G). defined as the average reward over multiple sampled trajectories per instance (Kwon et al. 2020). The policy gradient is estimated as:
V (OlG = Epstria) [(((r) - b(G)) Vo log (r)))
(1)
METHODOLOGY
This section presents the proposed combination-of-expee with knowledge sharing (CoEKS), which is tailored for cross-task generalization for VRPs with basic constraints and their combinations. The overall model structure is shown in Figure 2 where CoEKS is employed in the encoder (see Ap- pendix B for details). CoEKS addresses this challenge through two complementary components: 1) a combination-of-xxppeee combinations of constraint-specific experts handle diverse VRP tasks; and 2) a multi-view knowl- edge sharing strategy that enhances the model's 's learning of transferable general knowledge across different constraints, thereby improving cross-task generalization for VRPs.
(CoE) model that learns specialized knowledge and enables adaptive
4.1 CoE MODEL
The CoE model extends the transformer-basse architecture by introducing expert and combiner modules in the encoder. Each expert specializes in a basic constraint and adaptively aggregate their expertise through combiners, enabling efficient handling of VRPs with combinations of constraints.
Constraint-specifie expert. standard transformer block, the FFN processes node embeddings to capture complex relationships. CoEKS replace the FFN into a pool of constraint-specific experts (i.e., FFNs), where each expert (for j E = (C, O, B, I., TW3) }) specializes in a specific constraint: capacity (Ec), open route (Foo), backhaul ( F..). duration limit (Fz), and time window (Frw). For a given VRP instance with constraint set CS C ε, only the corresponding experts are activated, with outputs defined as:
(h), (h) - 0,
if E CS. otherwise,
where (h)) (FN E Rd is the output of the j-th expert, and E Rd is the input embedding. Since the capacity C' is fundamental constraint underlying all VRP tasks, shared expert mech- anism (Dai et al. 2024) is employed, where the expert Ec is always activated as a shared expert corresponding to CVRP. This design ensures universal problem-solving capability while facilitating expert specialization.
Combiner. To adaptively the combine outputs of activated experts, we introduce a combiner for each expert Ey. parameterized by W, E I The raw values of activated combiners are sy)h) = W,- h, j E CS, and then weights corresponding to the experts are calculated by softmax normalization:
exp(s; (h))
S,(h) = Exeesccp(ssall j E CS,
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pollcy
Salladio
In-diaribbttenn training vrt
Out-of-dittrbarinn
Decoder
generalizatio
Aill lnsver layee
Expert
Add & Norn
Comblnnr
DVRPIL
TTW
Tx
E
Malli attention
OVRPBLTW
FW
Friheiding
I
TW
Task
Sampling
OVIE
Task: (OVRF)
Figure 2: Workflow of the cross-task VRP method with CoEKS: the training (gray parts indicate inactive constraints), the encoder generates node embeddings. and the decoder constructs a feasible solution. CoEKS output is determined by the activated experts and combiners, with mutual distillation (MDis) among the activated experts. The trained policy then generalizes to OOD tasks. Add represents residual connections and Norm denotes normalization.
Sampling an OVRP instance from
with S& (h) for inactive experts (k & CS). Therefore, the final output of CoE can be obtained by weighted combination of active experts:
O(h) Σ Eh) Sh)h).
(4)
JECS
4.2 MULTI-VIEN KNOWLEDGE SHARING STRATEGY
The CoE model effectively specializes experts in distinct constraints, serving as foundation for han- dling diverse VRP tasks. To complement this, a multi-view knowledge sharing strategy proposed to enhance the model's learning of transferable knowledge across different constraints, thereby im- proving OOD generalization. This strategy operates in two views: expert-view and cmbbnnervveww
Expert-view knowledge sharing. To broaden the expert vision and strengthen their comprehensive understanding of VRPs, we introduce mutual distillation (MDis), where active experts exchange knowledge to capture shared patterns across constraints. Unlike traditional knowledge distillation, which transfers knowledge from a teacher to student model, MDis encourages peer-to-pest peer learn- ing among experts (Xie et al. 2024). An auxiliary loss Lmdd incorporated to facilitate this process. The overall loss function is defined as follows, where Lp denotes the primary loss of the task, and a controls the distillation strength.
L Lo a Linds
Lind calculated as:
0.
K 1,
Lmd
MS)(EE Ez(h)), EE MSE(E,(h), Eavg()),,
K 2,
(6)
K 2,
where K is the number of active experts, (h) E Rd is the output of the i-th expert, Eng (h) >K 1==1 (h) is a virtual expert averaging active expert outputs, and MSE is the mean squared error. For K = 1 (i.e., CVRP with only Ec), Lmd = 0. The virtual expert simplifies computation for K. > 2 by reducing the complexity from O(K2) (pairwise comparisons) to O(K), while still guiding experts toward consensus by minimizing the variance of their outputs.
K
To realize the trade-off between specialized and general knowledge, MDis is employed in the lower encoder layer (e.g., the first layer). This design choice is inspired by the property that lower lay- ers neural networks tend to capture general features, while higher layers focus on specific knowledge (Long al. 2017). By localizing knowledge sharing to these early representations,
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method promotes the exchange of broadly useful information among experts without causing ho- mogenization of their expertise. The similarity of expert representations in the lower encoder layers is empirically validated via the t-SNE analysis in Appendix E.2
Combiner-view knowledge sharing. To further enhance generalization to unseen constraint combi- nations, we introduce combiner-view knowledge sharing mechanism. Specifically, shared trans- formation layer fa is applied to the input embedding h before it reaches the combiners introduced in Section [4.1 Therefore, fa can inject cross-task knowledge for all combiners, enabling them to make informed weighting decisions across diverse VRP tasks. Given the importance of nonlinearity in modeling complex functions, we introduce nonlinearity to improve the representation of f.- For simplicity, fo is implemented as low-rank multilayer perceptron (MLP) with a residual connection:
(h) W2 ReLU(WW1 h) + h,
(7)
where Wi dyxr and W22 Rx are weight matrices forming bottleneck structure with < d, enhancing parameter efficiency while preserving expressiveness. The final output of CoEKS is:
OCOEKS(11) = Σ
Ey(h) S,(a()))
JECS
4.3 INFERENCE FOR CONSTRAINT COMBINATIONS AND ADAPTATION TO NEW CONSTRAINTS
During inference, CoEKS addresses diverse VRPs by activating experts corresponding to specific constraint combinations, as illustrated in Figure 2 It achieves zero-shot OOD generalization for unseen combinations by flexibly combining constraiin-speccffc experts. When encountering unseen basic constraints, new experts are plugged into the trained model and fine-tuned in isolation, with all existing parameters frozen to prevent catastrophic forgetting. This design maintains acquired knowl- edge and enables continuous rapid adaptation to new constraints, facilitating scalable deployment.
EXPERIMENTS
In this section, extensive experiments are conducted on 48 VRP tasks. All experiments are carried out on an NVIDIA RTX 3090 GPU and an AMD Ryzen 5 3600. Our code and data are publicly available at https://githibe com yonnn/uzziii ManzziikngC//oo.
We aim to answer the following research questions: Q1. Does CoEKS achieve superior ID and OOD generalization for tasks with unseen constraint combinations? Q2. Can CoEKS show superior scalability in OOD tasks with new constraints? Q3. Is universal CoEKS consistently effective across different backbones? Q4. How effective is the knowledge sharing strategy in CoEKS?
Baselines. 1) Traditional methods. Two heuristic solvers are employed in this study: the state-of- the-art VRP (Wouda et al,, 2024) and Google OR-Tools (Furnon & Perron 2023). Both methods use single CPU core to solve each instance. For node sizes limits are 10 and 20 seconds. 2) Neural methods. Recent representative cross-tas VRP methods are considered, including POMO-MTL (Liu al. (024b), RF-TE (Berto et al. 2024). MVMoE (Zhou et al. 2024a), CaDA (Li et al. 2025a), and ReLD-MoEL (Huang et al. 2025). RF-TE and ReLD- MoEL are the strongest variants reported in RouteFinder (Berto et al. |l. 2024 and ReLD (Huang al.,
50 and
100, the time
2025). CoEKS is implemented on the SOTA ReLD backbone (see Appendix B for more details).
Training. Our settings mostly follow RouteFinder (Berto et al., 2024). Each model is trained for 300 epochs, with each epoch containing 100K VRP instances. The Adam optimizer is used with learning rate of 3 x 10-4 and batch sizes are set 256 and 128 for n 50 and n 100, re- spectively. The learning rate is multiplied by 0.1 at epochs 270 and 295. Our training task set is similar to MVMOF, including CVRP, OVRP, VRPB, VRPI., VRPTW, OVRPTW, and OVRPI. (see Appendix C for further discussion). CoEKS adopts the mixed batch training and reward regular- ization scheme from RouteFinder, with the distillation strength a set to 0.01. MDis is employed in the first encoder layer. For all neural methods, the rest of the settings follow their original papers.
Inference & Metrics. For all neural methods, a greedy rollout with x8 instance augmentation (Zhou et al. 2024a) is employed. The test set is obtained through random sampling, with 1000 instances Per VRP RP task to reduce the impact of randomness. We show the average results of the tests, including the objective value (total cost), the gap to the best traditional solver, and the total test time.

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Published as a conference paper at ICLR 2026
Table 1: Performance on 1K test instances of ID VRP tasks.
Metiod
Obj.
Gap
30
Time
Okj.
Gap
10il
Time
Methidd
Obg.
Gap
50)
Time
Obj.
21 - 10I1
Cup
Time
PVPS-PVRPP
10.572
10:4m
15,638
20.8m
RGS--PVVRP
16:31
10:4m
25.433
20.5rm
OR-Toelst
10.572
1.507%
16.280
4.178%
OR-Toolsf
16.09
0.547%
25.814
1.506%
PUMU-MILL
10,502
1.257%
1.615%
76
POMU-MIL
16,428
2.471%
26,487
4.173%
76
MVMoE
10,482
1.05%
26
15.341
1.299%
9
MVMoE
16,439
2.550%
2s
25.472
4.113%
9
RF-TE TC
10,497
1.213%
1s
15.829
1.327%
66
RE
16.390
2.217%
Is
26.283
3.263%
7s
CaDA
10:49]
1,148%
36
15822
1.277%
lls
CaDA
16:297
1651%
2s
26.119
2.721%
12s
Rel.D-MoFI.
0.929%
15.707
1.116%
ReIDD
2.58696
36.
3.782%L
COEKS
10.464
0.891%
15.787
1.057%
9%
COEKS
16.361
2,050%
26,300
3.433%
BIGS-Py PRPP
6.507
10, Am
9.725
20.8m
VRPP
10.587
10.4m
15.766
20.8cm
OR TooTo
6.553
0.696%
10,4m
3.995
2.732%6
20.8m
OR OR-Toode
10.570
2,343%
10.4m
16.166
5.202%
20.8mm
OVRY
MVMoF
POMO-MTT
6.706
6.685
2.697%
3.025%
14
10.138
10.173
4.276%%
4.592%
66
MVMoF
POMO-MTI
10.756
10:736
1.362%
1.550%
1s
16063
160000
2,0598
76
6.678
2.595%
14
10.02
5813%
10.742
1.33%%
1s
16017
CaDA
6.683
2,6885
24
10.105
3.882%
11s
CaDA
10.729
1.317%
26
16014
1.584%
11s
ReLD MoEL
6,661
2,343%
2s
10,003
3,55916
9
ReLD MoEL
10.713
1.153%
25
15998
1.482%6
9
CoEKS
6.648
2.13%%
2x
10/046
3.290%
8s
CoEKS
10,712
1.12%
15,997
1.466%
9
HGS-PYVRPS
10.510
16926
OR-Toolla
9,802
1.159%
10.4m
14:933
5355%
OR-Toolaf
10.519
10.4m
17.027
0.583%
20.5
POMO-MIL
9,985
3.177%
1s
14,999
4.279%
76
POMO-MIL
10,991
1.700%
2s
17,500
3.2671
75
VI
MVMeE
9.966
2.857%
2s
14,952
4.025%t
9
MVMoE
10,696
1.747%
2s
17.165
3.278%
10s
RF-TE
9,984
3,050%
1x
14.926
3.838%
66
RF-TE
10.675
1.542%
1s
17.363
2.555R
76
CaDA
2.860%
14:906
6.00%
CaDA
1004%
17.267
1.990%
ReI.D-MoEL
9.936
2.557%
14:877
3.496%
9%
ReI.D-MBFL.
10482
1.613%
17,49
2.9455
10s
COEKS
9,9.0
2.497%
24
14,554
3.338%
9
COEKS
10,399
1.393%
24
17.376
2.633%
10s
IIGS VRPP
6.507
10,4m
9.724
20.8m
IIGS- PRRP
10.029
10.4m
15.367
20.8mm
OR-Tools
6,552
0,668%
10:4m
10.001
2.791%
20.8m
OR-Toolse
10.094
1.574%
10.4m
15.788
2.9925
20.8rm
POMO-MTT
6.709
3.070%
10.177
4.625%
POMO-MIT
10:255
2.521%
15,800
3.530%
OVRR
MVMoE
6.687
2.737%
10.140
4.244%
MVMoE
10.242
2.146%
153869
3.301%
5%
RE-E
6.678
2.600%
1s
10.096
3.303%
C6
RETE
10:235
2.097%
1s
15,802
2.501%
06
CaDA
6,681
2.685%
2s
10.006
3.900%
11s
CaDA
10.211
1.910%
2s
15.763
2.722%
11s
ReLD MoEL
ReED CoEKS
6.651
2.341%
2x
10.075
3.583%
9%
ReLD-MoEL
MoEL
10.219
1.902%
15,805
2.852%
9
2.135%
10,047
3.298%
10,203
1.751%
15.773
Avg.J I 2024
Average performance across VRP tasks. bold: Best results among learning based methods. Results are adopted from Barto for the convenicnce of comparison. * Best traditional method, taken the baseline for gap calculation.
hassing-hused
ct
QQ1) Generalization for ID and OOD VRPs. Table presents the results on ID tasks. Across different problem scales, CoEKS outperforms all neural methods on ID average gap (ID Avg.) and achieves the smallest gaps in 10 out of 14 cases. To evaluate their zero-shot OOD generalization performance, all methods are examined on 9 VRP tasks with unseen constraint combinations. As shown in Table 2 CoEKS consistently achieves the best performance, demonstrating its ability to effectively handle unseen constraint combinations by adaptively combining experts. For = 50 and 100, CoEKS outperforms all other neural methods in the overall OOD average gap (OOD Avg.), with relative improvements of at least 18.3% and 13,, respectively. Moreover, compared with the traditional solvers, CoEKS achieves competitive results with substantially lower solving time, offering notable efficiency for practical applications. We further conduct supplementary experiments on the OOD large-scale real-world instances of the CVRPLIB benchmark dataset (n > 500), where CoEKS consistently delivers superior generalization results (see Appendix E.4)
(Q2) Scalability for adaptation to unseen constraints by plugging in new experts. An unseen constraint MB is considered in VRP tasks, following the challenging setting in RouteFinder. Co- EKS plugs new MB-specific expert into the trained model and only fine-tunes this expert. Lin et al. (2024) introduce task-specific adapter layers (AL) for fine-tuning. RouteFinder proposes efficient adaptation layers (EAL) (Berto et al. 2024). which extend the weight matrix with zero- padding to support new constraints. We compare: 1) AL-based methods, including RF-TE-AL and ReLD-MoEL-AL: 2) EAL-based methods, including RF-TE-EAL and ReLD-MOEL-EAL. 3) two variants of our method. CoEKS+ where the new expert and combiner are randomly initialized; and Co which reuses shared modules (E, and Sc) for initialization to accelerate learning, inspired by Jiang et al. (2024a). For simplicity, our variants both use EAL to adapt to new constraint attributes. Aligning with RouteFinder, fine-tuning is conducted for 10 epochs with 10K instances per epoch tasks including VRPMB and VRPMBTW (see Appendix C.2 for more discussions).
As shown in Table 3 our methods consistently outperforms all baselines, where C exploits the knowledge of shared modules in CoE and performs best. This demonstrates that CoEKS has su- perior scalability in adapting to a new constraint by flexibly plugging in a new expert and combiner. This advantage is further amplified in OOD tasks with more constraints, where the extended model capability allows CoEKS to better handle complex generalization challenges (see Appendix E.3 for further validation on scalability to new Multi- (MD) constraint).
7

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Published as a conference paper at ICLR 2026
Table 2: Performance on IK test instances of OOD VRP tasks.
Mothod
5i1
100
Methad
- 5i1
R 2011
Okj.
Cap
Time
Gap
Time
Oky
Gap
Time
Obj
GGi
Time
TIGS- VYPP
6,898
10:1m
10.335
20.8m
I[GS- JIGS-PyVRPM
10.186
10.4m
14.779
20.8m
OR-Toolsk
6,938
0.412%
10.4m
10.577
2.315%
20.8m
OR OR--ookkk
10.331
1.390%
10.4m
15:426
4.338%
20.8m
OVRY8
POMO MTL
7,447
7.38615
12.091
169955
75
POMO MTL
10.743
5.28495
15.875
7.246%
7s
MYMol
11.720
13.305%
MVMoD
10,709
15.792
RF-TE
7.378
6.000%
1s
118840
14.520%
RE-TE
10.777
5689%
17.011
15.149%
CHDAA
2,701
11.3499%
25
11.796
14.074%
12s
CaDA
10.794
5.705%
15.883
1.399%
12s
ReLDDMMELL MoEL
7,335
6.272%
2s
11.446
10.6915
10s
ReLD-MMELL MoEL
10,621
4.130%
15.636 2x
15.639
5.694%
9
COEKS
7.341
4.913%
24
11.251
8811%
10c
CoEKS
10,650
4.387%
5.691%
9%
HGIS-P VRPS
HGS-PPVRPP
16.356
25.757
OR-Toollk
0.183%
1.597%
OR-Toolk
0.499%
26.259
POMO-MIT
19.105
4.09
18
31.419
6.992%
POMO-AITI.
16,864
3,677%
23 17.041
4,9799
7%
MYMoE
18:976
5.7175
24
31.441
6.675%
MVMoE
16.368
30888
26
26.99%
4.765%
10s.
RF-TE
19.029
4.0115%
Is
31.383
6.479%
76
RE-TE
16.365
3.069%
Is
27.055
4.963%
76
CEDA
19.018
4.4915%
34
31.568
7.117%
136
CaDA
16,780
2.514%
26.853
4.194%
13x
ReI.D-MoEL
31.218
5.92096
RcI.D-MoHL
3.599%
27.223
5.663%
10s.
COEKS
18,82
3.210%
31.108
5.746%
10h
COEKS
16.847
2.9494
26,891
4.555%
10%
HGS--PVRRP
10
10.335
HCPP-PYRPP
11,69
10.4m
19.155
20.8az
6.927
0.38611
10.4m
10.592
2.363%
20.8m
OR- Toolot
11,682
OLEUMEE
10.4m
19.303
0.757%
20.
ONTONN
POMO MTL
7A51
7.93716
1s
12.135
17.341%
7.
POMO MTL
12.094
3,5969
1s
20.255
3.690%
3s
MYMoF
7440
7.7789%
24
11823
14.289%
10h
MVMsE
12,027
3,038%
24
20.270
5,7508
11s
RETE
10663216
113876
14.64%
12:088
3.54866
CaDA
11.50595
25
11.791
14.024%
12:143
6.430%
13s
ReLD-MMoLL
7,341
6.380%
26
11:427
10,508%
10s
ReLDD ReIDD-MolLL
12,009
3.125%
20.159
5.192%
11s
COEKS
7,230
4.747%
24
11.245
N.755
96
CoEKS
11,972
2.566%
26
20.114
4.957%
10s
TIGS VYPP
10.51
10.4m
16.926
20:8m
IGGS PyVRE*
18.361
10.4m
29.026
20.8m
OR-Toolst
10497
0,1145
10
17.03
0.728%
20.8m
OR TooTTook
184422
0.3325
10,4m
29.830
2,770%
20,8m
POMO MTL
10,695
1.73516
17.508
3.415%
75
POMO MTL
19,482
4.74815
31,940
7,081%
7s
MYMEB
10.725
3.300%
MVMoL
19.361
7.256%%
10s
RF-TE
10.706
1.837%
1s
17.500
3.363%
74
RE-TE
19:400
4.379%
Is
31,936
7,089%
7%
CADA
10.627
1.097% 5
24
17:273
2.027%
134
CaNA
19:472
4.713%
36
33.447
8.8355
14s
ReLD-MoEL MoEL
10.702
1.7915%
26
17 17.462
3.141%
10s
ReID-MoEL MoEL
19,563
5.212%
31.738 25
32,222
8.056%%
11s
CoEKS
10.681
1.603%
24
17,402
2.785%
10s
CoLKS
19,263
3.714%
6.423%
10s
HGSSPP VRP
19.156
HGS-PyVRPS
12.315
11:681
0.10616
10.4m
0.767%
20,8m
12,364
0.415%6
10.4m
19.806
1.9425%
20.8a
POMO-MIT
12:101
3.59%
14
20.287
5.85%
POMO-MII.
12.587
4.701%
70.9%0
8.341%
RA
MVMoE
12:100
3.655%
24
20.306
5.949%
10k
MVMGE
12,842
4.349%
26
20.870
7,559%
10s.
RF:T
12,063
3.342%
1s
20.291
5.887%
74
RE-TE
12,554
4.223%
21,023
8. 4.7025
7%
CaDA
12.129
3,889%
34
20.356
6 1.666%
13%
CaDA
12,940
5.496%
20.030
7.8125%
136
R.I.D-MoRL
3.490%
5.747%
R:L.DMMdH
12,848
4.302%
29.784
6.735%
11s
CoEKS
11.999
2.797%
20.158
5.186%
10h
COEKS
12.754
3.432%
20.623
5.857%
10s
(OOD Ave.): Average performance across OOD VRP tasks. bold: Best results among learning-based methods. #: Results are adopted from Berto al. 2024 for the convenience of compurison. "; Best traditional method, Laken as the buseline for gup calculation.
(Q3) Universality across backbone models. we implement CoEKS on both the classic POMO (won et al. 2020) and the SOTA ReLD (Huang et al. 2025) backbones. The POMO- based methods include POMO-MTI (Liu al. 2024b). RF-TE (Berto et al. 2024h, MVMoR. (Zhou et al. 2024a), and our POMO- COMO-COEKS. The Re ReI.D-baee methods include ReLD-MTL, ReI.DRF, ReLD-MoEL (Huang et al. 2025), and our ReLD-CoEKS (i.c., the original COEKS implementa- tion). MVMoË and ReLD-MoE retain the MoE modules in their original decoders. Experiments are conducted on n = 50, and all new methods are trained using the original settings of their corre- sponding baselines. As illustrated in Figures (a-b), CoEKS consistently delivers the best average gap on both ID and OOD tasks under both backbone models. These results highlight the universality and consistent superiority of CoEKS.
(Q4) Effectiveness of knowledge sharing strategy. This section investigates the impact of the proposed multi-view knowledge sharing strategy. Ablation experiments are conducted on 16 VRP tasks with n = 50, under consistent training settings. The SOTA ReLD-MoEL serves as the baseline for comparison. Specifically, three ablated variants are examined: without (w/o) MDis, w/o shared transformation layer, and w/o both. Figures 3 (c-d) show the average gap for all variants on ID and OOD instances. It demonstrates that the multi-view knowledge sharing strategy significantly enhances model performance, particularly in OOD generalization. The results verify that multi- view knowledge sharing contributes to the learning of transferable knowledge across constraints.
Position of MDis. The effect of mutual distillation (MDis) among experts at different encoder layers is studied. Starting from CoEKS variant w/o MDis, the MDis mechanism is gradually introduced from lower to higher encoder layers, until enabled throughout. As shown in Figures 4 (a- b), incorporating MDis is generally beneficial for ID generalization unless applied to all layers. This suggests that moderate expert collaboration may promote their comprehensive understanding of the training task. For OOD generalization, improvements observed only when MDis is applied the
8

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Published as a conference paper at ICLR 2026
Table 3: Fine-tuning VRPMB and VRPMBTW at 7=
OVRPMB
VRPMBL
Metbed
Cost
Gxp
Cost
Gap
Cost
Gap
Cost
Gop
Cost
Gup
Cost
Gap
Cout
Cup
Cost
Gap
RE TE
11.31
24,69%
9.03
47.9215
22.32
137.2216
19.20
20.321
13.75
125.0495
14.22
36,375
19.95
22.81%
14,48
38.90%
2,54%
10.71
2.24%
Rel -AM
10.65
17.365
3.45
584455
11:37
19:919
18.18
13,65%
8.61
41.055
12.88
23,2553
18.54
15.9%
13.04
34.78%
ReLD MoLl- CAL
9.52
2.59%
6.43
5.09%
9.71
2.35%
15.38
2.30%
6,43
5.17%
10.67
1.88%
16.93
3.30%
10.00
2.14%
ColKS
4.19%
2.23%
10.64
1.57%
16.30
2975
COEKS
9:23
1.56%
6.33
3.52%
9.70
2.12%
16.33
1.9%%
6.33
3,48%
10.65
1.62%
16.79
2.92%
10.66
1.74%
22.30
2:40
56.10
7.000
g- 1:95
2.40
82220
62.00
2:00-
5.0
2:3
8.45
91.90
3.40
4.00
1.30-
0002
<
3.50
1.70
1400
Mrva
min
1,20
Mad BRO
univrnalty
Fuiivsselty
(DrAliaiio stayy
Figure 3: Left two panels: universality tests both (a) ID and (b) OOD tasks. Right two panels: ablation Study both (c) ID and (d) OOD tasks.
1.88
3.98
1.86
3.90
1.36
2 3390
g
33,00
3.85
8 1.84
3392
3.75
3,66
1.78
3.58
1.78
1.76
3.50
3.50
1.45
1.34
so M0
(1-21
(1-3)
(1-45
(h-h
(p-)
3.42
MBN
(b) of (145)
(1-21
(1-4)
(1-43
(t-)
1.51
5:40
0.0 (d) strength
Position
(e Dusilittion strongth
Figure 4: Left two panels: effect of MDis position on (a) ID and (b) OOD performance, The horizontal axis is the encoder layer number. Right panels: effect of MDis strength on (c) ID and (d) OOD performance. The x-axis is the weight a.
lower layers. The findings imply that deeper layers encode more task-specific patterns. Applying MDis on these layers may lead to homogenization of knowledge specialized constraints, thus preventing model generalization validate our choice of applying MDis only at the first encoder layer, which achieves a good balance between expert interaction and specialization.
VRPs with unseen constraint combinations. These results
Distillation Strength. We further assess the impact of the distillation coefficient a by measuring average gaps both ID and OOD tasks. Figures 4 (c-d) show that a properly chosen enhances overall performance. However, when a increases beyond a certain threshold, the experts tend to produce overly similar outputs, resulting in no performance gains. This supports our use of a 0.01 as simple and effective setting.
CONCLUSION
This paper presents CoEKS, novel model that leverages the structural characteristic of VRPs to address cross-task challenges. CoEKS integrates two complementary components: combination- of-experts architecture that adaptively combines constraint-specific experts for diverse VRPs, and multi-view knowledge sharing strategy that automatically learns transferable knowledge to enhance cross-task generalization. In addition, new experts can be seamlessly plugged into the trained model to handle unseen constraints. Extensive evaluations on 24 VRP tasks demonstrate that CoEKS greater gains OOD scenarios. includ-
achieves SOTA performance on ID tasks and yields ing unseen constraint combinations and new constraints. Furthermore, CoEKS exhibits consistent superiority across backbone models, highlighting its universality.

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Published as a conference paper at ICLR 2026
A current limitation is that handling more constraints inevitably increases the number of parame- ters. However, this trade-off is natural, since more complex problems with more constraints demand stronger model capabiliity A promising future direction is to explore more efficient expert-sharing mechanisms enabling a single expert to serve multiple similar constraints, or more efficient param- eterization strategies to scale model capability.
ACKNOWLEDGMENTS
This work is supported by the National Natural Science Foundation of China (62472461), and the Guangdong Basic and Applied Basic Research Foundation (2025A1515010122),
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Combination-of-Experrs with Knowledge Sharing for Cross-Task Vehicle Routing Problems (Appendix)
A DETAILS OF VRPs
This section provides details on the 6 basic constraints described in Section 3.1 By combining these constraints with the base CVRP task, a total of 48 VRP tasks are constructed. CoEKS is evaluated on 16 of these tasks as introduced in MVMoE (Zhou al. 2024a). To further assess the scalability of the model to new constraints, scalability experiments are conducted on the remaining VRP tasks with Mixed Backhauls or Multi- et al. 2024). The data generation process for VRP tasks is detailed below.
both (see Table 4, following RouteFinder (Berto
Node Coordinates. The single depot and all customer nodes uniformly sampled within the unit square [0, 1J2.
Capacity (C). Following RouteFinder and MVMoE, the vehicle capacity C is set to 40 for 50 and 50 for n = 100. For customer i, the linehaul demand d, sampled from the integer set (1, ............. 9).
Backhaul (B). 20% of customers are randomly selected to sample their backhaul demand from the integer set ( 9), while the rest to O. For the selected customers, the linehaul demand is set to O. As result, each customer has only one type of demand.
For non-backhaul instances, all backhaul demands are set to O. Before passing into the model, all linehaul and backhaul demands are normalized by vehicle capacity C to [0, 1].
Duration Limit (L.. This constraint imposes maximum route length per vehicle. Following RouteFinder, L is sampled from (22 max C where coi is the distance from the depot to customer i.
Time Window (TW). Following RouteFinder, the time window and service time for customer are generated through multi-step process ensure feasibility and diversity:
1. Service time: Sample service time 8; ~ (0, 0.18).
2. Window length: Sample time window length At, 0/0.18.0.2..
3. Upper bound: Calculate the upper bound for start time as uy Tm 1, where
T max the maximum allowed duration for a route.
4. Start time: Set the start time = (1 (w: - 1) r.) Coi. where ~ U(0,1).
5. End time: Compute the end time 4 = e, + At,.
For the depot node, the time window is fixed to [0, T max) and the service time is set to O. In addition, the vehicle speed 1.0.
Open Route (O). The O constraint alters the route structure, allowing vehicles to finish at any customer node instead returning to the depot. It is implemented by setting binary indicator o = 1, without additional data. When combined with other constraints, feasibility checks are adjusted dynamically:
With I., the route length is computed without the return-to-depot distance.
With TW, arrival time calculations omit the depot return segment.
With C, B, MB, the constraint logic remains unchanged, while the depot return require- ment is removed.
Mixed Backhauls (MB) The demand configuration follows the same setup the backhaul straint. For instances involving MB, a binary flag μ is set to and otherwise. This flag is used to distinguish between instances with and without the MB constraint.

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Table 48 VRP tasks with 7 constraints. 32 VRP tasks with Mixed Backhauls Multi-Depots
both are used to evaluate model's scalability in adapting to unseen constraints.
VRP usk
Capucity (C) Open Rouse (0) Backhaul (B) Mixed Backhauls (MB)
Duration Limit (L)
Time Window (TW) Multi Depet (MD)
CVRP
VRPB
VRFTW
OVRPTW
OVRYL
OVRPB
VRPOL
VRPBTW
VRPLTW
OVRPEL
OVRPETW
OVRPLTW
VRPBLTW
OVRPBLTW
VRPMB
OVRPPE
VRPMBL
VRPMHIW
OVREMIL
OVRPMBIW
VRPMBLIW
OVRPMBLTW
MDCVRP
MDOVRP
MOVRRPB
MDVRPL
MIVRRFTW
MIDORRFTW
MDOVRPL
RUB
MIVRRPBL.
MDVRPEIW
MIVRRLIW
RUBL.
MDOVRPETWW
MDOVRPLIW
MOVRPBLII
MIDVRPBUITW
MIVRRPAR
MDOVRPMB
MIVRRPABI
MIVREMBTW
MDOVRPMBIL
MIDORPMBITW
MIVRPMBIIW
MDOVRPMBLTWW
Multi- Mupti- (MD) The single-depot setting is extended to multi-depot configuration. Vehicles may start from any depot but must return to the depot they depart from. Appendix E.3 reports results 24 variants with the MD constraint. Following RouteFinder, the number of depots is fixed to three.
B DETAILED ARCHITECTURE OF COEKS
B. ENCODER
The encoder transforms static node features into embeddings for various VRP tasks. For the i-th (i E {I.....n.) n.) customer node, the static feature is defined as F; = {z,, yi, de Pt, e; 4, s.). where Ie, V/s represent coordinates, d;, denote linchaul and backhaul requirements, e,, 4 specify the time window, and s; indicates service time. The depot node F6 = {*a, Yo. μ}, where p is a binary flag indicating the presence of mixed backhaul. These features projected into an initial embedding h p (nxx through linear layers:
A Concat(W,, W., We,Fe Fw)
(9)
W., E 3axd and W., E 7xd are learnable parameter matrices, where = 128. N (N = 6) encoder layers process 4 produce the final embeddings N.N. Each encoder layer comprises
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two components: multi-head attention (MHA) layer followed by feedforward network (FFN) layer. Both are integrated with residual connections and instance normalization (IN) to stabilize training (Vaswani et al. 2017). Formally, for the f-th layer (e E [0, - 1)):
h2 (N + MHA (4, kD),
(10)
h +++ = IN + FFN(
(11)
Multi-head attention (MHA). The MHA mechanism employs A (A= 8) attention heads to com- pute diverse node interactions in parallel. Their outputs are then aggregated into unified represen- tation. For an input embedding hg E IRd each head E 21.2,..., A} computes query (Q), key (K), and value (V) vectors:
W - = = Vi = Wyn
(12)
where W W W& E Rd xd are learnable parameter matrices, and d2 d/A 16. These projections are computed for all nodes i E {0, (o....... n). with node as the depot. The compatibility between nodes i and j E (.......,, n) is measured via scaled dot- dot-product attention, followed by softmax:
(Q,")) f.
"
Softmax
Vd
(13)
weighted sum over the value vectors is then computed for each head:
L Mig
(14)
j=0
The outputs from heads are concatenated. Finally, a linear transformation applied to obtain the output of the i-th node in the MHA layer:
MH M4, hf) Concat (2) ,....,2" L Wo:
(15)
where Wo E R is a learnable parameter matrix.
Feedforward network (FFN). The FFN layer contains two linear layers with ReLU activation:
FFN( = WA, ReL.U(W,, M).
(16)
where W E Rd and W E Rddxd are learnable parameter matrices, and d, 512 denotes F. the hidden dimension. In our framework, each FFN layer in each encoder layer is replaced with CoEKS layer, comprising & FFNs (FFN,.... FFN k 5 aligns with the number of VRP constraints considered in our experiments. can be flexibly extended to adapt new constraints. According to Eq. (8) Eq. (11 can be rewritten as:
M IN + FAN, -,)((())))
(17)
JECS
where CS is the set of constraints activated for the current instance, S)(+) represents the j-th acti- vated combiner function, and f-) is the shared transformation layer.
B.2 DECODER
The decoder constructs solutions by sequentially selecting nodes based on static embeddings h. (produced by the encoder) and dynamic features D, (ct, 1,, de, ot, be). where Ct l,, de, ot, by rep- resent the remaining linehaul capacity, current time, current route length, binary open route indicator and remaining backhaul capacity, respectively. At decoding step t, the context embedding is com- puted as hē W. Concat(h," D.). where We E dxx((+5) h denotes the node embedding visited at step
learnable parameter matrix and The context embedding is then updated via the MHA layer and the identity mapping functton (IDT) (Huang et al. 2025):
h (HA N NM + IDT(h;).
(18)
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In MHA, hy is used to compute queries, and h is used to compute keys and values:
Qae W heg k W vo Wy""" h
(19)
where T E (d+5) Wx WV E Rdxxd are learnable parameter matrices of the a-th attention head. Then, the output of the MHA layer obtained by Eqs. (3)) The IDT function explicitly injects context information into ht", complementing the attention based update,
(h,) h + WIDIDD,
(20)
where WiDT RRdx is learnable parameter matrix. h: is passed through FFN layer (see Eq. 16) with a residual connection to generate the query oi The logits for all nodes are then computed as:
(ATTT Vd
C- tanh
i
(21)
otherwise
where C = 10 is a clipping hyperparameter that bounds the logits to promote exploration. Fo represents the set of feasible nodes at step t, defined by task-specific constraints. Finally, the node selection probability is computed using softmax over the logits.
C TRAINING AND FINE-TUNING DATASETS
C.I TRAINING DATASET
The training task set (see Section 5) follows the configuration in MVMoE Zhouu et al. 2024a) covering CVRP, OVRP, VRPB, VRPL, VRPTW, and OVRPTW tasks. In addition, we include the OVRPL task, which is motivated by two key reasons: (I) More complex instance generation. In- stead of applying a fixed duration limit L as in prior work (Liu et al. 2024b Zhou et al.) 2024a Huang et al. 2025), , adopt sampling-based strategy (Berto al. 2024) generate L, resulting in more diverse and challenging instances. (2) Complex interaction between constraints O and L. During training, the VRP tasks involving O and L constraints consistently degrade generalization performance (see validation curves in Figure 5). This instability arises from complex interaction be- tween the two constraints: O enlarges the solution space by removing the depot-return requirement, while I. restricts it with strict route-length bounds, resulting in convergence difficulties. To address this, the OVRPI. task is included in the training set and all models are retrained to ensure a fair and consistent comparison.
C.2 FINE-TUNING DATASET
This section explains the rationale for including the VRPMB and VRPMBTW tasks in the fine- tuning task set. Initially, all methods are fine-tuned solely on the VRPMB task, treating the remain- ing VRP tasks with MB cuttt--diistibutionn (OOD) generalization targets. As shown in Figure 6 the generalization performance of VRP tasks involving both TW and MB degrades significantly. The results may arise from the spatio-temporal conflict between the flexible routing requirements of MB and the strict deadlines imposed by TW. To mitigate this, VRPMBTW added the fine- tuning task set. As result, our method consistently improves performance across all tasks, whereas RF-TE and ReLD-MoEL exhibit convergence failures on several tasks. These findings underscore the challenges of adapting to unseen constraints and demonstrate the superior scalability of CoEKS.
D EFFECTS OF DIFFERENT TRAINING SETS
To further investigate the impact of CoEKS under different training datasets, we consider incorpo- rating all possible combinations of basic constraints into the training set for VRP tasks, reflecting RouteFinder's 's philosophy of establishing a foundational model for VRPs (Berto et al. 2024). To ensure fair comparison, all methods trained under the same settings.
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(TARI ART 00RP
CMRPL WUY
140
130
100
30
50
80
20
Mn Mir
40
20
c0och
0
50
103
150
200
250
100
150
200
250
OVRPBLTW
DHAPL wib GURPL
CWPPL OWR
40
120
30
Mn Andid
20
60
400ch-
50
100
150
200
50
100
150
200
250
Figure 5:
The validation curves of ReLD-MoEL trained without (w/o) OVRPL and with (w.)
OVRPL on n 50.
D.1 TRAINING ON 16 VRP TASKS
The ID generalization results are reported in Table where task types of the test set are identical to the training set. Among the neural methods, CoEKS delivers the best overall performance. CaDA shows strength on TW-constrainedd tasks but struggles to maintain competitive performance on the remaining variants. In contrast, CoEKS consistently achieves top ranks across all tasks, highlighting its superior ability to balance diverse VRP variants within a unified model. These results represent an in-distribution (ID) scenario, all constraint combinations are included in the training set.
Beyond this ID setting, CoEKS significantly outperforms CaDA in the OOD scenario (see Table 2. which underscores its potential foundation model capable of generalizing unseen tasks.
D.2 FINE-TUNING ON ALL VRP TASKS WITH MB
To evaluate scalability new constraints, we previously fine-tuned CoEKS on small set of tasks with new constraints and evaluated generalization to all tasks. Building on this, we further con- sider fine-tuning on all VRP tasks with new constraints, aligning with RouteFinder. Following this setup, all methods are fine-tuned on all VRP tasks with MB. The results are presented in Table 6 where CoEKS consistently demonstrates superior performance across all tasks. Furthermore, the relative gap with comparative methods widens, suggesting that the new expert can further refine itself through interaction with diverse experts. This highlights CoEKS's exceptional scalability to adapt new constraint.
E ADDITIONAL EMPIRICAL RESULTS
E COMPARISON WITH MoE
Difference from MoE. 1) Semantically grounded routing: MoE architectures typically use learned gating to select top-k experts per node embedding, lacking semantic alignment or cross-task reuse. In contrast, CoEKS better leverages prior knowledge to combine experts, which is both interpretable and efficient. 2) Broader expert vision: Gating mechanism of MOE-based methods (Zhou et al.
2024a Huang et al. 2025) restricts expert vision to a narrow node subset, which weakens experts cognition of level knowledge. In contrast, CoEKS effectively learns constraint-leva knowledge through dedicated experts and understand task-level knowledge via combination of experts. 3) Sta- ble utilization and load balancing: Existing MOE-based methods (Zhou et al. 2024a Huang al.
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VRPMB
VRPMBL
CIDAS AND AEPMITH
HETMET & MOD, REPMET SMP/0TT
CGEAS WID VREMOTH
-SADD HUC.- with VRPMITT
10
20
30 40
50
10
20
30
40
50
OVRPMB
-O-PPAT wit ARPHIIT
ROLD MOEL VAPMOTTH
DEPMET SI-TE WI-IPNETT DA.D-WEL 1RPHITWW VRPMETH VRRMSTT
18
16
14
12
10
20
40
20
30
40
VRPMBLTW
OVRPMBL
VIT9RITH INLD PUEL
Ri.D HUGL, VAPMOTT
12
16
10
14
12
10
-CCERS SAAPPAAT OVRPMBTW
OVRPMBLEW
PULD
--GEKS WPPOTM
CCDDS v. VSPHITW
DALD MUCL 3RETMETM
RE-T2 w/ NRTTMSTW
Re.D MuC. VITVUTM
10
Figure 6: The validation curves of CoEKS, RF-TE, and ReLD-MoEL trained without (w/o) VRPMBTW and with (w.) VRPMBTW on 50.
typically rely on gating mechanisms route data to experts, which may lead to load balancing issues. In contrast, CoEKS explicitly activates experts based on task constraints, ensuring more sta- ble and balanced expert utilization. 4) Scalability via plugging in new experts: New constraints can be handled by adding and fine-tuning a dedicated expert without modifying the rest of the trained model. as validated in Section 5 (Q2). This structural modularity offers practical advantages over MOR-based methods.
2025)
Parameter efficiency analysis compared with MoE. methods: When compared with MoE-based methods (e.g., ReLD-MoEL (Huang et al. (025)), CoEKS shares the same decoder architecture, but CoEKS's s encoder has one additional expert. To validate CoEKS's parameter efficiency, we use five experts in ReLD-MoEL-ES, resulting in total of 4.6 million parameters, which matches that of CoEKS. During training. CoEKS activates an average of two experts, aligning with ReLD-MoEL-ES that activates the top-2 experts. During inference, the number of activated experts in E5 is dynamically adjusted based on task constraints to maintain consistent parameter usage with CoEKS. The results are presented in Tables 7 and 8 Given the same total and activated parameters,

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Table 5: Performance on 1K test instances of 16 VRPs (the training set includes all 16 VRPs).
Method
Chij
Clpp
Time
Chhj
Cupp
Time
Citj.
Cupp
Time
Chhj
Glap
Time
HGS PYPP
10.372
10.4m
15.028
20.8m
HGS PVRP
16.031
10.4m
25,423
20.8ca
OR Toolsk
10,572
1.907%
10.4m
16.280
4.178%
20.8m
OR OR-Toolsh
16.089
0.347%
10.4m
25.814
1,506%
20.8c
POMO-MTL
10,520
1.429%
1s
15,910
L8445
7s
POMO-MTI
16.421
2.4328
1s
26.417
3,896%
7s
MVMOF
10.501
15.880
1.611%
94
MVMOI
16.397
2.787%
26 189
3,780%
10.509
1.33
1x
15.861
1.539%
74
RF-T
16.362
2.0601%
26.267
3.3044
74
CaDA
10.505
1.2811
36
15.855
1,899
1Is
CaIAA
16.291
1.611%
25
26.078
2.560%
12s
ReLD-MoEL MoËL
10.422
L06216
26
15.832
96
ReLD MoËl
MoBL
16.381
2.1711%
25
26.320
3,515%
9s
CoEKS
10.477
1017%
25
15,816
1.42%%
9
CoBKS
16.352
1.873%
26,209
3,0701
HGS-PyVRPA
6.507
HGS-PyVRPP
15,766
20
OR-Toolst
6,553
0.68645
10
9.995
2.792%
OR-Toolst
10.570
2.343%
10,4
16466
5,3025
20.8/8
OVRR
POMO-MTI.
6.716
1858%
Is
10.195
4.766%
64
POMO-MTI
10.774
1.7225L
1s
16.132
2. 3249
66
MVMOF
6,702
2,967%
26
10.164
4.4901%
94
MVMOF
10.749
1.4911%
2s
16.088
2,047
9%
RF-TE
6.687
2.731%
4.031%
RF-E
1.51449
16.057
CaDAA
6.684
2.679H6
26
10.116
3.98%
12s
CaDAA
10:745
1.465%6
16.055
1.768%6
11s
ReLD-MoEL MoEL
6.679
2.618%%
26
10.101
3.811%
95
MoEL
10.228
1.303%
25
16,032
1,655%
%
CoEKS
6.607
2.424%
2s
10.073
3.562%
8s
CoBKS
10.724
1.266%
28
16J023
1644%
9
HGS-PyVRPA
9.687
Am
14.377
20.
HGS-PyVRRP
10,510
10.4
16.926
20,8
OR-Toolst
9.802
1.1599
3.853%
OR-Toolst
17.027
0,5835
POMO
10:032
3.55616
15,054
4,725%
6s
10.673
1.52616
17,418
76
VI
MVMOE
10.008
3.298%
26
15.012
4.132%
85
MVMOE
10.671
1.511%
25
17,029
10s
RF-TE
9.986
3.0835
1s
14.934
1.8918
65
RF-TE
10.654
1.350%
1s
17:333
2.377%
76
CIDA
9.978
7.9879
2x
14.932
18759
11+
CIDA
10,622
1.41%%
24
17.330
1.972%
17x
ReI.D-MoEL.
2,8758L
14.921
ReI.D-MoHL.
10,658
1.3915
2.5939
10s
COEKS
9.948
2.674%
14.884
3.546%
96
COEKS
10,638
1.19258
17.313
2,26315
10s
HGPPPPYRPP
10.186
A0
14.779
20.
HGSPPyVKPP
18.361
10.4a
29,022
20.540
OR-Toolst
10.331
1.39016
10.4m
15:426
4.338%
20.8m
OR-Toolst
18,422
0.33216
10.4m
29,830
2.77016
20.8
POMO MTL
10.675
4.733%
1s
15.688
6.1033
7s
POMO MTL
19.001
2.186
30.934
3.740%
7s
MVMOE
10,622
4.309%
15,621
5.6433%
MVMOE
30,905
3,648%
10s
RE-TE
10.584
3,856%
16
15.515
4.500%
74
RF-TE
18.942
8898
1s
30.719
3.0266L
74
CaDA
10:569
3.708%
26
15.506
4.872%
116
CIDA
18858
1.325%
26
30,531
2.393%
136
ReLDDMMOLL
10.567
3.67518
26
15.408
4 4.828%
96
ReLD-MoBL
18,959
1.9669
2s
30,8001
3,299%
10s
COLKS
10.546
3,480%
15.466
4.621%
COBKS
18.93
1.728M6
30,680
2.8961
HRP-
HGGS-PyRRP
16.356
25.757
8cm
18.366
0.3836
4m
29.945
1.597%
OR-Tralsk
16.441
0.499%
10,4m
26,259
1,8095
8rm
POMO--MIL
18.647
1.915%
Is
30.447
3.3244%
76
POMD-MTIL
16.855
2886
Is
26.895
4.379%
76
MV MOME
18.637
1.86396
26
30.439
3. 3.292%6
10s
MVMOE
16,804
2.712%
25
26.858
4.2344
9%
RF-IE
18.604
1.68515
30,265
2.70216
75
RF-TE
16,751
2.38916
26.717
3.6901
76
CaDA
18,519
%
30,080
2.064%
12s
CaDA
1.964%
26.525
2.945%
1.s
ReLD-MoEL
18.611
1.725%
26
30.349
2.9865
10s
ReLD-MoEL
16.767
2.4968
28
26.768
3,8945
10s
CAFKS
18.567
489%
24
30.213
2,5238L
10s
CHIRSS
16.737
2.7988
26
26.673
3.5175
10+
VRP-
6.898
1m
10.335
20 8m
HGS-PyVRPA
6.899
10.4m
10.335
20.8ra 80
OR-Toelsa
6.928
0.412%
2.315%
OR-Toolst
6.927
0.3868
10,582
2,363RL
PUMO-MTL
7.106
2.989%
1s
10,852
4.933%
74
POMO-MTL
7.111
3.046%
10,65
5.081%
74
MVMOE
7,086
2.69618
26
10.825
4.707%
9s
MYMOE
7.096
2818%
25
10882
4.778%
9
TE
7,075
2.538%
Is
10.769
4.179%
65
RF-TE
7.079
2.580%
Is
10.772
4.207%
75
CaDA
7,062
2.341%
26
10,745
3.941%
12s
CaDA
7.064
2.362%
10.748
3.9629
Lls
ReI.D-MoEL
7.064
7.3755L
10.754
4.0345L
ReI.D-MoHL.
2.3548
10.750
3,990
CHEKS
7.048
2.139%
10.710
3.611%
94
CARKS
7.050
2.155%
10.711
3.616%
HGS PRRP
11.668
10.4m
19.156
20.8m
HGS PVPRP
11.669
10.4m
19:156
20.8ca
OR Toolsk
11.61
0.106%
10.4m
19,305
0,767%
20.8m
OR OR-Toolsk
11.682
0.109%
10.4m
19,303
0.757%
20.8cm
POMO-MTL
118825
1.304%
Is
19.635
1-482%
74
POMO-MTL
11.821
1.2938
Is
19:631
2.464FL
7s
MVMOF
11.115
1.244%
1.609%
10x
MVMOR
11.814
19.654
2.587%
10%
RF-TE
11.804
1.145%
1s
19,552
2.049%%
74
11.804
1.140%
19.551
2.0455
CaDA
11.767
0.332%
36
19.454
1.430%
136
CaDA
11.766
0.828%
36
19.435
1.32%
13s
ReLD-MoEL MoËL
118007
1.170%
26
19,585
2.22816
11s
ReLD MolLL
11,808
1.177%
25
19.586
2.2301%
11s
COEKS
11,789
1.024%
1.990%
10s
CORKS
1.980%
PYPP
6.507
10.4m
9.724
Py VRRM
10.510
10.4mh
Smn
OR Toolsk
6.552
0.668%
10
10.001
2.791%
2m
OR-Tools
10.497
0.114%
10.4m
17.023
0.728%
20.8mm
OVREL
POMO-MTI.
6.719
3.230%
1s
10.195
4.705%L
6s
POMO-MTIL
10.672
1.520%
1s
17.421
2.896%
76
MVMOF
6.701
2.949%
24
10.169
4.140%
94
MVMOF
10.673
1,5311
26
17.432
2.967%
10s
RE-E
6.685
2.701%
10.119
4.033%
RF-E
1.361%
2.370%
CaDA
6.684
2.679M6
26
10.118
4.007%
12s
CaDA
10.022
1.047%
17.230
1.768%
12s
ReLD-MoEL
6.627
2.588%
26
10.103
3.875%
96
ReLD McEL
10.661
1.411%
25
17.368
2.589%
10s
CoEKS
6.666
2.40%%
2s
10.073
3.568%
9s
CoBKS
10.639
1.204%
26
17.319
2.293%
10s
bold: Rest resultss among learning- hased methods.
underline: Sccond-best results among learnin-bbased methods. #. Results are adopted from Berto et al. (2244) for the convenience comparison.
CoEKS consistently outperforms ReLD-MoEL-ES across all tasks, in both ID and OOD scenarios. This underscores that COEKS achieves superior parameter efficiency through its model architecture.
E.2 T-SNE VISUALIZATION ANALYSIS
To gain insights into how the experts in CoEKS learn and specialize, we visualize their embedding tokens using t-distributed Stochastic Neighbor Embedding (t-SNH). We performed t-SNE analysis
7

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Published as a conference paper at ICLR 2026
Table 6: Fine-tuning performance all VRPs with MB.
Methed
VRPMB
DVRPMB
VRPMBL
VRPMBTW
OVRPMBL
ONRPMETW
VRRMBLTW
OVREMBLLT
Cool
Cool
Cxp
Clos
Clup
Crol
Gxp
Cool
Cap
Clad
Cup
Cool
Cost
HKIS-Py VIP
9.00
6.11
16.31
0.40
6:11
10:47
16.01
10:47
RE TE AL
11.76
29.66/s
9.44
54.52%
11.27
18.92%
18.46
15.59%
40.00%%
13.27
27.14%
16.00%
13.29
27.33%
RE-TEALL
9.36
2.08%
6.26
2.385
2.42%
10.66
1.73%
1.76%
BoLD-MMBL-AN
2.34%
2.73%
COLKS"
9.25
1.73%
6.21
1.08%
9.68
1.97%
16.35
2,07%
6.22
1.75%
10.63
1.44%
16.74
2.65%
10.64
1.50%
COEKS
9.34
1.6%%
6.30
1.50%
9.66
1.81%
16.35
2.07%
6.21
1.57%
10.63
1.4%%
16.73
2.56%
18,03
1.50%
Table 7: Parameter efficiency comparison. (In-distribution tasks (n2 50))
Capp
CVRP
OVRP
VRPB
VRPL
VRFTW
OVRPTW
OVRPL
(ID Avg.)
ReLD MoEL-ES
1.06%%
2.324%
2.509%
L1808
2.308%
1.68%%
2.465%
2.069%
CoLKS
0.891%
2.188
2.49%
1152%
2050%
1.39%
2.13%
1.751%
Table 8: Parameter efficiency comparison. Oout-of-distriuttin tasks (n 50))
Method'
OVRPB
VRPB1
VRPBTW
VRPETW
OVRPBL.
OVRPBTW
OVRPLTW
VRPBLTW
OVRPBLTW
(OOD Avg)
RoLL-MoEL-ES
3.026%
CoEKS
4.91%%
4.397%
3.210%
2.949%
4.747%
2.566
1.03%
3.714%
2.797%
3.432%
on the expert embeddings for the full-constraint task OVRPBLTW. For each expert each encoder layer, we sampled 5,000 embedding samples and projected them into a 2D space for visualization.
As illustrated in Figure 7 the t-SNE plots reveal significant overlap in the embeddings of all experts, except for the capacity expert (Ec), within the lower layers (layer 1). This indicates that lower layers capture shared, transferable representations. aligning with our design where knowledge sharing is applied at the lower layers. The capacity expert (Ec), which is active across all VRP tasks, learns in the model. As we move to deeper layers, the clusters become clearly separated, showing that experts gradually specialize and align with their assigned constraints.
universal representations, resulting in its distinct and stable representation early
We also visualize the expert embeddings for the variant without the knowledge sharing strategy in Figure [ In this case, experts' representations begin to separate even in the first layer, indicating early specialization and a lack of transferable knowledge. This confirms that our knowledge sharing strategy necessary for building meaningful cross-task representations.
E.3 SCALABILITY TO CONSTRAINT MULTI-DEPOTS
To further verify the scalability of CoEKS new constraints, introduced the Multi-Depot (MD) tasks, which is an extension of the single-depot tasks. Following the configuration of RouteFinder, we set the number of depots 3. total, we added 24 new VRP tasks, which include:
16 tasks that incorporate only the MD constraint.
8 tasks that combine the MD with the new MB constraint.
The experimental results are presented in Tables 10 Table and 12 where CoEKS continues to achieve the best performance under both few-shot fine-tuning and zero-shot generalization. These findings strongly support our claim that CoEKS is robustly scalable to diverse and previously unseen constraints.
E.4 CVRPLIB BENCHMARK
On the CVRPLIB benchmark dataset, we conduct additional experiments to evaluate the general- ization ability of the model real-world instances. Several representative cross-task methods are compared as a supplementary validation of OOD generalization performance. Each model is trained uniformly distributed instances with
100. Additionally, the original POMO model (Kwon (et al., 2020) trained on a single task also reported. The evaluation primarily focuses on large-scale datasets (n 500) in the classic Set-X (Uchoa et al. 2017), following the setup in MVMoE. (Zhou
8

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Published as a conference paper at ICLR 2026
Layer
Layer
Capett
Exnet
Expt Expet
Srcett
Dpett
Iasert
Exper IW
becent
20-
-109
D
Tasert
Eyec
Secert
Dpart 0
Layor
Layer
Expert
fazett
bact TW
troot TW
-25
-30
Figure t-SNE visualization of experts" latent representations across encoder layers.
Table 9: Zero-shot generalization performance (the gap to the best traditional solver) on 24 VRPs
with MD. These methods do not add new experts and are only allowed to activate experts corre- sponding to previously known constraints.
Gapp
MDCVRP
MDOVRP
MDOVRPB
MDOVRPBL
MDOVRPBLTW
MDOVRPBTW
MDOVRPL
MDOVRPLIW
RF-EAI.
37.514%
29.887%
37.9035
40.128%
38.314%
45.111%
29.9345
34 305%
RoLD-MoEL
39.359%
20.266/
13.501%
32,543/
CoEKS
38.174%
15.678%
22.690%
21.883%
21.920%
22.901%
15.616%
20.949
Method':Gapp Gapp
MDOVRPIW
MDVRPB
MDVRPBL
MDVRPBLTW
MEVRPBTW
MDVRPL
MOVRPLTW
MDVRETW
40.836%
55.689%
95.955
42.723%
46.764%
43,181%
42.167%
ReLD-MoEL
28.6341%
18.087%
47.975%
30.31
39.091%
39.211
30.257%
36.833%
CoEKS
21.730%
38.612%
38.188%
27.884%
31,994%
36.288%
26.986%
28.731%
Table 10: Zero-shot generalization performance (the gap to the best traditional solver) 8 VRPs with MB with MD. These methods do not add new experts and are only allowed to activate experts corresponding to previously known constraints.
Method Clap)
MBOVRPMBLIW
MIOVRPMEET
MDVREMBL
MOVRPMBLITW
MOVRPMBEW
MoLD CAEKS
47.741%
45.58116
26.22% 24,03%%
29,828%
64.000%
60.381%
31,7766
38.6469
35.186%
33,701
25.365%
53.428%
50.735%
36,000%
32,874%

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Published as a conference paper at ICLR 2026
Layer
Layer
Eapett
Exnet
Expet
Secott
Expet
Dpect
Sasert
Expe IW
Sxcert
-2
-00
-90
-90
-90
-20
Dipc
Tasert
Eyec
dxcent
Dpart 0
Dapert
-22
Layer
Expert
Sazert
Exet TW
Stcot TW
Figure 8: t-SNE visualization of 5 experts' latent representations across encoder layers (without
knowledge sharing strategy)
Table 11: Fine-tuning performance (the gap to the best traditional solver) on 24 VRPs with MD.
Gapp
MDCVRP
MDOVRPBL
MDOVRPBLIW
MDOVRPBTW
MDOVRPLIW
RF-EAL.
15.2515
14.250%
18.906%
19.479%
19.8419
19.315%
15.419%
17.97%%
ReED-MoEL
15.235%
13.525%
17.125%
16,9339
11,1059
11,2789
13.3339
10.758%
MDOVRPTW
MDVRFB
MDVRPBL
MDVRPBLTW
MDVRPSTW
MDVRPL
MOVRPLTW
MDVRETW
RF-EAL
17.300%
21.943%
23, 149%
25.549%
24.098%
15.537%
21.952%
20. 20.244%
20.009%
CoEKS
49788
12.093%
12.487%
9.532%
9.324%
9.109%
8.970%
8.422%
Table 12: Fine-tuning performance (the gap to the best traditional solver) on VRPs with MB with MD.
Gaps
MDOVRPMETW
MOVREMBLIW 150015 9.122%
18.378% 17.543%
ReLD MoEL
17.790% 6.99%
11430% 5.250%
11.206% 5.130%
23.672%
22.1019
14.417%
CoEKS
7.061%
12.128%
12032%
8.00%
et al., 2024a) and RouteFinder (Berto et al. 2024). CoEKS achieves the best OOD generaliza-
tion, surpassing the state-of-the-art Rel.D-Moll. (Huang al. 2025). The performance gains be-
10

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Published as a conference paper at ICLR 2026
come more pronounced as the problem size increases. Notably, the single-task training method (i.e., POMO) demonstrates limited generalization ability on diverse real-world benchmarks, potentially due overfitting the uniform training distribution. Conversely, cross-task training substantially enhances model generalization.
Table 13: Results on large-scale CVRPLIB instances. # Results are adopted from MVMoFF (Zouu Zhou et al. 2024a), with the model trained single task.
Set-X
POMO#
RF-TE
POMO-MTL
MVMcB
ReLD-MoEL
CoEKS
Instance
Obj.
Gap
Obj.
Gap
Obj.
Gap
Obj.
Gap
Obj.
Gup
Obj.
Gap
X-n502-339
75617
9.232%
72098
4.149%
84021
21.372%
81611
17.891%
73073
5.557%
73947
6.820%
X-n513-k21
30518
26. 102%
30330
25. 32.5
29022
19.921%
27368
13,086%
27063
11.826%
27124
12,078%
X-n524-k153
201877
30.586%
168473
8.978
173838
12.449%
174427
12,830%
12,832%
174325
12.764%
X-n536-A96
106073
11.837%
102520
7.880%
106851
12.657%
105167
10.882%
102548
8.121%
103497
9.121%
X-n548-A50
103093
18 908%
102078
17.737
102217
17.897%
107767
24. 24.299%
99121
14.326%
108115
18. 18.933%
X-n561-442
49370
15.575%
49632
16.188%
48553
13.662%
47759
11.803%
47022
10.078%
46838
9.647%
X-n573-30
83545
64.871%
55296
9.123%
60870
20.123%
66831
31,295%
57249
12.977%
$4699
7.945%
X-586-k1599
229687
20.79296
208397
9.501%
211421
11.089%
214247
12.574%
206793
8.658%
205612
8.037%
X-n599-192
150572
38.839%
117226
8.091%
122028
12.519%
126915
17.025%
116463
7.388
116547
7.465%
X-n613-X62
68451
14.976%
68066
14.329%
82141
37.971%
67964
14.124%
67272
12.996%
66050
10.943%
X-n627-k43
84434
35.825%
69046
11,071%
70923
14.090%
70572
13.526%
68141
9.615%
67571
8.698%
X-6611-335
75573
18.672%
73071
14.740%
72378
13.652%
70445
10.616%
8.913%
68650
7.798%
X-m655-1131
127211
19.1345
112355
5.221%
123144
15.325%
126352
18.329%
120650
12,989%
113905
6.73%
X-n670-k130
2080779
42 197%
167786
14.661%
167131
14.214%
168834
15.377%
169163
15.602%
167007
14.129%
X-m685-A75
79482
16.534%
77681
13.893%
99452
45.813%
78080
14.478%
78050)
14,493%
76402
12.018%
X-4701-144
19.433%
92541
10.205%
9.664%
7,275%
7.249%
Xa7766435
51381
18.463%
50333
16.047%
49420
13.942%
50218
15.782%
47981
10.624%
10.191
X-n733-k159
159098
16.823%
162059
18.997%
184714
35.633%
153087
12.409%
153884
12.995%
150508
10.516%
X-n749.988
87786
13.611%
85623
10.8124
88493
14,526%
86961
12.543%
86380
11.791%
84974
9.972%
X-n766-k71
135464
18.395%
132819
16.083%
127674
11.587%
129107
12.839%
126139
10.245%
125801
9.950%
X-4783-148
24.733%
864-15
19.422%
84220
16.318%
82163
13.507%
10.890%
79411
9.751
X-a801-140
124278
69.536%
921-19
25 66.69699
96138
31.5166%
88091
20.161
85315
16.374%
86417
17.959%
X-n819-1171
193451
22.3449%
187863
18.810%
188537
19.236%
187714
18.715%
175282
10.853%
173464
9.703%
X-n87--1422
237884
22.787%
209629
8.203%
218437
12,7495
223912
15.575%
210889
8.853%
208673
7.709%
X-8856495
182528
71.447%
99082
11.372%
157894
77.479%
175004
96.7909
100320
12.763%
98740
10.987 %
X-8876-159
119764
20. 6099
100566
10.339%
110488
11,268%
115516
16.331%
106631
7.384%
106684
7.437%
X33855-37
70245
30.421%
67995
26.244%
67527
25.375%
64649
20.032%
62172
15.433%
61740
14.631%
X-n96-4207
399372
21.324%
354011
7.544%%
382125
16.084%
372237
13.060%
355853
8.103%
352206
6.995%
X-996-k151
237625
79.049%
164931
24.275%
193030
45.447%
16064S
21.047%
160460
20.906%
158551
19.467%
X-89574887
130850
53. 104%
110516
29.311
108401
26.837%
127388
49.053%
101629
18.913%
103700
23.3366
X-a979-158
147687
24.15
133825
12.481%
134759
13.266%
132546
11.406%
129738
9.046%
129074
8.487%
X-n1001-k43
100399
38. 38.759%
92837
28, 28.30818
89098
23.140%
86107
19.006%
81081
12.060%
80458
11.199%
Avg. Gup
29.66%
14.931%
21.482%
19.252%
11.590%
10.832%
F THE USE OF LARGE LANGUAGE MODELS (LLMS)
In this research, we employed Large Language Models (LLMs) general-purposs tool to assist with writing polish. These LLMs were utilized to enhance textual clarity without contributing to research conception methodological development.
11


Reviews:
[{"review_id": "PD0PfvdGo8", "reviewer": "Reviewer_nRLf", "summary": "This paper addresses the problem of vehicle routing problem in a multi task setting.\nThis work modifies the Mixture of Experts (MoE) model into a Combination of Experts (CoE) framework, where experts are constraint-specific and are combined later. Knowledge sharing with MSE loss is applied between the experts. A combiner is introduced to regularize the model. Experiments are conducted to show the validitty of the method.", "questions": "Please see weaknesses", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The idea of combining experts is intuitive and well-aligned with the CVRP multi-task setting.\n2. The use of separate heads for each constraint is straightforward and easy to understand.", "weaknesses": "1. **Writing needs improvement:**  \n   The description of the combiner is presented in two separate sections (lines 203–211 and lines 263–275), with no clear connection between them. It is unclear whether these sections refer to the same concept or different ones. If they are the same, the repetition needs to be justified. If they are different, the components should be clearly defined.\n\n2. **MD loss for \\( K = 2 \\) vs \\( K > 2 \\):**  \n   The rationale behind the different MD loss for \\( K = 2 \\) and \\( K > 2 \\) is not provided. There is no explanation for this distinction.\n\n3. **Distillation loss (MD) adoption from (Xie et al., 2024): [1]**  \n   The MD loss is adopted from (Xie et al., 2024), but it is unclear whether this approach has been extensively studied in other works besides Xie et al. (2024). Regardless, the reasoning behind using distillation when both heads are learning simultaneously is not clear. It appears to be primarily a regularization technique, restricting the output heads to remain similar. Even with this interpretation, it is not clear why this regularization is combined with the Mixture of Experts model (or the Combination of Experts model, in this case), which involves a weighted sum of outputs. The main motivation behind MoE (or CoE) is to capture distinct modes across different network heads and combine them. This regularization approach would work against that motivation, as it encourages all experts to output similar embeddings.\n\n4. **Comparison with SHIELD:**  \n   A relevant work, **SHIELD**[2], presented at ICML, shares a similar motivation of generalizing across different tasks and distributions, but incorporates tasks as part of the input features. This work is neither compared nor discussed in the paper.\n\n## Overall:\nThe idea of combining different heads for different tasks is reasonable. However, the framework appears somewhat incremental, with the primary modification being the adaptation of the Mixture of Experts model to fit experts to different subtasks, along with the addition of regularization techniques.\n\n## References:\n1. Zhitian Xie, Yinger Zhang, Chenyi Zhuang, Qitao Shi, Zhining Liu, Jinjie Gu, and Guannan Zhang. Mode: A mixture-of-experts model with mutual distillation among the experts. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pp. 16067–16075, 2024\n1. Goh, Yong Liang, et al. \"SHIELD: Multi-task Multi-distribution Vehicle Routing Solver with Sparsity and Hierarchy.\" Forty-second International Conference on Machine Learning.", "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": "MRpWxMPqom", "reviewer": "Reviewer_zFcW", "summary": "This paper proposes CoEKS, a new method to encode problems in multi-task learning for VRPs. Instead of employing MoE in encoder side, the authors propose to assign an expert to each constraint. For problems with multiple constraints, all the respective experts of each constraint will be activated at the same time, and a sharing mechanism enhances performance. The model can also be adapted for new problems by adding a new expert for each constraint. CoEKS demonstrates SOTA results among neural methods.", "questions": "1. What if instead of experts we just had all MLPs work at the same time, or a larger MLP with the same number of parameters?\n    \n2. I am not too clear about the relevance of OOD generalization to “unseen” tasks. As the tasks in Table 2 are merely a combination of constraints, in practice, why not simply train a method that incorporates them already?\n    \n3. Why do you finetune only the added experts? Wouldn’t the performance improve if you finetune all of the models as well?\n    \n4. It would be interesting to see what the learned representations of experts are in the latent space. Do they align with the specified constraints, or do experts learn generic representations? It would be interesting to see more about this in say a t-SNE analysis similar to POMO-MTL or RF", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper is well-written, easy to follow and with good motivation\n    \n2. CoEKS is novel to my knowledge and tackles the problem in an interesting manner\n    \n3. Good performance", "weaknesses": "1. My major concern is the following: given that we have multiple experts, why not simply train on all of them at the same time and always activate them? In other words, why bother with specialization when one can have multiple MLPs at the same time (or a larger MLP with the same number of parameters with the sum of experts)? I believe the bump in performance might be simply due to having more parameters\n    \n2. The method will need to activate all the parameters for problems, including all of the constraints, which is however acknowledged by the authors\n    \n3. Overall, the work is a bit incremental, in the sense that it just adds a specific mechanism for the encoder layers with a relatively small improvement in performance\n    \n4. Code is not provided", "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": "JkV3YfWtZy", "reviewer": "Reviewer_dA3D", "summary": "A neural \"cross-task\" vehicle routing problem (VRP) solver is presented. The main insight is that model components are introduced for each constraint type, e.g., time windows or backhauls. These are called experts, and the authors define a strategy for combining their outputs to deal with problems with combinations of multiple outputs. The approach is compared against other multi-task approaches for the VRP on a part of a standard benchmark of VRP tasks. It outperforms the current state-of-the-art approaches for this multi-task setting.", "questions": "1. RouteFinder has 48 problem variants, this paper only has 24. What happened to the other variants and why are they not used?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The idea is generally simple and well-founded, I think it basically makes sense even if it has scaling issues in the number of constraints supported.\n2. The performance is good, but I do have a very important question regarding the benchmark (see below)\n3. The paper is generally easy to understand.", "weaknesses": "1. The experimental section is written very short, and important details are pushed to the appendix multiple times. This is unfortunate, as the paper wastes space on page 6 providing what I can only describe as a marketing rehash of the introduction. That entire paragraph ought to go towards describing the results. Perhaps the authors can address this.\n2. The results are only incrementally better than previous work; this paper is a small step forward, but certainly not a breakthrough.\n3. I do not think the idea is one that will scale in general; the concept of adding a new expert for each type of constraint will eventually just overload the model with parameters that are likely unnecessary. The approach reminds me of the Poppy approach from Grinsztajn et al. (2023) in NeurIPS in which multiple decoders are used. The extra decoders in Poppy were shown in the Polynet paper (Hottung et al. (2025) in ICLR) to not be necessary, a vector input can be used instead. The same likely holds here, so I feel like this paper is not quite finished.\n\nSome other comments that the authors should consider addressing:\n1. The contributions in the introduction are vague, it would be better to be more precise about what is offered in this work.\n2. The first paragraph of the related work has some grammar issues, let an LLM fix it.\n3. The research questions offered in the experimental section are well done.\n4. I would like to see more about the ablation and sensitivity analysis in the main text.", "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": "c6Of8EBw2p", "reviewer": "Reviewer_qhjy", "summary": "This paper proposes CoEKS, a novel neural routing solver for cross-task VRPs. The core idea is to leverage the structural property that VRP variants are defined by combinations of basic constraints. The model consists of two main components: 1) a Combination-of-Experts (CoE) architecture, where individual constraint-specific FFNs are activated via a prior assignment based on known constraints, and 2) a multi-view knowledge sharing strategy to enhance the learning of transferable knowledge. The authors claim this method achieves state-of-the-art performance on ID tasks and demonstrates superior generalization on OOD tasks, including unseen constraint combinations and rapid adaptation to new constraints by plugging in new experts.", "questions": "1. It would greatly enhance the paper's contribution to evaluate the zero-shot generalization capabilities on unseen constraints(e.g., Mixed Backhauls and Multi-Depot) without applying additional expert networks and fine-tuning operations.\n\n2. For the scalability for adaptation to unseen constraints, the authors conducted fine-tuning experiments on unseen constraints with 50-node VRP variants. I am curious about how the performance of CoEKS would compare with existing methods if the problem size increases to 100?\n\nReferences:\n\n[1] Huang, Z., Zhou, J., Cao, Z., & XU, Y. Rethinking Light Decoder-based Solvers for Vehicle Routing Problems. ICLR, 2025.\n\n[2] Zhou, J., Cao, Z., Wu, Y., Song, W., Ma, Y., Zhang, J., & Xu, C. MVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts. ICML, 2024.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The paper is well-written and easy to follow.\n\n2. The experimental design is rigorous and comprehensive, the inclusion of strong SOTA baselines provides a high-quality empirical evaluation.\n\n3. The implementation on both POMO and ReLD backbone networks demonstrates the universality of the CoEKS architecture.", "weaknesses": "My main concerns stem from the model's core assumption, CoEKS must rely on **prior assignment of constraint-specific experts**, which limits its technical novelty and the general applicability of its claims. \n\n1. The contribution is relatively incremental. The model essentially replaces a learned gating mechanism in standard MoE-based multi-task solvers (ReLD-MoEL[1], MVMoE[2]) with a handcrafted external gating mechanism. While the idea of constraint-specific experts is effective, the implementation offloads the most complex gating selection logic to the user (or what we might call the real expert with domain knowledge). \n\n2. For the scalability for adaptation to unseen constraints, Appendix C.2 mentions the spatio-temporal conflict between MB and TW. Although the model's overall performance is promising, the process requires the inclusion of a TW  constraint to mitigate performance degradation.  I also notice that CoEKS is less effective than ReLD[1] on some tasks when only the MB constraint is applied.\n\n3. The claim of robust scalability to new constraints is supported by a weak experiment. Mixed Backhauls (MB) is merely a relaxation of the Backhaul constraint, which is already part of the training set. Both constraints deal with the same problem type (linehaul and backhaul demands). This is an easy test case and does not sufficiently support the claim that the method can adapt to any new constraint.\n\n4. The authors acknowledge that handling more constraints increases parameters. This one-to-one mapping of constraints-to-experts seems inefficient.", "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)