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REVIEW 3 major objections 5 minor 31 references

SHIELD: Multi-task Multi-distribution Vehicle Routing Solver with Sparsity and Hierarchy

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read SHIELD claims a single neural solver can generalize across 16 VRP variants and nine real-world maps by learning when to spend computation and how to group customers.

desk verdict A solid empirical extension of multi-task VRP to realistic map distributions; the main issue is an underspecified data-generation step that needs clarification, not a fatal flaw. read the letter →

arxiv 2506.08424 v2 pith:C4IOAXPT submitted 2025-06-10 cs.AI

Yong Liang Goh , Zhiguang Cao , Yining Ma , Jianan Zhou , Mohammed Haroon Dupty , Wee Sun Lee This is my paper
classification cs.AI
keywords vehicleroutingproblemmulti-taskmulti-distributionVRPneuralcombinatorialoptimizationmixture-of-depthssoftclusteringout-of-distributiongeneralizationreinforcementlearningfoundationmodels
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to show that a single neural network can solve many vehicle routing variants across many real-world customer distributions at once, and that two architectural principles, sparsity and hierarchy, are what make that generalization work. It introduces the Multi-Task Multi-Distribution VRP (MTMDVRP) setting, where instances are generated by sampling subsets of customers from fixed country maps under multiple route constraints. The proposed SHIELD solver adds token-level sparsity to a deeper decoder, so each decoder layer processes only a fraction of candidate nodes, and a context-based soft-clustering layer that summarizes unvisited customers into a handful of task-aware centroids. On 9 maps and 16 VRP variants, SHIELD reports lower optimality gaps than existing unified solvers on both seen and unseen tasks, seen and unseen maps, and transfers zero-shot to 200-node instances and CVRPLib benchmarks. If the result holds, a logistics firm could train one model on a few cities and tasks and deploy it on new geographies and problem types without retraining.

What carries the argument

The load-bearing mechanisms are two. First, Mixture-of-Depths (MoD) in the autoregressive decoder: each decoder layer's router scores every candidate node and only the top $\beta$ percentile of tokens (here 10%) is processed as query through the attention layer, while the rest skip via a residual connection, so the model dynamically decides how much computation each decision gets. Second, context-based soft clustering: an EM-inspired layer iteratively computes attention weights between node embeddings and five learnable cluster centers, with the cluster centers concatenated with a task-constraint prompt vector, then feeds the resulting centroids into the decoder and updates them by subtracting visited nodes. A Mixture-of-Experts encoder supplies task-adaptive node embeddings. These pieces together are what the paper argues create compact, shared representations and better out-of-task and out-of-distribution generalization.

What would settle it

Train SHIELD with the MoD router replaced by random token dropout at the same 10% budget, all else equal; if out-of-distribution gaps match SHIELD's, then the learned sparsity selection is not what drives the reported generalization.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is that learned sparsity and learned hierarchy, combined in one decoder, are the right inductive biases for a unified VRP solver. SHIELD trains a single policy with a Mixture-of-Experts encoder, a three-layer Mixture-of-Depths decoder that routes only the top 10% of query tokens through each layer, and a soft-clustering layer with five cluster centers that runs five expectation-maximization-style iterations using task-constraint prompts. The experiments compare against POMO-MTVRP, MVMoE, MVMoE-Light, and a deeper MVMoE-Deeper, and report that SHIELD achieves the best average optimality gaps in every aggregate block, in-task and out-task, in-distribution and out-distribution, for both N=50 and N=100, with zero-shot gains at N=200 and on CVRPLib Set-X. The authors interpret the compute savings from Mixture-of-Depths as a regularizer in the VC-dimension sense: less computation and fewer parameters push the model toward representations shared across tasks and distributions instead of overfit ones.

Load-bearing premise

The central argument assumes that sampling a subset of customers from a fixed map preserves that map's geographical structure; if the subset sampling instead changes the effective distribution, then the maps labeled out-of-distribution are not truly novel and the reported transfer gains are overstated.

Editorial extensions

If this is right

  • A single SHIELD model trained on six tasks and three country maps can be applied zero-shot to ten unseen tasks and six unseen maps, with the reported gaps still below all compared baselines.
  • Because MoD skips most tokens, a three-layer decoder is computationally feasible at N=100 where the dense deeper baseline runs out of memory, so depth is no longer the bottleneck for larger instances.
  • The sparsity level acts as a regularization dial: raising the token budget from 10% to 40% improves in-distribution fit but degrades out-of-task and out-of-distribution gaps, so compute can be traded against transfer.
  • Keeping the number of cluster centers small, five instead of 10 or 20, preserves the generalization benefit, meaning the hierarchy mechanism works by abstraction rather than by adding capacity.
  • SHIELD also retains its advantage when reduced to a single task, CVRP, across distributions and when trained on the uniform MTVRP setting, so the design is not specific to the multi-task multi-distribution scenario.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the subset-sampling procedure preserves map geography, then the model's success implies a transferable notion of map motif: countries with similar density patterns should share router and cluster behavior; the paper's layer-usage heatmaps already hint that two similar maps behave alike, so one could test this by clustering maps by their learned layer-usage signatures.
  • The MoD router scores could serve as an interpretable difficulty map of which customers require deeper reasoning, which the paper does not exploit; a planner could use those scores to flag unusual nodes or allocate human review.
  • A natural stress test the paper does not run is temporal drift within one map: if a city's layout changes over years, the fixed-map training distribution becomes stale, and online adaptation of the cluster centers would be needed for SHIELD to remain reliable.
  • The VC-dimension argument is invoked as motivation rather than proof; the practical claim that compute budget regularizes generalization would be strengthened if the paper provided a formal link between MoD's token dropout and a known complexity measure.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a new problem setting, Multi-Task Multi-Distribution VRP (MTMDVRP), in which VRP instances are generated from real-world country maps rather than from a uniform distribution, and introduces SHIELD, a neural constructive solver that combines a Mixture-of-Experts encoder, a Mixture-of-Depths decoder with token-level sparsity, and a context-prompted soft clustering layer. The model is trained with REINFORCE on six VRP tasks from three country maps and evaluated on ten unseen tasks, six unseen maps, N=50/100, with additional zero-shot experiments on N=200 and CVRPLib. The paper claims state-of-the-art optimality gaps against POMO-MTVRP, MVMoE, MVMoE-Light, and MVMoE-Deeper in all four in-task/out-task by in-distribution/out-distribution groups, and attributes the gains to sparsity and hierarchy acting as regularizers.

Significance. If the empirical claims hold, the paper makes a useful contribution to the emerging foundation-model direction for neural routing: it broadens the MTVRP benchmark to realistic distributions and provides a concrete architectural recipe (MoD sparsity plus contextual clustering) that appears to improve both efficiency and zero-shot generalization. The experimental effort is substantial, covering 9 maps, 16 VRP variants, two training sizes, ablations of MoD capacity and cluster count, matched-epoch comparisons, a uniform-only training ablation, and size generalization to CVRPLib. The earlier-checkpoint comparison and the uniform-only ablation are valuable controls that partially address the concern that the gains come only from added capacity. The main weakness is that the data-generation protocol underlying the out-distribution claim is never defined, and the reported numbers are single-run point estimates without a validation protocol or variance information.

major comments (3)
  1. [Section 3.1, Appendix B] The load-bearing data-generation step for MTMDVRP is not specified. The text says only that instances are generated by "selecting smaller subsets of V from the fixed set of M locations such that the geographical characteristics of M are retained," and Appendix B adds only that coordinates are not drawn uniformly. If the subset is drawn i.i.d. from the M locations, the resulting N=50/100 point sets are sparse samples whose empirical density is dominated by sampling noise, so the claim that country geography is retained is not correct; if the subset is constructed to preserve density (e.g., via stratified sampling or cluster representatives), then the generator itself injects clustered structure, which is exactly the structure SHIELD's clustering layer is designed to exploit. The two readings have opposite implications for interpreting the out-distribution results in Table 1 and the size-generalization results in Tables 8-10. Please provide the exact sampling algorithm, including pseudo-code, seed, and (ideally) released code, and discuss which of these regimes the benchmark actually implements.
  2. [Section 4, Tables 2 and 3] The central claim that sparsity and hierarchy improve generalization is supported only by single-run point estimates, and the main hyperparameters (MoD capacity beta=10%, number of clusters Nc=5) appear to be selected on the MTMDVRP50 test set without a described validation split. Since the ablations in Tables 2 and 3 are evaluated on the same MTMDVRP50 setup used for model selection, the reported optimality gaps may be optimistic, and the conclusion that "sparsity is crucial" would be more convincing with a held-out validation protocol and with mean and standard deviation over at least three random seeds for the main comparisons and the ablations.
  3. [Table 1 and Tables 14-22] Several reported optimality gaps against OR-tools are negative (e.g., VRPL and VRPTW rows in Tables 14-22), which indicates that the OR-tools configuration used as the reference solver is not optimal on these instances. This does not invalidate the neural-solver-to-neural-solver comparisons, which are the primary claim, but the absolute gap numbers should be interpreted as gaps to a heuristic reference, not to an optimum. Please state this clearly when reporting the aggregate gaps or consider tightening the OR-tools configuration for the affected variants.
minor comments (5)
  1. [Equation (4)] There is a typesetting error in Equation (4): the text "rl ifi" appears inside the first case of the MoD update; this should presumably be a router score multiplied by the layer function, and should be rewritten for clarity.
  2. [Algorithm 1] In Algorithm 1, the residual connection "C_OUT = C_hat + C" is ambiguous because C is overwritten in the same loop iteration; please rename the quantities (e.g., C_old and C_new) so that the residual update, normalization, and the B-iteration loop are unambiguous.
  3. [Section 3.2] The VC-dimension theorem from Goldberg and Jerrum is used only as informal motivation, and the paper itself notes it is not strictly applicable to networks with exponential functions; this is acceptable, but the paragraph should be framed explicitly as heuristic rather than as a theoretical justification for the architecture.
  4. [Appendix M] In Table 8, one runtime entry for MVMoE is written as "45.23" without the "s" unit, and several tables (e.g., Tables 16-22) contain truncated model names such as "MVMoE-L"; these should be corrected for consistency.
  5. [General] The paper does not state whether code, trained models, or the MTMDVRP dataset will be released; given that the benchmark construction is central to the contribution, a release statement or a detailed data-generation appendix would substantially improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: SHIELD's generalization claims rest on held-out maps/tasks, external baselines, and ablations; the only flagged item is an under-specified data-generation step, which is a reporting gap rather than a circular reduction.

full rationale

I walked the paper's claimed derivation chain: the MTMDVRP setup, the VC-dimension motivation, the MoD sparsity mechanism, the context-based clustering layer, and the empirical comparisons. No step reduces to its own inputs by construction. (1) The central claim is empirical: SHIELD is trained on 6 tasks and 3 distributions and evaluated on 10 unseen task combinations and 6 unseen country maps, plus CVRPLib instances, with optimality gaps measured against HGS and OR-tools. No test-set quantity is used to fit a parameter that is then reported as a prediction; the SHIELD-400Ep and SHIELD-600Ep checkpoints are earlier training checkpoints used as control points, not refits. (2) The architecture choices (MoD routing with 10% capacity, Nc=5 clusters) are design choices tested by ablations in Tables 2 and 3 and by the SHIELD-MoD variant; they are not fitted to the held-out gap. (3) The clustering mechanism is motivated by a self-citation, 'Goh et al. (2024) first showed that one can apply a form of soft-clustering to summarize the set of unvisited cities into a handful of representations' (Sec. 3.4), but the paper supplies its own equations and algorithm for the clustering (Eqs. 5-8, Algorithm 1) and validates it internally via SHIELD versus SHIELD-MoD, so the self-citation is not load-bearing. (4) The VC-dimension discussion cites an external theorem (Goldberg & Jerrum, 1993) and explicitly notes the theorem is not strictly applicable, so it is used only as motivation, not as a forced derivation. The one legitimate concern is not circularity: Section 3.1 says instances are generated 'by selecting smaller subsets of V from the fixed set of M locations such that the geographical characteristics of M are retained,' without specifying the subsampling algorithm. This under-specification affects reproducibility and the interpretation of out-of-distribution generalization, but it does not make any reported result an equation-level restatement of its inputs. I therefore find no significant circularity.

Assumptions & free parameters 4 free parameters · 4 assumptions · 2 invented entities

The central claims rest on model hyperparameters chosen via ablations and on domain assumptions about map sampling and solver references. No new physical entities are posited; the invented entities listed are internal architectural components.

free parameters (4)
  • MoD capacity beta = 10%
    Selected via ablation on MTMDVRP50 (Table 2); choosing 10% maximizes out-distribution/out-task generalization, while 20-40% improve in-task but hurt generalization.
  • Number of cluster centers Nc = 5
    Selected via ablation (Table 3); Nc=10 and Nc=20 reduce out-task generalization, especially Nc=20 on out-task.
  • Number of clustering iterations B = 5
    Set by hand (Appendix I); the paper does not ablate B, but this parameter controls the clustering layer depth and is load-bearing for the hierarchy mechanism.
  • Number of MoD decoder layers = 3
    Chosen to match MVMoE-Deeper capacity while remaining trainable on N=100; the depth affects the sparsity-regularization tradeoff.
assumptions (4)
  • standard math VC-dimension theorem of Goldberg and Jerrum (1993): VC-dim(C_{k,n}) = O(kt)
    Invoked in Section 3.2 to motivate that limiting compute regularizes neural nets; the paper acknowledges it does not directly apply to networks with exponentials.
  • domain assumption Sampling V points from a map's M locations preserves the map's spatial distribution
    Section 3.1 states subsets are selected 'such that the geographical characteristics of M are retained'; no proof or algorithm specificity.
  • domain assumption HGS and OR-tools provide reference solutions accurate enough for optimality gaps
    Section 4 uses these solvers as ground truth; for several variants and out-distribution maps, neural solvers report negative gaps, meaning OR-tools may be suboptimal.
  • standard math REINFORCE with POMO-style augmentation and shared baseline is a valid training method
    Section 2/4: training follows Liu et al. (2024) and Zhou et al. (2024); standard for the field.
invented entities (2)
  • Context prompt vectors alpha_k
    purpose: Inject task constraint information into the clustering layer via a learnable map from one-hot constraint vector to latent prompt.
    Introduced in Section 3.4; no external falsifiable prediction, they are internal model parameters.
  • Soft cluster centers C
    purpose: Summarize unvisited nodes into Nc latent representations to guide decoding.
    Introduced in Section 3.4; learnable parameters, no independent empirical evidence outside the paper's experiments.

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Cite this review

Pith. "Pith review of SHIELD: Multi-task Multi-distribution Vehicle Routing Solver with Sparsity and Hierarchy." pith.science (2026). https://pith.science/paper/C4IOAXPT

@misc{pith2026250608424,
  author       = {Pith},
  title        = {Pith review of: SHIELD: Multi-task Multi-distribution Vehicle Routing Solver with Sparsity and Hierarchy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C4IOAXPT}},
  note         = {Machine review of arXiv:2506.08424}
}
read the original abstract

Recent advances toward foundation models for routing problems have shown great potential of a unified deep model for various VRP variants. However, they overlook the complex real-world customer distributions. In this work, we advance the Multi-Task VRP (MTVRP) setting to the more realistic yet challenging Multi-Task Multi-Distribution VRP (MTMDVRP) setting, and introduce SHIELD, a novel model that leverages both sparsity and hierarchy principles. Building on a deeper decoder architecture, we first incorporate the Mixture-of-Depths (MoD) technique to enforce sparsity. This improves both efficiency and generalization by allowing the model to dynamically select nodes to use or skip each decoder layer, providing the needed capacity to adaptively allocate computation for learning the task/distribution specific and shared representations. We also develop a context-based clustering layer that exploits the presence of hierarchical structures in the problems to produce better local representations. These two designs inductively bias the network to identify key features that are common across tasks and distributions, leading to significantly improved generalization on unseen ones. Our empirical results demonstrate the superiority of our approach over existing methods on 9 real-world maps with 16 VRP variants each.

Figures

Figures reproduced from arXiv: 2506.08424 by the authors.

Figure 1
Figure 1. Overall proposed approach for MTMDVRP. First, in-distribution maps are sampled uniformly, and a set of points is sampled from the map. Next, a task is sampled uniformly from the in-task set. These form a batch of problem instances and are passed to the network. SHIELD encompasses an MoE encoder, a context-based clustering layer, and the MoD decoder. The decoder is applied autoregressively to in-task/out-task out-dis… view at source ↗
Figure 2
Figure 2. Token is routed differently for each agent depending on the router. efficient generalization across tasks and distributions. 3.3. Going deeper but sparser Our proposed architecture is shown in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Left two panels: Plot of routes for OVRPBTW between SHIELD (left) and MVMoE (middle). Points denoted with a star are the top few points that SHIELD identified for more processing. Note that the initial routes from the depot are masked away for a better view. Right panel: Average number of layers used as the demand is being met for CVRP. fect, which confuses the decoder, resulting in poor decision￾making. Only attend… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Plot of all 9 World Maps and their points We utilize the following 9 country maps2 shown in [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Plot of layer usage for CVRP samples across three maps, with the x-axis as node IDs, y-axis as layer numbers, and values as average usage frequency during decoding. We conduct further analysis on the simpler CVRP to examine how the model generalizes across tasks and di…
Figure 6
Figure 6. Figure 6: Overall performance of SHIELD with varying sampling widths. For NCO solvers, we can allocate more test time to perform sampling and find better solutions during inference. In this experiment, we reduced the number of test instances to 100 instances per problem and perf…

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Works this paper leans on

31 extracted references · 19 canonical work pages

  1. [1]

    L., Almeida, D., Altenschmidt, J., Altman, S., Anadkat, S., et al

    Achiam, J., Adler, S., Agarwal, S., Ahmad, L., Akkaya, I., Aleman, F. L., Almeida, D., Altenschmidt, J., Altman, S., Anadkat, S., et al. Gpt-4 technical report.arXiv preprint arXiv:2303.08774,

    arXiv

  2. [4]

    Goal: A generalist combinatorial optimization agent learner.arXiv preprint arXiv:2406.15079,

    Drakulic, D., Michel, S., and Andreoli, J.-M. Goal: A generalist combinatorial optimization agent learner.arXiv preprint arXiv:2406.15079,

    arXiv

  3. [6]

    INViT struggles with the complexity of the MTMDVRP compared to SHIELD despite using some form of sparse attention

    Performance of INViT and SHIELD on the MTMDVRP50 and MTMDVRP100 scenarios. INViT struggles with the complexity of the MTMDVRP compared to SHIELD despite using some form of sparse attention. MTMDVRP50 MTMDVRP100 In-dist Out-dist In-dist Out-dist Model Obj Gap Time Obj Gap Time Obj Gap Time Obj Gap Time In-task INViT 6.4082 9.1437% 66.48s 6.7462 9.0992% 66....

    work page

  4. [7]

    Ant colony sampling with gflownets for combinatorial optimization.arXiv preprint arXiv:2403.07041,

    Kim, M., Choi, S., Son, J., Kim, H., Park, J., and Bengio, Y . Ant colony sampling with gflownets for combinatorial optimization.arXiv preprint arXiv:2403.07041,

    arXiv

  5. [8]

    SHIELD is the superior model even when tested on problem sizes larger than those it was trained on

    Performance of trained MTMDVRP100 models on MTMDVRP200. SHIELD is the superior model even when tested on problem sizes larger than those it was trained on. MTMDVRP200 In-dist Out-dist Model Obj Gap Time Obj Gap Time In-task Solver 13.7525 - 943.23s 14.8228 - 921.81s POMO-MTVRP 14.5695 5.4613% 19.80s 15.9036 7.0430% 20.01s MVMoE 14.6137 5.8753% 44.25s 15.9...

    work page

  6. [10]

    Neural combinatorial optimization for real-world routing

    Son, J., Zhao, Z., Berto, F., Hua, C., Kwon, C., and Park, J. Neural combinatorial optimization for real-world routing. arXiv preprint arXiv:2503.16159,

    arXiv

  7. [11]

    Sequence to sequence learning with neural networks.arXiv preprint arXiv:1409.3215,

    Sutskever, I. Sequence to sequence learning with neural networks.arXiv preprint arXiv:1409.3215,

    arXiv

  8. [14]

    Related Work Generalization Study.Joshi et al

    12 SHIELD: Multi-task Multi-distribution V ehicle Routing Solver with Sparsity and Hierarchy A. Related Work Generalization Study.Joshi et al. (2021) highlighted the generalization challenge faced by neural solvers, where their performance drops significantly on out-of-distribution (OOD) instances. Numerous studies have sought to improve generalization pe...

    work page 2021

Show all 31 references
  1. [15]

    However, their methods are tailored to specific settings and cannot handle our MTMDVRP setup, which considers crossing tasks and realistic customer distributions

    settings. However, their methods are tailored to specific settings and cannot handle our MTMDVRP setup, which considers crossing tasks and realistic customer distributions. While a recent work Goh et al. (2024) explores more realistic TSPs, their approach struggles with comple...

    2024

  2. [16]

    Zhou et al

    explored the training of a Multi-Task VRP solver across a range of VRP variants sharing a set of common features indicating the presence or absence of specific constraints. Zhou et al. (2024) enhanced the model architecture by introducing Mixture-of-Experts within the transfor...

    2024

  3. [17]

    Alternatively, Berto et al

    investigate this promising direction, but with different problem settings. Alternatively, Berto et al. (2024) improved convergence robustness by training on all tasks within a batch using a mixed environment. Single-task VRP Solver .Most research focuses on developing single-t...

    2024

  4. [19]

    improved upon AM by considering the symmetry property of VRP solutions. More recently, a wave of studies has focused on further boosting either the performance (Kim et al., 2022; Drakulic et al., 2023; Chalumeau et al., 2023; Grinsztajn et al., 2023; Luo et al., 2023; Hottung et al.,

    2022

  5. [20]

    or versatility (Kwon et al., 2021; Berto et al., 2023; Son et al.,

    2021

  6. [22]

    construct matrices, such as heatmaps representing the probability of each edge being part of the optimal solution, to solve VRPs through complex post-hoc search. In contrast,improvement solvers(Chen & Tian, 2019; Lu et al., 2020; Hottung & Tierney, 2020; Costa et al., 2020; Wu...

    2019

  7. [23]

    While constructive solvers can efficiently achieve desirable performance, improvement solvers have the potential to find near-optimal solutions given a longer time

    typically learn more efficient and effective search components, often within the framework of classic heuristics or meta-heuristics, to iteratively refine an initial solution. While constructive solvers can efficiently achieve desirable performance, improvement solvers have th...

    2021

  8. [24]

    For those interested, we refer readers to Bogyrbayeva et al

    of neural VRP solvers, which are less related to our work. For those interested, we refer readers to Bogyrbayeva et al. (2024). Apart from such single-task VRP solvers, there are alternative approaches to complex routing problems, such as the PDP, where travel times change ove...

    2024

  9. [25]

    To ensure feasible solutions, we ensure that all starting points are linehauls only unless all remaining nodes are backhauls

    whereby routes can have a mix of linehauls and backhauls without any strict precedence. To ensure feasible solutions, we ensure that all starting points are linehauls only unless all remaining nodes are backhauls. Duration limit (L):The duration limit is fixed such that the ma...

    2021

  10. [27]

    for CVRP and VRPTW instances, and Google’s OR-tools routing solver (Furnon & Perron) for the rest. For HGS, we use the default hyperparameters, while for OR-tools, we apply parallel cheapest insertion as the initial solution strategy and guided local search as the local search...

    2020

  11. [30]

    We retain a similar layout to Table 1 but all distributions are considered out-of-distribution in this case

    Performance of all models when trained on only Uniform data. We retain a similar layout to Table 1 but all distributions are considered out-of-distribution in this case. MTMDVRP50 MTMDVRP100 Model In-dist Out-dist In-dist Out-dist Obj Gap Obj Gap Obj Gap Obj Gap POMO-MTVRP (Un...

  12. [31]

    This suggests that while we can allocate more test time for inference, its effectiveness eventually saturates

    As shown, as we increase the sampling width, the general performance of the model increases (lower gap is better) in a logarithmic fashion. This suggests that while we can allocate more test time for inference, its effectiveness eventually saturates. 24 SHIELD: Multi-task Mult...

  13. [2014]

    Llama 2: Open foundation and fine- tuned chat models.arXiv preprint arXiv:2307.09288,

    Touvron, H., Martin, L., Stone, K., Albert, P., Almahairi, A., Babaei, Y ., Bashlykov, N., Batra, S., Bhargava, P., Bhosale, S., et al. Llama 2: Open foundation and fine- tuned chat models.arXiv preprint arXiv:2307.09288,

    arXiv

  14. [2015]

    and Yu, T

    Wang, C. and Yu, T. Efficient training of multi-task com- binarotial neural solver with multi-armed bandits.arXiv preprint arXiv:2305.06361,

    arXiv

  15. [2017]

    The overall feasibility of solution can be managed by the use of masking, whereby invalid moves are masked away during the construction process

    is popularly used, whereby attention scores govern the decision-making process in an autoregressive fashion. The overall feasibility of solution can be managed by the use of masking, whereby invalid moves are masked away during the construction process. Classically, neural con...

    2014

  16. [2018]

    A major breakthrough in AR-based methods came with the Attention Model (AM) (Kool et al., 2018), which became a foundational approach for solving VRPs

    in an autoregressive (AR) way. A major breakthrough in AR-based methods came with the Attention Model (AM) (Kool et al., 2018), which became a foundational approach for solving VRPs. The policy optimization with multiple optima (POMO) (Kwon et al.,

    2018

  17. [2019]

    C., and Santoro, A

    11 SHIELD: Multi-task Multi-distribution V ehicle Routing Solver with Sparsity and Hierarchy Raposo, D., Ritter, S., Richards, B., Lillicrap, T., Humphreys, P. C., and Santoro, A. Mixture-of-depths: Dynamically allocating compute in transformer-based lan- guage models.arXiv pr...

    arXiv

  18. [2020]

    PolyNet: Learn- ing diverse solution strategies for neural combinatorial optimization.arXiv preprint arXiv:2402.14048,

    Hottung, A., Mahajan, M., and Tierney, K. PolyNet: Learn- ing diverse solution strategies for neural combinatorial optimization.arXiv preprint arXiv:2402.14048,

  19. [2021]

    Berto, F., Hua, C., Park, J., Luttmann, L., Ma, Y ., Bu, F., Wang, J., Ye, H., Kim, M., Choi, S., Zepeda, N. G., Hottung, A., Zhou, J., Bi, J., Hu, Y ., Liu, F., Kim, H., 9 SHIELD: Multi-task Multi-distribution V ehicle Routing Solver with Sparsity and Hierarchy Son, J., Kim, ...

    arXiv

  20. [2022]

    K., Laurent, T., and Bresson, X

    Joshi, C. K., Laurent, T., and Bresson, X. An efficient graph convolutional network technique for the travelling salesman problem.arXiv preprint arXiv:1906.01227,

    1906 arXiv

  21. [2023]

    G., Hottung, A., Wouda, N., Lan, L., Tierney, K., and Park, J

    Berto, F., Hua, C., Zepeda, N. G., Hottung, A., Wouda, N., Lan, L., Tierney, K., and Park, J. Routefinder: Towards foundation models for vehicle routing problems.arXiv preprint arXiv:2406.15007,

  22. [2024]

    Multi-task learning for routing problem with cross-problem zero-shot generalization

    Liu, F., Lin, X., Wang, Z., Zhang, Q., Xialiang, T., and Yuan, M. Multi-task learning for routing problem with cross-problem zero-shot generalization. InProceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 1898–1908,

    1908

  23. [2025]

    of these solvers to handle more complex and varied problem instances. Beyond AR methods, non-autoregressive (NAR) constructive approaches (Joshi et al., 2019; Fu et al., 2021; Kool et al., 2022; Qiu et al., 2022; Sun & Yang, 2023; Min et al., 2023; Ye et al., 2023; Kim et al.,...

    2019

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.