RADAR: Learning to Route with Asymmetry-aware Distance Representations
Authors: Hang Yi
Ziwei Huang
Yining Ma
Zhiguang Cao
Venue: ICLR 2026 Poster (2026)
Keywords: Neural Combinatorial Optimization
Vehicle Routing Problem
Primary area: optimization

Abstract: Recent neural solvers have achieved strong performance on vehicle routing problems (VRPs), yet they mainly assume symmetric Euclidean distances, restricting applicability to real-world scenarios. A core challenge is encoding the relational features in asymmetric distance matrices of VRPs. Early attempts directly encoded these matrices but often failed to produce compact embeddings and generalized poorly at scale. In this paper, we propose RADAR, a scalable neural framework that augments existing neural VRP solvers with the ability to handle asymmetric inputs. RADAR addresses asymmetry from both static and dynamic perspectives. It leverages Singular Value Decomposition (SVD) on the asymmetric distance matrix to initialize compact and generalizable embeddings that inherently encode the *static asymmetry* in the inbound and outbound costs of each node. To further model *dynamic asymmetry* in embedding interactions during encoding, it replaces the standard softmax with Sinkhorn normalization that imposes joint row and column distance awareness in attention weights. Extensive experiments on synthetic and real-world benchmarks across various VRPs show that RADAR outperforms strong baselines on both in-distribution and out-of-distribution instances, demonstrating robust generalization and superior performance in solving asymmetric VRPs.

Published as a conference paper at ICLR 2026
RADAR: LEARNING TO ROUTE WITH ASYMMETRY-
AWARE DISTANCE REPRESENTATIONS
Hang Yi', Ziwei Huang', Yining Ma², Zhiguang Cao'
Singapore Management University. "Massachusetts Institute of Technology hang. 202 2024 48phdcs. . smu, edu. sg, ziweihuange smu edu. sg yiningmaemi t. edu, zgcaoesmu.
ABSTRACT
Recent neural solvers have achieved strong performance vehicle routing prob- lems (VRPs), yet they mainly assume symmetric Euclidean distances, restricting applicability real-world scenarios. A core challenge is encoding the relational features in asymmetric distance matrices of VRPs. Early attempts directly encoded these matrices but often failed to produce compact embeddings and generalized poorly at scale. In this paper, we propose RADAR, scalable neural framework that augments existing neural VRP solvers with the ability to handle asymmetric inputs. RADAR addresses asymmetry from both static and dynamic perspectives. It leverages Singular Value Decomposition (SVD) on the asymmetric distance ma- trix to initialize compact and generalizable embeddings that inherently encode the static asymmetry in the inbound and outhound costs of each node. To further model dynamic asymmetry in embedding interactions during encoding, it replaces the standard softmax with Sinkhorn normalization that imposes joint row and column distance awareness in attention weights. Extensive experiments on synthetic and real-world benchmarks across various VRPs show that RADAR outperforms strong baselines both in in-distribution and robust generalization and superior performance in solving asymmetric VRPs.
instances, demonstrating
INTRODUCTION
Vehicle Routing Problem (VRP) represents a classical NP-hard problem in combinatorial optimization with widespread applications such as transportation (Toro O et al. (016). It requires finding optimal routes to serve spatially distributed customers under various operational constraints. Traditional solvers (Helsgaun) 2017 Lawler & Wood, 1966) either suffer from exponential computational complexity or depend heavily on handcrafted heuristics, which often limit scalability on large- scale VRPs. These challenges have sparked growing interest in neural combinatorial optimization (NCO), which leverages deep learning to develop data-driven solvers that approximate VRP solutions efficiently with reasonable optimality gaps (Huang et al., 2025 Luo al. (023). Among those
models, constructive neural solvers (Kwon et al.) 2220; 2021 Drakulic et al. 2023 2025 Berto et al., 2025 have proven particularly efficient, as they sequentially select nodes based on partial solutions and instance features, i.c., an approach that closely mirrors the natural process of route construction.
Despite this progress, significant gap remains between research and real-world applicability. In practice, travel costs are shaped by factors such as road topology, traffic directionality, and physical barriers, introducing asymmetries like one-way streets and time-dependent congestion (Cubukeu & Hatcha 2016). Yet, most neural VRP solvers assume symmetric Fuclidean instances (Kool et al. 2019] Kwon et al. 2020) [uoo et al. 2023, Gao al., 2024 Berto et al. (025) and rely on coordinate inputs, making them not applicable when only pairwise asymmetric distance matrices are available, as is common in real-workh systems. This remains a major bottleneck for deploying NCO models in practical, asymmetric routing scenarios (Son et al. 2026).
To address this limitation, a key challenge is effectively encoding the relational structure of asymmetric distance matrices. In this paper, consider the modeling of asymmetry into two aspects: static asymmetry and dynamic asymmetry. Static asymmetry refers to directional discrepancies in the input distance matrix. Current neural solvers incorporate this primarily in places: encoder attention

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Published as a conference paper at ICLR 2026
and initialization. In attention, many methods concatenate dot-product scores with distance signals to inject static asymmetry information (Kwon et al. 2021). At initialization, however, encoding asymmetry is intrinsically more difficult: asymmetric costs are defined at the edge level, while most architectures operate on node-level representations ge.g.. (Drkullic et al. 2023 Luo et al. (023)). Unlike Fuclidean settings, where coordinates provide geometric scaffold that fully recovers the distance structure, asymmetric matrices lack such geometry, making directional patterns harder to learn. Prior attempts encode static asymmetry at initialization often lose global structure and thus underperform (Zhou et al. 2025; Son et al., 2026). Dynamic asymmetry refers to learned, layer-dependent interactions differences that emerge inside the encoder's attention. At each layer, the interaction score for i + j. fused with edge signals, need not equal that for + i, and these discrepancies evolve dynamically with context and depth. However, current solvers rely on row-wise softmax attention, which integrates information only over the neighborhood of node i. Consequently, A. reflects i's local context but ignores how j interacts with the rest of the graph, limiting the model's ability to capture global distance information, particularly in asymmetric settings.
We introduce RADAR: Learning to Route with Asymmetry-aware DistAnce Representations. RADAR tackles both static and dynamic asymmetries through two key components to learn asymmetry- aware embeddings. For static asymmetry, we propose an initialization scheme based on Singular Value Decomposition (SVD) of the cost matrix. By decomposing it into left and right singular vectors, RADAR learns compact node embeddings that encode each node's 's role as a source and destination, preserving global directionality. For dynamic asymmetry, we replace the softmax function in the attention mechanism with Sinkhorn normalization (Sinkhorn 1966), which jointly normalizes rows and columns of the attention matrix. This enforces balanced bidirectional flows, enabling the attention score to capture interactions that depend not only on each node's 's own neighborhood but also on the neighborhood structure of its counterparts.
Our contributions are as follows: (1) We investigate neural VRP solvers under realistic asymmetric distance matrices, advancing the applicability of NCO methods to real-world scenarios; (2) We introduce an SVD-based initialization that captures global directional relationships from the input distance matrix, improving generalization across instance sizes. We also show that Sinkhorn nor- malization yields substantial gains, underscoring the importance of capturing the full neighborhood context of both interacting nodes during attention. (3) We evaluate RADAR on 17 synthetic and 3 real-world VRP variants. Across all cases, RADAR consistently outperforms of-te-of-the-are baselines, demonstrating strong performance. (4) We provide several in-depth analyses, including the structural properties of the input, the role of coordinates under asymmetry, and the evaluation of initialization strategies across different asymmetry levels.
2 RELATED WORKS
Neural Solvers for VRPs. They generally fall into three categories. Constructive methods generate routes sequentially by selecting one node at time, such as AM (Kool et al. 2019) and POMO (Kwon)
ct al., 2020). Improvement methods (Wu et al.) [202]] Ma et al. 2021 2023) begin with an ini- tial solution and learn to iteratively refine it, often combined with classical techniques like local search (Hudson et al. (202)), beam search (Choo et al. 2022). or dynamic programming (Kool et al. 2022). Heatmap-based methods (Joshi et al., [021] Sun & Yangl 2023 Min al. 2023] Zhang & Cao 2025) predict pairwise connection probabilities via GNNs and extract solutions from these learned heatmaps using heuristics or post-processing. Among these, constructive approaches are most common but are typically designed for Euclidean VRPs and perform poorly in asymmetric settings.
Solvers Using Distance Matrices. Several recent methods (Meng et al. 2025) address the limitations of coordinate-centte neural solvers by directly encoding the distance matrix. MatNet (Kwon et al. 2021) reformulates ATSP (asymmetric traveling salesperson problem) as a bipartite graph, but its one-hot embedding limits scalability. UniCO (Pan et al. 2025) extends this with pseudo one- hot encodings for larger instances. ICAM (Zhou et al. 2025) replaces fixed embeddings with k-nearest distances, enabling generalization to instances with up to 1000 nodes. ReLD (Huang et al. 2025) improves MatNet's decoder with identity mapping and feed-forward layers. RRNCO (Son et al. 2026) incorporates context-aware gating, adaptive biases, and distance-basse probabilistic sampling, and further validates its effectiveness on real-world datasets. Post-in cost-improvemen methods like GL.P (Ye et al. 2024) and UDC (Zheng al. 2024) adopt divide-and-conquer strategies to
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improve scalability. Despite these advances, most models still fail to effectively capture both static and dynamic asymmetry, limiting performance on real-world asymmetric VRPs.
Graph Positional Encoding, To embed structural information into node representations, various graph positional encoding techniques have been developed. These methods fall into three main
categories: (1) Matrix factorization approaches Yang al. 2017 Song et al., 2009 Katz 1953)
compute a proximity matrix (e.g.. using spectral or diffusion metrics) and apply dimensionality reduction derive node embeddings; (2) "aandom-walk-bene
methods (Perozzi et al. 2014 Grover
& Leskovec 2016) simulate walks to capture local structure, assigning similar embeddings to frequently co-occurning nodes: (3) Autoencoder- based methods (Wang et al.) [016 Cao al., 2016)
learn low-dimensional embeddings by reconstrueting graph structure through neural networks. These techniques inspire embedding strategies in VRP solvers where structural signals, especially under asymmetric cost matrices, need to be effectively encoded.
PRELIMINARIES
typical VRP instance comprises set of nodes, including one or more depots and multiple customers, and is defined by either distance matrix or coordinates. The objective is to construct set of routes that (1) visit each customer exactly once, (2) start and end at a depot, (3) satisfy all task-specific constraints (e.g., vehicle capacity), and (4) minimize total travel cost such the distance. VRP encompasses a wide range of variants depending on the constraints, such as the Capacitated VRP (CVRP), VRP with Time Windows (VRPTW), and Asymmetric VRPs (e.g., ATSP). These variants reflect different practical requirements and significantly increase problem complexity. In this work, we study 17 asymmetric VRP variants. Among them, one is ATSP, and the remaining 16, whose symmetric versions are adapted from RouteFinder (Berto et al. (2225). ) and we replaced coordinates with asymmetric distance matrices to them. Full details are provided in in/ppendix A
4 METHODOLOGY
Figure 1 depicts the overall framework of RADAR. Given an asymmetric distance matrix D, we compute a truncated SVD U V. The left and right factors encode outgoing and incoming signals. We compute a truncated SVD and retain the top-k singular values with their left/right factors (Up, Eg, Vx). We form the distance feature by concatenating the left and right components (U,E2)²² and V,El/² then concatenate this with node features (e.g., demand) and project the result into the embedding space. We use 5 encoder layers; each has a multi-head attention, two add and normalization, and feed-forward blocks. In the attention block, D and D are concatenated with the dot product scores, then passed through two linear layers, and normalized by Sinkhorn to obtain doubly stochastic attention scores. At each decoding step, we mask visited nodes and pick the next node by sampling or greedily from the predicted probabilities, until the tour is completed.
4.1 STATIC ASYMMETRY: SVD-BASED EMBEDDINGS
A core principle of a successful neural architecture for VRPs is its ability to generate node embeddings that encapsulate both node-specific attributes (e.g., demands) and the relational features (e.g., distance matrix) of the graph. However, we emphasize a crucial point: most of such designs cannot effectively exploit relational structure unless the initial node embeddings are distinguishable and capture directional distance information. When all nodes start with identical embeddings, attention outputs remain identical regardless of attention weights, as they are convex combinations over identical value vectors. This renders the model unable to learn effective representations.
Unlike Euclidean problems, where the coordinates of nodes provide strong inductive priors for such node embedding initialization, asymmetric distance matrices lack such geometric structure, making difficult to generate initial node embeddings to capture their inherent directional patterns. In the absence of informative node features, alternative initialization strategies are required to introduce node-specific variation before relational reasoning begins. Two main approaches have emerged in the literature to address this challenge: uninformed initialization and informed initialization.
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Statie Artwmetry
Node Feature
Dynamir dsymmetry
Linese Layer
Lincar Lover
K
V
Asymmetric Distance Matris
Mulli-Hel Allezion
Encoder Layers
Matmul
Add & Norm
XW
Scale
Feed Porward
Cercal
Neem
Feol
Ue V CR
Sinchom
Crocil
Drooder
Matmil
Ouncel Nete
Probability
Distance Feature
Mubot
Layess Larmalle
Layers Wi Leurratle Punnetts
Figure 1: Framework of RADAR, which features two key designs for asymmetric VRPs: an SVD-based embedding method that captures static asymmetry in the distance matrix; and (2) Sinkhorn normalzzd attention in the encoder layers, replacing standard Softmax to enforce balanced incoming and outgoing attention, modeling dynamic asymmetry in representation learning.
Uninformed initialization introduces node dittndiitig signals without relying input structure. These methods provide synthetically distinguishable embeddings, c.g., node indices, fixed positional patterns, or randomly initialized vectors. For example, MatNet (Kwon et al. 202T initializes row and column embeddings with zero and one-hot vectors. As each node requires a unique one-hot vector in a d-dimensional space, the number of nodes must not exceed d. UNICO (Pan et al. 2025) introduces pseudo-hot encoding (POE) to relax the dimensionality constraint for large-scale problems. Although uninformed embeddings can assign distinguishing embeddings, they introduce random noise that lacks any semantic grounding in the input and may break the node symmetry w.r.t graph structure. As result, the model must learn ignore or override meaningless differences, which can hinder learning efficiency and misalign with the task's 's structural inductive bias.
Informed initialization constructs node embeddings by leveraging structural information embedded in input edge features, i.e., the distance matrix D E Ra*n This matrix encodes static asymmetry and captures the overall topology of the instance. Rather than injecting arbitrary signals, informed embeddings aim to derive node representations directly from this relational structure, allowing the network to start from relationally meaningful representations. However, informed embeddings also pose unique challenges. The values in the distance matrix D are mutually dependent and collectively define the instance topology. As a result, any projection from edge space to node space can become inherently tied to the number of nodes This entanglement often leads to size-specific embeddings that generalize poorly to problem instances with different sizes than those seen during training. A common strategy is summarize each node's neighborhood, e.g., by selecting top-k nearest neighbors (Zhou et al., 2025) or sampling neighbors based on distance-based probabilities (Son et al.) 2026). and project the local context into the embedding space. However, such direct use of raw distances as node features fails to capture the underlying topological structure of the graph.
To address the limitations of existing informed embeddings, we focus on the embedding itself and its capacity to encode the structural information of the distance matrices. The inherent asymmetry of the distance matrix called static asymmetry. Ideally, we hope that the initial embeddings could represent the static asymmetry. So, we introduce a formal definition to describe when an embedding can be said to represent static asymmetric relational information.
Definition 1 (Asymmetry-Aware Embedding). An embedding matrix X E Bank is said to be axymmetric-aware w.r.t. a feature matrix E Roxa if there exist two distinct linear transformations W1. W2 E 1 such that
XWW3X -D| O.
(1)
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This definition formalizes the capacity of embeddings to represent static asymmetry in form compatible with attention mechanisms. Attention computes pairwise interactions as QK" We adopt a similar bilinear form X (XW2)) with W1 # W2 to produce non-symmetric interaction matrix that reflects the asymmetry in the original distance matrix. To construct such embeddings, we introduce truncated SVD (TSVD) to reconstruct each node's relative coordinates:
D R V.
where Ug E RRn and V E R contain the top-k left and right singular vectors of D, respectively, and Σg E B is diagonal matrix consisting of the top-k singular values.
In the cost matrix, each entry Di,j represents the cost from node (departure) to node j (arrival). Accordingly, the rows of D correspond to departure nodes, and the columns correspond to arrival nodes. Based on this structure, we can construct two intermediate representations X₁, and X RR. which captures the row-wise features associated and the column-wise features. The distance feature X E R x22 is defined as the concatenation of X and XR along the feature dimension:
X1. UVV
Xn V
X = X XR).
The obtained X is asymmetry- w.r.t. D as there exist two projection matrices:
W1 = [4x 10] E R2ixk
W2 =0 [4] E 22
(4)
such that
=WI=Unm UVV
XW2 ER XXW2) = UKER ~ D.
(5)
Therefore, the model is theoretically capturing static asymmetry through a single embedding matrix. The process is shown in Algorithm [] We analyze the reconstruction quality of the input distance matrix under different truncation levels. The top 10 singular values could capture around 85% of the matrix information, while 20 and 30 singular values improves the retention to about 93% and 97%, respectively. We choose the top 10 singular values as trade-off between both in-distribution and out-of-distribution generalization. The effectiveness of this choice is further discussed in Section
4.2 DYNAMIC ASYMMETRY: SINKHORN NORMALIZATION
In asymmetric VRPs, directional dependencies should be preserved not only at initialization but also during representation learning. In cesenttoo-baseed encoders, node embeddings are updated dynami- cally layer by layer through weighted aggregation of information from other nodes. The asymmetry learned during encoder attention is called dynamic asymmetry. Existing works (Kwon et al. 1021 Drakulic et al. 2025) typically incorporate the distance matrix directly into the computation of attention scores. Formally, the attention from node i to its neighbors is given by:
A Softmax ([Sim(X,, Di.j (D.))))))))))))
(6)
where Sim(-) denotes similarity function measuring the compatibility between node i and node j conditioned on both their embeddings and pairwise distances. Compared to vanilla attention, this formulation explicitly integrates node-to-node relational signals (Di,, D5.) into the similarity function. The subsequent row-wise Softmax normalizes these scores across all neighbors of i, coupling local distance information with broader relational patterns in its neighborhood.
However, such modeling only makes the attention score Aij aware of distance information in the neighborhood of node i (i.e., D..: and D.,,), while remaining unaware of the complete neighborhood structure of node j. i..., D3.: and D. We hypothesize that this limitation weakens the encoder's ability to capture global directional dependencies, since the interaction between and j is evaluated without considering how j itself relates to the rest of the graph. Notably, this issue is absent in the 2D Euclidean setting, where the entire distance matrix D can be reconstructed from node coordinates, and thus its information is already embedded in the node representations X.
To address this, we propose replacing the row-wise Softmax with Sinkhorn normalization (Sinkhom 1966), which iteratively normalizes rows and columns of the attention matrix, Algorithm 2 This
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ensures that each attention score As, reflects a more complete characterization of both nodes i and by incorporating the full set of distance-basee relations directly connected to them.
Algorithm 1 SVD-based Initialization
Algorithm 2 Sinkhorn Normalization
Require: Distance matrix D E R rank k
Require: Score matrix S R iterations T
Ensure: Node embeddings X
Ensure: Normalized matrix P
I: μ Mea a Std(D)
1: P s exp(SS
2: D +- (D p1/a
3: [U, SVD.oowank(D.
2: for | = to do
4: @+ U.5
3: P Conl())
5: V
4:
P P/E Erow (rom(() (P)
6: x [Q]]
5: end for
7: X Fimal Linear(X) X)
8: return Nimal
6: return P
5 EXPERIMENT
We compare RADAR against baselines synthetic single-task instances, multitask setting covering 16 asymmetric VRP variants, and real-world datasets from RRNCO (Son et al.] 2026). We then provide several in-depth analyses, examining the effects of coordinates, varying asymmetry levels, and demand distributions, followed by ablation studies. All experiments are run on a single NVIDIA RTX 3090 GPU (24 GB). More details and results are provided in [Appendix B and Appendix C Our code is available at wtlhtpss:/// /tttps::/giieee giggtthuubb.....g wwt.hb...ooooooooo
CRRDDR
5.1 SYNTHETIC ATSP AND ACVRP
Setup. We evaluate RADAR on synthetic instances of sizes 100, 200, 500, and 1000. All instances
are randomly generated following (Kwon al. [021] Luo et al. 2023 Kwon al. 2020). We train RADAR and all baselines on size 100, and evaluate zero-shot generalization to instances of size 200, 500, and 1000 without finetuning. Training takes 39.31h for ATSP and 54.7 74h for ACVRP
Baselines. (I) Traditional solvers. LKH3 (Helsgaun 2017) with 100, 1k and 10k search trials (10k omitted for ATSP as performance saturates by 100). HGS (Vidal 2022) under two different
time budgets aligned with LKH-1000 and LKH- 1-HH10000 10000. (2) Constructive neural solvers. We retrain MatNet (won et al. 202T), ICAM (Zhou et al.) 2025), ELG (Gao al. 2024), and ReLD (Huang et al. (225)) under setup; all are evaluated with z-score normalization. We also report MatNet trained for 2100 epochs under its original setting (no z-score). UNICO (Pan et al. (2025) is evaluated using its released checkpoint. We further include MatNet variants: Single (One-hot) replaces the dual embeddings with one-hot vectors, and Single (Random) uses a Uniform(0, 1) scalar mapped by linear projection layer. Since ELG does not natively support asymmetry. we adapt by replacing its encoder with MatNet using random embeddings and removing Euclidean-specift components in the local policy; we then retrain on synthetic data and apply local policy from epoch 1750. In ACVRP, MatNet (Demand) uses only demand as the initial node feature while retaining the original dual eeal-embedding design; MatNet-Single (Demand)+ is a Langl--brnchh version that initializes solely from demand. MatNet-Single (Random) follows the same configuration as in ATSP setup. (3) Improvement Neural Solvers. GL.OP (Ye et al. 2024) is evaluated on our retrained MatNet with z-score but shows improvement over the authors' original setting. UDC (Zheng et al., (024) is also retrained in our setup. For fairness, mixed-size training is disabled for both ICAM and UDC.
Results. As shown in Table I RADAR consistently outperforms prior learning-based baselines. On ATSP, RADAR achieves the lowest objective values and gaps among neural methods, and the gap remains small even the problem size increases 1000. On ACVRP, RADAR achieves the best performance among all learning- baaning-bbsed methods, and even surpasses LKH on ACVRP200. In Appendix C.T. we also compare RADAR with Matnet and its variants with augmentation ATSP.

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Table 1: Performance on ATSP and ACVRP.
Testing (Ik instances)
Generalization (Ik instunces)
ATSP100
ATSP200
ATSP500
ATSP1000
Method
Otg.
Gap
Time
Obj
Gap
Time
Ohj
Gap
Time
Ohj
Gup
Time
LKH-100
1 1,5643
0.009
1.08m
1.5721
0.00%%
2,44m
1.5763
0.00%
7 7.32m
1.5739
0.00%
21.92m
LKH-1100
1 1.5645
5.91m
1,5721
16,40m
1,5763
44,45m
1,5739
1,21h
GLOPP
18848
70.40%
2.75m
2,0584
30,93%
3,07m
2.2345
41.7698
3.72m
2.3429
48 869
7.29m
UDC-12500 50)
1.9911
1.78%
1,02h
1.7577
11.81%
2.04h
2.3810
51.05%
6.4hh
3.0734
95.27%
19.12h
Motnet
16473
5.31%
0.03m
3.1928
103.07%
0.14m
Motnest
16161
3.32%
0.03m
1.9111
21.56%
0.14m
Single-Single (Che-hoon*
1.5995
2.25%
6.40%
0.12m
Mitnes-Single (Randons)*
2,08%
5.23%
0.13m
1,8610
1.34m
2.1821
38.64%
11.11m
MasOOENee-sx
18779
19.67%
0.54m
2/2033
71, 71.95%
2.33m
4.1189
161 30%
21.87m
5.5072
249.91%
2.98h
ICAM"
1,6500
5.99%
0.01m
1.8471
17.49%
0.06m
2.4592
56.01%
0.73m
2.9069
$4.69%
9. 9.80m
ELG
1.9822
2.17%
0.00m
1.6425
4.47%
0.27m
1.7456
10.74%
2.60m
1.8441
17.17%
22.31m
1,5000
1.64%
0.03m
1,6310
3.75%
0.15m
1.7873
13.39%
1.49m
2.0723
31.67%
12.18m
RADAR
1.5756
0.72%
0.04m
1.5879
1.01%
0.15m
1.6098
2.13%
1.45m
1.6899
4.13%
11.57m
Testing (Ik instinces) ACVRP100
Generalization instances)) ACVRP500)
ACVRP200
ACVRP1000
Method
Obj.
Gap
Time
Obj.
Gap
Time
Obj.
Gap
Time
Obj.
Gap
Time
LXH-100
2.2526
6.05%
2.20m
2.2345
2.77%
3.24m
2,4155
3.20%
11. 11.58m
2,5092
2.84%
1.00h
-KE--100
2.1635
186%
17.64m
0.75%
2.3605
0.86%
1.06h
2.4569
0.69%
2,86h
LXII-10000
2.1240
0.00%
2.7%
2.1615
0.00%
4.25h
2.3805
0.00%
10.29h
2.4100
0.00%
36,25h
HGS-Shorr*
2.1614
1.76%
16.67m
2,0806
-3,88%
25.58m
2,5011
-1.68%
1.08h
2.1094
-13.55%
2.99h
HGS-Long
2,0942
-1.40%
2.78hs
1.9733
-8.83%
2,1451
-8.35%
10.47h
1.9792
-18.89%
36.33h
Matnet (Demand)"
2,1968
3.42%
0.04m
3,1620
46. 10%
0.20m
4,4847
91.58%
1.92m
5.7523
135.76%
13.50m
Matnet- Single (Demandyt
2,1821
2.73%
0.03m
2,6283
21 433%
0.15m
3,4994
49 49.466%
1.67m
4.3736
79.25%
11. 11.79m
Matnet-Siagle (Random)*
2,1813
2.70%
0.03m
2,2372
3.36%
0.15m
3.3697
43.98
1.73m
3.9904
63.52%
11.38m
ReLD*
2.1656
1.96%
0.04m
2.1635
-0.05%%
0.18m
3,4507
47.40%
1.96m
4,7424
94.45%
13. 33.5
RADAR
2,1588
1.64%
0.05m
2,1483
-0.75%
0.14m
2.4118
3.39%
1.75m
2.4634
0.96%
11,60m
Note. denotes evaluation using the authors' official chockpoints;
indicates training and resting with 2-score normalization. indicates that HGS yiclds infeasible solutions under the given time budgets. Consequently, we do not use it as the baseline for gap computation. and the reported Appendix[] Results for UDC and ICAM differ from their original reports because we disubled their
detailed infeusible mixed size training scheme to ensure faimess. MaiNet with one hat node embesddings fails to generalize to [500, 1000). as the identity embedding is ried to fixed 256-dimensional space. The boldface indicates the best result among learning-hused methods.
5.2 MULTI-TASK LEARNING ON VARIOUS ASYMMETIC VRP VARIANTS
Setup. We further integrate our RADAR into the multi-task framework in RouteFinder (RF) (Berto et al. 20255) and follow its experimental setup to demonstrate its strong generalizability. The model is trained on 16 asymmetric VRP variants and evaluated on 1,000 test instances per variant.
Baselines. We adopt the traditional solver HGS (Vidal 2022). For neural baselines, we adapt ROUTEFINDER with two changes: (i) replace its encoder attention with MatNet attention; and (ii) modify the initialization, yielding two variants, i.e., RF, which uses no distance features and initializes solely from node attributes (e.g., demand, time windows), and RF-NN, which uses top-k neirest distances concatenated with the node features.
Table 2: Average performance.
Method
Avg. Obj.
Aug. Gap (%)
HGS
2.4709
OR-To
2.5409
3.0863
RF
2.5330
2.4688
RF-NN
2.5216
1.9875
RADAR
2.5044
1.3331
Results. Table summarizes average performance over 16 asymmetric VRP variants in the multitask setting. Despite the diversity of constraints, RADAR remains competitive and achieves the lowest average gap among all neural methods. This validates RADAR's asymmetry-centric design across broad spectrum of asymmetric routing problems. See Table 8 for more results.
5.3 REAL WORLD DATASETS
We evaluate RADAR on weal-world asymmetric henchmarks introduced by RRNCO (Son et al. 2026). Following their framework, we train our model on the same datasets using Min-Max normalization. Since the test sets remain unchanged, we directly reuse the GCN and MatNet results reported in their paper, excluding other baselines due to incompatible settings. Unlike synthetic datasets, RRNCO provides both realistic node coordinates and asymmetric distance matrices (Section [5.4] further analyzes their individual impact). Table 3 shows the results across three real- wolld tasks. RADAR consistently achieves lower costs and smaller optimality gaps across all tasks and distribution settings. It also generalizes well to both in-distribution and out-of-distribution instances, highlighting
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Table Performance on Real-woril Datasets for ATSP, ACVRP, and ACVRPTW.
In-distribution
Out-of-distriutionn (city)
Out-of-distribttion (cluster)
Task
Method
Cost
Gap (%)
Time
Cost
Gap (%)
Time
Cost
Gap (%)
Time
LKH3
38.387
1.6h
38.903
1.6h
12.170
1.6h
AISP
MatNet
39.915
3.98
27s
40.548
4.23
27s
12.886
5.88
27s
RRNCO
39.077
1.80
21s
39.783
2.26
21s
12.450
2.30
21s
RADAR
38.671
0.74
21s
39.272
0.95
21s
12.314
1.18
21s
PyVRP
69.739
7h
70.488
7h
22.553
7h
OR-Tools
72.597
4.10
7h
73.286
3.97
7h
23.576
4.54
7h
ACYRP
GCN
90.546
99.84
17s
90.805
28.82
17%
34.417
52.61
17%
MaiNet
74.801
7.26
30s
75.722
7,43
30s
24.844
10.16
30s
RRNCO
72.145
3.45
23s
72.999
3.56
23s
23,280
3.22
238
RADAR
71.557
2.61
23s
72.336
2.62
23s
23.137
2.59
23s
ACVRPTW
PyVRP
118.056
7h
118.513
7h
39.253
7h
OR-Tools
119.681
1.38
7h
120.147
1.38
7h
39.903
1.66
7h
RRNCO
122.693
3.93
32s
123.249
4.00
32s
41.077
4.65
32s
RADAR 121.260
2.71
32s
121.841
2.81
32s
40.459
3.07
32s
Note. The boldface indicates best result among learning- basedd methods.
Table 4: Study on the effect of coordinates in asymmetric settings (ATSPI00).
In-distribution
Out-of-distributie (city)
Out-of-distribun (cluster)
Method
Cost
Gap
Time
Cost
Gap
Time
Cost
Gap
Time
LKH
38.387
1.6h
38.903
1.6h
12.170
1.6h
RRNCO (wio coords)
39.624
3.22%
7s
40.670
4.53%
7s
12,982
6.68%
RRNCO (w/ coords aug)
39.309
2.40%
21s
40.020
2.87%
21s
12.567
3.26%
218
RADAR (w/o coords)
38,958
1.49%
7s
39.551
1.66%
7s
12.413
2.00%
7s
RRNCO (w/ courds)
39.385
2.60%
7s
40.383
3.80%
7s
12.717
4,50%
7s
RRNCO (w/ coords aug)
39.077
1.80%
21s
39.783
2.26%
21s
12,450
2.30%
21s
RADAR (w/ conrds)
38.972
1.52%
7s
39.765
2.22%
7s
12.487
2.61%
7s
RADAR (w/ coords + aug)
38.671
0.74%
21s
39.272
0.95%
218
12,314
1.18%
218
its robustness under real-world variability. These results underscore RADAR's practicality and effectiveness in handling complex, asymmetric routing in real-world scenarios.
5.4 COORDINATES VS. DISTANCE MATRICES
We study whether coordinates provide meaningful signals in asymmetric routing tasks. Following the ATSP setup from RRNCO (Son et al. 2026), we can isolate the effect of coordinates. We compare RADAR and RRNCO under various coordinate usage conditions, applying the augmentation strategy from POMO (won et al. 2020). Since RADAR employs deterministic initialization, the (w/o coords + aug) case yields identical outputs and is therefore omitted. Results show that RRNCO degrades noticeably without coordinates, while RADAR maintains strong performance. Remarkably, even without coordinates, RADAR outperforms RRNCO with coordinate augmentation, indicating that our distance- based embeddings effectively capture structural information without positional cues. When coordinate augmentation is applied, both methods improve, with RADAR being the best. This shows that, in asymmetric tasks, the main value of coordinates may lie in enabling augmentation and promoting diversity, rather than encoding structure.
5.5 EFFECT OF INITIALIZATION UNDER DIFFERENT ASYMMETRY LEVELS
In real-world scenarios, distance matrices often violate geometric assumptions such the triangle inequality. As asymmetry increases, structural regularity deteriorates, making harder for models to extract meaningful representations. Since initialization influences early learning dynamics, it important to understand how varying asymmetry levels impact embedding quality. To simulate this, generate node coordinates uniformly in (0, 1]2 and compute pairwise Fuclidean distances
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Table Cost and relative GAP (%) under varying asymmetry levels. RADAR serves as the bascline.
Low Asymmetry
Medium Asymmetry
High Asymmetry
Method
GAP
100
50
GAP
GAP
GAP
100
GAP
MatNer
4.6446
0.54%
6.2992
1.83%
4.8124
9.55%
6,7598
11.84%
4,7877
21.93%
6,9756
24.04%
Random
4.6851
1.41%
6.3244
2.24%
4.5128
2.72%
6,2680
3.70%
4.0722
3.70%
5,9836
6.41%
UNICO
0.08%
6.4936
4,98%
4.4896
2.19%
7:2094
19.27%
4.0759
3.80%
6.5876
17.14%
ICAM
4.6966
1.66%
6.3632
2.87%
4.4840
2.06%
6,2293
3.05%
4,0636
3.48%
5,9471
5.75%
RRNCO!
4.6995
1.73%
6.4048
3.54%
4,5005
2,44%
6.3436
4.95%
4.1282
5.13%
6.0893
8.28%
RADAR
4.6198
6.1858
4.3933
6.0444
3,9268
8
5.6235
Note.
indicates uninformee embedding:
indicates informed embedding.
form a symmetric matrix. Asymmetry introduced by multiplying each entry Dij with a noise
coefficient Og sampled from ((11, a where a controls asymmetry: weak (a 0.1), moderate (a = 0.2), and strong (a = 0.3). We compare initialization strategies from RRNCO (Son et al.) 2026), UniCO (Pan et 2025), MatNet (Kwon al., 2021), ICAM (Zhou et al. 2025), and random initialization. For RRNCO and ICAM, we use single-embedding variants without coordinate inputs to isolate initialization effects. All methods are evaluated using a unified MatNet-style attention architecture, trained on 50-node instances and tested on 50- and 100-node sets. Training runs for 1200 epochs, with performance recorded at epoch 1101, Additional settings follow Section 5.1
The results in 5 presents the performance of different initialization strategies under increasing levels of asymmetry. Overall, performance tends to degrade as asymmetry intensifies, but the rate of degradation varies significantly across methods. Approaches such as MatNet and UniCO (*) exhibit sharp increases in relative gaps, particularly at larger problem sizes and higher asymmetry levels. In contrast, informed initialization methods (f) degrade more gradually and maintain lower overall gaps. Notably, RADAR demonstrates consistently strong performance across all settings.
5.6 DIFFERENT DEMAND DISTRIBUTION
In VRP models, customer demands are typically sampled from discrete uniform distribution over (I..... 9}. While this demand setting is convenient for training and evaluation, it provides only a limited view of the demand landscape. A model that fits well under this single distribution may lack sensitivity to distributional shifts, leading to degraded performance once the demand law deviates from the training regime. See Appendix C.3 Table 9 for more details.
ABLATION STUDY AND DISCUSSIONS
6.1 SVD-BASED INITIALIZATION
Current Initialization Methods. We evaluate the proposed initialization against ICAM Zhou et al. 2025). UniCO (Pan al. 2025), Random, MatNet (Kwon et al. (02I). and RRNCO (Son et al. (026) under the same experimental setup described in Section [.5] UniCO could not run on ATSPI000 due to memory constraints. As shown in Figure 2 the proposed initialization yields consistent gains on in-distribution instances and exhibits stronger -fit----ditriiuution generalization to larger problem sizes, outperforming all baselines both regimes.
Alternative SVD Initialization Methods. We compare our SVD-based initialization with eigenvalue decomposition (EVD), multidimensional scaling (MDS) (Beals et al. 1968), QR decomposition (Fran- cis. 1961), and random approximation (RA) variants ATSPI00- 1000. As evidenced by Table I0 and the theoretical analysis in Appendix D.2 our method aligns best with the asymmetric structure of the distance matrices, leading superior performance over existing alternatives.
Effect of l in Informed Embedding. We further examine the effect of k. the number of informative neighbors used during initialization. We apply this analysis to ICAM, RRNCO, and method with & {10, 30, 50). Figure 3 shows radar plots of generalization performance. We report an Efficiency Score, defined as max Gap, 0), capture both accuracy and stability. Increasing

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Published as a conference paper at ICLR 2026
Table 6: Ablation on SVD and Sinkhorn for RADAR.
Method
SVD
Sinkhorn
Gap(St
Time
Gap(S))
Time
Gap(S)
Time
Gap(S)
Time
RADAR
X
1.5909
2.08
0.02m
1.6543
5.23
0.13m
1,8610
18.06
1.34m
2.1821
38.61
11.1
RADAR
V
X
1.5928
1.82
0.03m
1.6275
3.51
0.14m
1.7418
10.50
1.44m
1.9342
22.89
11.45m
RADAR
K
1.5829
1.19
0.03m
1.6007
1.82
0.13m
1.6379
3.91
1.43m
1.6878
7.24
11.37m
RADAR
0.72
1.01
2.13
4.13
11,57m
improves in-distribution accuracy but tends to degrade generalization to larger instances. Since our method remains low-cost and stable across all tested k, we fix top-10, which offers the strongest generalization while retaining competitive in-distribution performance.
Runtime Ananlysis. We profile the SVD step to characterize its computational cost. Our implementa- tion employs randomized, truncated SVD that is GPU-accelerated and supports batched evaluation across instances. As shown in Figure. and discussed in Appendix [.4] end-to-end runtime scales smoothly with problem size, and SVD becomes progressively less dominant at larger scales.
6.2 SINKHORN NORMALIZATION
Sinkhorn Normalization. To isolate the impact of Sinkhorn normalization, compare RADAR with and without it. As shown in Table incorporating Sinkhorn significantly improves performance across all instance sizes, highlighting importance modeling directional flow consistency.
Sinkhorn VS. Softmax. To highlight the advantage of Sinkhorn over the standard softmax normaliza- tion, we conduct an additional comparison on ATSP100. Specifically, we contrast the first 10 epochs and the last 10 epochs of training, as reported in Appendix D.5] The results show that Sinkhorn enables faster convergence and yields better performance on asymmetric problems.
Runtime Analysis. We further profile the Sinkhorn normalization on ATSP100 (see Fig. and Appendix D.6). Our measurements separate end-to-end wall time from module breakdowns. The GPU-batched implementation keeps Sinkhorn's 's overhead modest and stable across problem sizes.
Number of Iterations. In main experiments we set the number Sinkhorn iterations to 10. In Appendix D.7 we conduct a sensitivity study on ATSP with iterations.
7
CONCLUSION
This work introduces a framework for effectively incorporating asymmetric distance information into neural VRP solvers. By leveraging SVD-assisted structured initialization and Sinkhorn-norman attention, our approach achieves strong and consistent performance across wide range of asymmetric VRP variants. More importantly, demonstrates improved generalization and robustness in modeling asymmetric structures, making it well-suited for the decision- decision-making in real-world VRP scenarios. In future work, we aim to steer NCO toward more realistic settings. First, we will extend RADAR to broader class of combinatorial optimization problems in which relational (edge) features are central. Our recipe use a SVD-based initialization to turn edge signals into compact node-level anchors. We expect this to improve cold-start quality. Second, richer paradigms, such as improvement heuristics and search-bare based solvers, where RADAR provides asymmetric, cost-aware priors to guide neighborhood selection and move acceptance. We believe these directions will further enhance robustness under distribution shift and strengthen the application of NCO models in solving real-wolrd VRPs.
plan to integrate RADAR with
ACKNOWLEDGMENT
This research is supported by the National Research Foundation, Singapore under its Al Singapore AI Research Fundamental Research Collaborative (US-NSF Researcher Call) (AISG Award No: and its Al Singapore Programme (AISG Award No: AISG3-RP-2022- 031). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of National Research Foundation, Singapore. This research/project was supported by the Singapore Ministry of Education (MOE) Academic Research Fund (AcRF) Tier grant (Grant Approval No: 22-
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A PROBLEM DETAILS
This section defines the five key constraints used in the 17 asymmetric VRP variants. These variants are commonly used in previous work (c.g., (Zhou et al. 2024 Berto et al. (025)) as benchmarks to evaluate the applicability and multitask performance of learned VRP solvers. We follow the setup in recent work, RouteFinder (Berto et al. 2025), to generate instances for each variant. To better reflect real-world conditions such as one-way streets and traffic patterns, all problems are made asymmetric, meaning the travel cost from node i to node j may differ from the cost from j to i, consistent with the MatNet setting (Kwon al. 2021 Note that the term ACVRPTW used in Section 5.3 is equivalent to AVRPTW in this table.
Open route (0). The vehicle does not need to return to the starting point after visiting all customers.
Capacity (C). Under this constraint, each customer is associated with fixed nonnegative demand, representing the quantity of goods to be delivered. Each vehicle has predefined capacity limit, which denotes the maximum total demand it can serve on a single route. A route is considered feasible only if the sum of the demands of all customers assigned constraint ensures that vehicles do not operate beyond their load limits.
does not exceed the vehicle capacity. This
Backhauls (B). In the backhaul constraint, customers are divided into two groups: linchaul customers. who need goods delivered from the depot (positive demand), and backhaul customers, who need goods picked up and brought back to the depot (negative demand). The constraint requires that all deliveries to linehaul customers must be completed before starting any pickups from backhaul customers on the same route. This rule limits the order in which customers can be visited.
Duration limit (L.. Each vehicle is limited by maximum travel time or distance. The total time or distance of the route must be within this limit. If the route exceeds the limit, it is considered infeasible.
Time window (TW). The time window constraint assigns each customer a time interval during which the visit must occur. A vehicle only serve customer if it arrives within the time window. If it arrives early, it must wait until the window opens; if arrives late, the service is considered infeasible.
B EXPERIMENT DETAILS
B.1 DETAILS OF ATSP AND ACVRP
Problem setting. We construct distance matrices following MatNet (Kwon et al. 2021 and enforce the triangle inequality to ensure metric consistency. We perform Z-score normalization distance matrices, where each entry is transformed by subtracting the mean and dividing by the standard deviation of pairwise distances. Customer demands are randomly sampled as integers from 1 to 9, based on LEHD (Luo et al.] 2023) and POMO (Kwon et al. 22200) and normalized by their total sum. Vehicle capacities are set at 50, 80, 100, and 250 for ACVRP instances of sizes 100, 200, 500, and 1000, respectively, following LEHD [uoo et al. (202)).
Training Setting. The embedding dimension is set to 256, with 8 attention heads and 5 encoder layers. We use the Adam optimizer with an initial learning rate of 4 x 10-4 and a weight decay of 10-6, The model is trained for total of 2100 epochs. A learning rate scheduler is employed, maintaining a constant learning rate before epoch 2001, after which a decay factor of = 0.1 is applied. Each epoch consists of 10,000 instances and a batch size of 64. Compared to MatNet (Kwon et al. 2021). which employs 16 attention heads and 12000 training epochs, fewer attention heads and epochs are used our RADAR to reduce memory usage and improve training efficiency. addition, bias terms are introduced in the projection matrices W,, and Wa. As result, training process achieves a 89% reduction in training time compared to MatNet.
Testing Setting, A greedy decoding strategy is used for inference. Following MatNet (Kwon et al. 2021), the number of trajectories is set equal to the number of nodes. Each test set contains 1000 instances, and the reported inference time reflects the total wall-clock time for the full test set. The test batch sizes are configured to fully use the memory of one GPU card as follows: 1000 for ATSP100, 400 for ATSP200, 50 for ATSP500, and 3 for ATSP1000
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Baseline. The selected baselines represent both strong traditional and tfae-of-thee-r reinforcement learning-based baased methods, each reporting performance surpassing that of MatNet (Kwon et al. 2021) to handle asymmetric VRPs. All constructive neural solvers retrained using our training setup, except UNICO (Pan et al.; 2025), where the released checkpoint is used. Since UNICO was trained test data, the provided checkpoint remains representative for fair
on the same distribution comparison. These baselines reflect the representative design choices of learning- approaches to handle asymmetric VRPs in the current literature. MatNet and MaiPOENet-8x adopt dual embedding architectures, which separately encode source and target nodes, and can naturally capture directional distance information. Therefore, they do not require the transposed distance matrix in encoder attention. Other constructive neural solvers use single embeddings and typically incorporate the transposed distance matrix to model asymmetry.
Our Model. We build upon Matnet (Kwon et al. 2021 by replacing its original dual embedding with our SVD-based initialization and substituting the softmax with Sinkhorn normalization in the encoder.
B.2 DETAILS OF MULTI-TASK VRP
Problem Setting. All problem instances are generated following Routefinder (Berto et al., 2025). except for the coordinates, which are replaced by an asymmetric distance matrix constructed based on MatNet (Kwon et al. 202T) and standardized via z-score normalization. However, enforeing strict satisfaction of the triangle inequality across all datasets becomes computationally expensive during instance generation in the training phase. To mitigate this, relax the constraint by applying a correction: for each node triplet, the direct distance is updated as Dis min(Dss Dix + Dkş) over n iterations. This approximation is reasonable in the context of hard benchmarks, as real-world routing data often deviates from a strict triangle inequality. Service times are uniformly sampled in the range [0.12, 0.15). and time window lengths are drawn from [0.15, 0.3). Other problem settings follow Routefinder (Berto et al. 2025).
Training Setting. We follow the same training setting as Routefinder Berto et al. (025)) 5). We train each model for 300 epochs using dynamically generated VRP instances, with 100,000 samples per epoch. The optimization is performed by the Adam algorithm with an initial learning rate of x 10-1 and batch size of 256. The learning rate is reduced by factor of 0.1 at epochs 270 and 295.
Testing Setting. We use the same evaluation setting as RouteFinder with greedy decoding. For each instance, we generate many solution trajectories the number of nodes (Kwon et al. (2200) and (Kwon et al. 202T). Each variant's test set consists of 1000 instances of size 100, and the reported inference time is measured over the entire set.
Baseline. Since the original RouteFinder architecture is not designed for asymmetric vehicle routing problems, we introduce two modified variants, RF and RF-NN, which represent the minimal changes required to adapt it to our setting. Both variants are implemented within the RouteFinder framework and incorporate MatNet attention model asymmetric relationships. In RF, each node is initialized using only its own node features, such as demand and service time window, without access to any pairwise distance information. RF-NN extends this by incorporating the distances the 50 nearest neighbors as additional node features, following the initialization used in ICAM (Zhou et al. 2025).
Our Model. To verify the effectiveness of our method, we build upon the RouteFinder framework by replacing its original coordinate embedding with our SVD-based initialization and modifying the attention mechanism to Matnet attention with Sinkhorn normalization.
B.3 DETAILS OF REAL-WORLL DATASETS
Problem Setting. The RRNCO dataset (Son et 2026) used to evaluate the performance of neural solvers in real-world instances. It is constructed from 100 cities distributed across six continents: 25 in Asia, 21 in Europe, 15 in North America, 15 in South America, 14 Africa, and 10 in Oceania. For each city, a 3x3 km region is extracted from OpenStreetMap (OpenStreetMMap contributors 2017). The travel distances and durations between locations are calculated using a locally deployed OSRM server (Luxen& & Vetter 2011). VRP instances subsequently generated by subsampling locations and assigning demand vectors and time windows, while the original spatial topology and routing constraints are preserved.
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Table 7: Comparasion with Matnet Augmentation on ATSP.
ATSP100
ATSP200
ATSP500
ATSP1000
Method
Obj.
Cap
Time
Cap
Time
Obj.
Gap
Time
Obj.
Cap
LKH-1000
1,5643
5.91m
1.5721
16.40l
1,5763
44.45m
1,5739
1.71h
Matnet (X12X)
1.6002
2.29%
4.39m
3.0272
92.56%
20. 34m
Matner" (x128)
1.3863
1.41%
4,40m
1.8428
17.22%
20.34m
Matnet Single (One- (One-hoty* (x126)
1.5778
0.86%
3.16m
1,6305
3.71%
16.33m
Single (Random)* (x126)
0.88%
3.14m
1,6173
2,88%
16.33m
1.806]
2.95h
2,0836
32.38%
RADAR*
1.5756
0.72%
0.04m
1.5879
1.01%
0.15m
1,6098
2.13%
1.45m
1,6389
4.13%
11.57m
Note.
indicates training and lesting with 2-score normalization. The boldiace indicates the best result among learning- beseiing--eeeddee methods.
Training Setting. We follow the same training setting as RRNCO for a fair comparison. We train the model for 200 epochs using the Adam optimizer with an initial learning rate of 4 x which decayed by a factor of 0. at epochs 180 and 195. Each epoch processes 100,000 instances with batch size of 256. The model employs 128- di-dimensional node embeddings, 8 attention heads, 5 512-dimensional feedforward layers, and a total of 12 Transformer layers.
Testing Setting. We follow the same testing setting as RRNCO. The test batch size is set to 32. Data augmentation is made possible by the inclusion of coordinate features. Thus, we apply a data augmentation factor of 8 to the input coordinates. Each test set consists of 1280 instances of size 100, and the reported inference time is measured over the entire test set.
Baseline. Given that our experiments follow the same datasets and evaluation protocol RRNCO, we directly adopt the baseline results reported in their work. The selected baselines include Mat- Net (Kwon et al., 2021) RRNCO, and GCN (Duan et al., 2020). Other models are not included in the comparison, mainly due to, 1) RRNCO already achieved state-of-the-arr performance among neural solvers as reported in (Son et al., 2026), and 2) most of the baseline methods used in (Son)
et al.] 2026). such as LEHD (Luo et al. (202)) and POMO (Kwon et al. 2020). do not support explicit distance matrices as input, making the comparison unfair.
Our Model. To verify the effectiveness of our method, we build upon the RRNCO framework by replacing its original distance embedding with our SVD-based initialization and modifying the attention mechanism to Matnet attention with Sinkhorn normalization.
C MORE RESULTS AND DISCUSSION
C.I METNET AUGMENTATION
MatNet's one-hot node initialization introduces stochasticity in the constructive policy. To make use of this, the original paper introduces an augmentation scheme that repeats each instance 128 times and returns the best solution among the independent runs (we denote this as MatNetx128). In this section, we report MatNet X128 on ATSP and also include new variants for comparison. For ACVRP, MatNet variants in our setup employ demand-based projections (deterministic), so the 128-repeat augmentation is not applicable. As indicates in Table 7, matNet augmentation reduces the gap but incurs a substantial runtime increase and still fails to surpass RADAR across scales.
C.2 MULTI-TASK LEARNING ON VARIOUS ASYMMETIC VRP VARIANTS
This section reports detailed multitask results across 16 asymmetric VRP variants. Across all problems, RADAR consistently surpasses the learning- based baselines, achieving the lowest gaps and strong overall efficiency; see Table for per-varian numbers.
C.3 DIFFERENT DEMAND DISTRIBUTION
In VRP models, customer demands are typically sampled from a discrete uniform distribution over {1....,9) 99. While this demand setting is convenient for training and evaluation, it provides only limited view of the demand landscape. A model that fits well under this single distribution may lack sensitivity to distributional shifts, leading to degraded performance once the demand law deviates from the training regime. To probe robustness and distribution awareness, we further evaluate on two oft-of-distrinee distiitiinttion demand distributions. Skewed-small. 80% of customer demands are sampled
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Table Performance on 16 VRP variants at problem size N
100 (each with IK test instances).
ACVRP Gap
AOVRP Gap
AVRPB Gup
AVRPL. Gap
Method
Obj.
Time
Obj.
Time
Obj.
Time
Obj.
Time
HGS
2.197
5.93m
1.587
5.81m
2.285
9
5.90m
2.191
6.21m
OR-Tools
2.326
5.87%
6.01m
1.665
4.91%
6.07m
2.397
4.90%
5.99m
2.308
5.34%
5.83m
2.246
2.23%
2.39s
16655
3.02%
1.98s
2.363
3.41%
1.8%s
2.242
2.37%
1.91s
RF-NN
2.232
1,59%
3.11s
1.626
2.46%
1.99%
2.348
2.76%
1.90s
2.229
1.73%
1.948
RADAR
2.222
1.14%
3.51s
1,618
1.95% %
2.39%
2.334
2.14%
2.31s
2.218
1.23%
2.34s
AOVRPB
AOVRPL
AVRPBL
AVRPTW
HGS
1.763
5.90m
1.594
6.03m
2.286
8
5.89m
3.281
5.87m
OR-Tools
1.820
3.23%
6.01m
1.665
4.45%
5.98m
2.402
5.07%
6.09m
3.328
1.43%
6.00m
1.804
2.33%
2.00s
1.643
3.07%
1.99s
2.364
3.41%
1.92s
3.386
3.20%
1.96s
RE-NN
1.797
1.3%%
2.02s
1.634
2.51%
2.00s
2.351
2.76%
1.92s
3.374
2.83%
1 1.98s
RADAR
1.784
1.19%
2.43s
1.625
1.9%%
2.42s
2.335
2.14%
2.35s
3.344
1.2%%
2.38s
AOVRPBL
AOVRETW
AVRPBTW
AVRPLTW
HGS
1.760
6.03m
2.443
5.87m
3.581
5.83m
3.277
5.93m
OR Toolsoos
1.831
4.03%
5.87m
2.457
0.57%
6.00m
3.666
2.37%
6.01m
3.372
2.90%
5.90m
RF
1.804
2.50%
2.01s
2,467
0.98%
2.12s
3.699
3.30%
1.99s
3,388
3.39%
2.00s
RF-NN
1.795
1.99%
2.02s
2.459
0.65%
2.00s
3.681
2.79%
1.92s
3.374
2.96%
1.98s
RADAR
1.782
1.25%
2,43s
2.444
0.00%
2.55s
3.649
1.90%
2.41s
3.350
2.23%
2.42s
AOVRPHTW
AOVRPLIW
AVRPBLTW
AOVRPBLTW
HGS
2.624
6.06m
2,435
5.86m
3.604
5.87m
2.627
5.97m
OR-Tools
2.653
1.11%
5.87m
2.462
1.11%
6.00m
3.667
1.75%
6.01m
2.636
0.34%
6.05m
RF
2.650
0.99%
2.15s
2.458
0.94%
2.12s
3.727
3.41%
2.02s
2.652
0.95%
2.16s
RF-NN
2.640
0.61%
2.02s
2.449
0.57%
2.00s
3.710
2.94%
1.92s
2.646
0.72%
1.98s
RADAAR
2.627
0.11%
2.564
2.435
0.00%
2.524
3.676
2.00%
2.632
0.19%
Note. The boldiace indicates the best result among learning bused methods.
Table 9: Performance under skewed demand distributions of ACVRP100.
Skewed-Large
Skewed-Small
Method
Obj.
Gap
Time
Obj.
Gap
Time
LKII-100
2.3626
4.97%
2.19m
1.8474
2.05%
2.04m
LKH-1000
2,2507
18.08m
1.8105
17.55m
MatNet
2,2806
1.33%
0.04m
1.8937
4.60%
0.04m
MatNet-Single (Demand)
2,2641
0.60%
0.03m
1.8211
3.95%
0.03m
MatNet Single (Random)
2,2631
0.55%
0.03m
1.7600
3.63%
0.03m
ReLD
2,2480
-0.12%
0.04m
1,8555
2.60%
0.04m
RADAR
2.2428
-0.35%
0.04m
1.8536
2.38%
0.03m
Note. The holdface indicates the hest result among learning based methods.
from (1113)) and the remaining 20% from (444, 10). Skewed-large. 80% of customer demands are sampled from (4, 10), and the remaining 20% from (), shown in Table 9 our method consistently attains the lowest average cost under both skewed distributions.
D ABLATION STUDY
D.1 CURRENT INITIALIZATION METHODS
Figure 2 sssooss the generalization performance of different initialization methods trained on ATSPION 100 and evaluated on larger sizes. MatNet and UNICO use dual embeddings, while others single embeddings. ICAM includes distances to the 50 nearest neighbors, and RRNCO samples 25 neighbors based distance-derived probabilities.
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ATSP100
ATSP200
ATSP500
ATSP1000
4.99
5.--
5,20%
22
5.0
20
2.00%
18
1.0%
4.0
1.5
KGMM
UNSCO
Ranioo
10
Motoer
Randor
RANCO
ICMM
UNNCO
Random
ICAd
Random
meteoo
RADH
Figure 2: Different initialization performance under varying size of ATSP.
The figure shows that the performance differences between initialization methods become more pronounced the problem size increases. While several methods exhibit significant degradation, our initialization remains stable across scales. This indicates that well-designed initialization improves solution quality on large instances and contributes to better generalization across problem sizes.
D.2 ALTERNATIVE SVD METHOSS
To substantiate the effectiveness of our SVD-based embedding, we compare it against several common matrix-factorization baselines. Below we summarize the intuition behind each family and why their behavior differs in our setting.
Eigenvalue Decomposition (EVD). EVD factors matrix QAQ It captures global spectral structure well on symmetric distances. Analogous to our truncated setup, we retain the top-k eigenpairs (here k=10). Because for symmetric decompositions the left and right eigenvectors coincide, we form the embedding with a single factor Qs and A 11/2
- J_J with I= 1 111 and recovers an embedding by the EVD B. Compared with plain EVD, MDS improves the geometry by (i) removing the effect of global translation via J, and (ii) operating squared distances so that B behaves like a Gram matrix of an Euclidean embedding.
Multidimensional Scaling (MDS). Classical MDS applies double-
QR Decomposition. QR factorizes A QR with Q orthonormal and R upper- triangular. Since QR decomposition provides no spectral ordering. we take the first columns of Q and the first & rows of R and their product to go through projection layer to obtain the embeddings.
Random Approximation (RA). We uniformly initialize a now-dimensional vector for each node. To encourage asymmetry- awaree embeddings, add an information loss that penalizes the mismatch between the predicted scores and D:
Cinfo
|| (W - 11
where we use an elementwise & distance. The overall objective is + X Linfos
Listal
Lissk
(8)
with A > 0 balancing task performance and the asymmetry -aware aware constraint.
From the results Table several key observations can made.
bigenvalue-based methods. EVD and MDS consistently achieve competitive results across different problem sizes. Their advantage lies capturing global spectral structures, which are beneficial for encoding relational information. However, since EVD is fundamentally designed for symmetric matrices, its direct application to asymmetric distance matrices leads to degraded performance, as reflected in the moderate gap compared to the RADAR baseline.
QR decomposition. In contrast, the QR based approach yields the weakest performance across all scales. Unlike EVD, QR decomposition does not produce eigenvalues and therefore lacks natural criterion for ordering feature significance. When truncated to fixed dimension, this absence of ranking mechanism results in substantial structural loss.
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Table 10: Comparison with other SVD-based methods.
ATSP 100
ATSP200
ATSP500
ATSP1000
Melhod
Obj.
Time
Obj.
Time
Obj.
Time
Obj.
Time
EVD
1.5960
0.04
1.6420
0.16
L8447
1.47
2.2576
11.81
QR
1.6137
0.03
1.6822
0.13
2.0650
1.34
2.7875
11.47
MDS
1,5983
0.05
1,6371
0.19
1,7893
1.59
2.0737
12.05
RA1-LI
1,6003
0.03
1.6577
0.12
1.956
1.30
2.5125
11.11
RA2-12
1.5971
0.03
1.6448
0.13
1.8844
1.30
2.2518
RAI-LS
1.5973
0.03
1.6548
0.13
1.9087
1.30
2.3916
11.11
RADAR
1.5928
0.04
1.6273
0.15
1.7418
1.45
1.9422
11.57
Note. injecting RAI-L5 injects the information loss after the fifth encoder layer with À 1.
layer
RA2-L2 same placement 2;
Random Approximation (RA) variants. The RA family shows partial improvements when stronger geometric guidance is injected, either by increasing the loss coefficient or by pushing the injection point deeper into the encoder. Nevertheless, the overall performance of the RA variants remains infe- rior to RADAR. This suggests that the information loss supervision, while theoretically encouraging asymmetry awareness, also increases the optimization burden.
D.3 EFFECT OF INFORMED EMBEDDING.
Figure illustrates the top-k selection distribution across different instance sizes for three informed initialization methods: ICAM, RRNCO, and RADAR. Each subplot corresponds to a different top-k threshold (10, 30, 50), showing how frequently each method selects high-quality solutions different ATSP problem sizes (100, 200, 500, 1000). The radar plots provide visual summary of how each method behaves under varying values and problem scales. The efficiency score is defined as 1 - GAP, where higher values indicate better initialization quality.
TOP-30
TOP-50
1000 200
000
200
3000
500
500
500
Figure 3: Efficiency Score of informed initialization across different top on ATSP.
We can observe that as k increases from 10 to 50, a general decline in performance observed across all methods, suggesting reduced generalization under larger candidate sets. RADAR consistently outperforms ICAM and RRNCO across all settings, with the performance difference becoming more pronounced the problem size increases. Based on this trade-off between candidate flexibility and generalization performance, we adopt & = 10 as the default setting in our experiments.
D.4 SVD RUNTIME ANALYSIS
To quantify the runtime impact of the SVD step, conduct two complementary experiments.
Remove SVD. We retrain a variant with the SVD step removed and compare end-to-end training time and inference latency against full RADAR. The ablation shows modest reduction in training
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Method
Train Time
ATSPI00
ATSP200
ATSP500
ATSP1000
RADAR
39.3
0.04m
0.15m
1.45m
11.57m
SVD
35.67h
0.03m
0.13m
1.43m
11.37m
Table 11: Ablation on SVD. Removing SVD shortens training by 9% with negligible differences in inference latency across sizes.
time, while inference latencies across XTSP-100/200/500/1000 remain essentially unchanged. These results indicate that SVD is not the main runtime bottleneck at scale.
Profiling of the SVD step. We profiled the runtime overhead of SVD during inference. Specifically, we reused the RADAR test setup and evaluated 1,000 instances per setting across multiple problem sizes. As shown in Fig. while the total runtime increases with problem size, the relative cost of SVD diminishes and is no longer the dominant components at larger scales.
Trtal time (s) SVD time 65 Sindhhom time
SVD
Swithor
Time teecorox log
Stare tooa tive C51
1.7
3.79%
10"1-
1.00
200
200 000
900
1000
SCD
272
340
1003
Figure 4: The left panel shows wall-elock time versus ATSP size (log scale) for the total pipeline, SVD, and Sinkhorn. The right panel reports the percentage share of the total runtime attributable to SVD and Sinkhorn at each size. Each measurement averages 1,000 test instances.
Finally, considering that many real world routing problems may often involve fewer than one thousand nodes owing to logistical, geographical or operational limitations, we believe that the computational burden of current SVD step remains well within acceptable bounds for practical applications.
D.5 SINKHORN V.S SOFTMAX
To test whether Sinkhorn can serve as drop-in alternative to softmax, we compared the first 10 epochs and the final 10 epochs (2091-2100) on ATSPI00 under identical settings except for the attention normalization. Sinkhorn reduces the score from 3.06 to 1.69 by epoch 10, whereas softmax goes from 3.05 to 1.79. In late training, Sinkhorn stabilizes around 1.577-1.579, while softmax remains around 1.591-1.596 Thus, Sinkhorn both converges faster early on and attains lower final score, supporting its use as substitute for softmax in ATSP.
D.6 SINKIOORN RUNTIME ANALYSIS
We measured the cumulative runtime of SINKHORN across ATSP test sets of increasing size (see Fig. 4). Empirically, its share of the overall wall-clock time tends to decrease as problem size grows and remains modest the largest size we evaluate. This suggests that, under our implementation, the normalization step scales favorably and is unlikely to dominate end-to-end runtime.
D.7 NUMBER OF SINKHORN ITERATIONS
We additionally conducted sensitivity study on the ATSP with the number of Sinkhorn iterations set to T E {1, 5, 10). Table 12 indicates that increasing T consistently improves solution quality, while the associated runtime overhead remains marginal. Therefore, a larger T yields clear accuracy gains with almost additional computational cost.
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Setmoe
1 59000
15875
2.4
13556
19925
2.
13500
15775
2042
2095
2048
Sporh
Figure 5: ATSP-100: Softmax Sinkhorn. Left: Early training (epochs 1-10) showing faster convergence with Sinkhorn. Right: Late training (epochs 2091-2100) zoo m-in, where Sinkhorn stabilizes lower objective than Softmax.
Table 12: Effect of Sinkhorn Iterations on RADAR.
Method
lteration
100
GAP
Time
200
GAP
Time
500
GAP
Time
1000
GAP
Time
RADAR
0.14m
1.7576
11.67%
11.42m
RADAR
1.5764
0.77%
0.04m
1.5909
1.20%
0.14m
1.6210
2.84%
1.44m
1.6590
5.41%
11.50m
0.72%
D.8 STATISTICAL SIGNIFICANCE EXPERIMENT
To validate the training stability and performance advantage of our model, we conduct statistical significance experiment based on multiple independent runs. We select ReLD (Huang et al. (025). the strongest baseline model, as the comparison target. For each model, we run three independent training processes on the ATSP100 dataset and record the training scores at each epoch.
Figure shows the training curves of RADAR and ReL.D on ATSP100. The shaded regions denote the standard deviation over multiple runs, indicating variance across training trajectories. The left panel corresponds to the first 10 epochs (early stage), and the right panel corresponds to the final 10 epochs (late stage), highlighting differences in both convergence behavior and final stability.
ReLD
1.5950
ReLD
RADAR
4.0
1.5925
1.5900
3.5
1.5675
1.5850
1.5825
1.5800
2.0
1.5775
2095 2096 2009 Epoch
Epoch
Figure 6: Training curves of RADAR and ReLD ATSP100: early-stage (left) and late-stage (right) comparison over the first and last 10 epochs.
RADAR achieves lower scores than ReI.D in both stages. Moreover, its training curves exhibit smaller variance, indicated by the narrower shaded regions. This suggests that RADAR not only converges faster, but also maintains more stable performance with less fluctuation during training.
To further validate the robustness of our approach, compare the test performance of RADAR and ReLD. Figure 7 shows the score distributions of RADAR and ReLD on four ATSP test sets with 100. 200, 500, and 1000 nodes, respectively. Each boxplot summarizes the results over 1000 distinct test instances. Statistical significance is assessed using two sample t-tests, with significance levels denoted by asterisks: *p < 0.05, **p 0.01, and *** < 0.001.
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2.4
2.b
2.2
2.0
2.0
2.0
1.8
3.8
1.6
14
1.8
Meinod
Nathod
Mathod
Melhed
Figure 7: Statistical comparison of RADAR and ReL. on ATSP test instances. Each subplot presents the score distributions over multiple runs on ATSPI00, ATSP200, ATSP500, and ATSP1000 test sets, respectively. Statistical significance is assessed using independent two-sample r-tests. Asterisks indicate significance levels: *p < 0.05, *pp < 0.01, and ***p p < 0.001.
Table 13: Performance of Different Normalization on Synthetic ATSP.
Method
100
GAP
Time
200
GAP
Time
500
GAP
Time
1000
GAP
Time
LKH
1,5613
1.05m
1,5721
4.43m
1.5763
29. 9m.
1,5399
130.15m
RADAR-
L,5800
1.26%
0.04m
1,6939
7.75%
0.15m
3,7112
137,31%
1.45m
6.0139
281.12%
11.57m
RADAR-
1,5776
0.85%
1.5917
16756
6.27%
RADAR*
1,5756
0.72%
0.04m
1.5W79
1.01%
0.15m
1.6098
2.13%
1.45m
1,6889
4.13%
11.57m
ReL.D
diverge
diverge
diverge
diverge
Rel D2
1,5976
2.13%
0.33m
1,6403
4 3444
0.15m
1,8207
15,50%
1.49m
2.4648
56.32%
12.18m
ReLD*
1,5900
Single (Random)-
diverge
diverge
diverge
diverge
Saine--Single (Randomie
L.6245
3.85%
0.02m
1.6907
7.54%
0.13m
1.9651
24.67%
1.34m
2.8334
79.70%
11.11m
Mainet-Singge (Randoms"
I.5969
2.08%
0.02m
1.6433
5.23%
0.13m
1,8610
18.10%
1.34m
2,1821
38.64%
11.11m
ICAM-
L6440
5,09%
0.01m
1,9715
25.41%
0.06m
2.6485
68.02%
0.73m
3.3871
114.82%
9.50m
1.9735
92.91%
4.3716
ICAM
L6680
5.99%
0.01m
188711
17.49%
0.06m
2.4592
56.01%
0.73m
2.9069
84.69%
9.30m
One Joo-
diverge
diverge
diverge
diverge
boot
L6008
2.53%
0.02m
16601
5.60%
0.12m
1.5995
2.25%
Mainet-
L6173
5.31%
0.03m
3.1925
103.07%
0.1
Mainet+
L6158
3.29%
0.03m
3.1493
100.32%
0.1 0.14m
Mainet"
L6158
3.29%
0.33m
1,9111
21.56%
0.14m
EL ELG
1,6080
1.8492
3066
2.5286
661 60.37%
1,5907
1.69%
0.06m
1.6287
1.7404
1.9380
22.91%
ELG*
1.5982
2.17%
0.06m
1.6423
4.47%
0.27m
1.7456
10.74%
2.00m
1.8441
17.17%
22.31m
Note. suffix no normalization,
normalization, "sm denotes z-score normalization. Boldface indicates
the best performing normalization variant within each method.
RADAR consistently produces lower scores than ReLD across all instance sizes. The performance gap increases with problem scale, and the statistical significance of the difference becomes more evident on larger instances, reflecting RADAR's stronger scalability and effectiveness asymmetric settings.
E EFFECT OF NORMALIZATION
The use of z-score normalization in Table I was guided by standard deep-learning consideration: normalizing input scales reduces sensitivity to magnitude variations and stabilizes gradient flow. This is especially important for attention based models that consume raw distance matrices, whose values can differ substantially across instances. To verify that imposing a uniform z-score setting on all baselines does not bias the comparison, re-evaluated all constructive neural solvers on the ATSP benchmarks under three strictly matched input regimes (z-score, min max normalization, and no normalization), while keeping architectures, training schedules, and evaluation protocols identical. Table indicaaes z-score normalization remained consistently competitive and, in most cases, yielded the best performance relative min-max or unnormalized inputs.
22

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Published as a conference paper at ICLR 2026
0.00
0.000
000
(a) Softmax. Clear attractor patterns.
(b) Sinkhorn. Uniformly distributed weight map.
Figure 8: Visualization of attention weight matrices produced by Softmax and Sinkhorn.
Table 14: Infeasible rate on ACVRP instances.
Infeasible Rate ACVRP100
ACVRP200
ACVRP500
ACVRP1000
HGS-Short
25.9%
48.7%
92.2%
6.6%
HGS-Lonng 2.5%
0.1%
4.6%
0.00%
F SINKHORN ATTENTION HEATMAP
Sinkhorn normalization improves generalization because it introduces an explicit column- communication mechanism. With standard Softmax attention, each row is normalized independently. so the model can assign disproportionately large probability mass to few "attractor" nodes that naturally receive high incoming attention from many rows. This creates strong column imbalance: small set of nodes dominate the attention matrix, while most columns stay uniformly low. Dur- ing decoding, once those high-attention nodes are visited and removed from the candidate set, the remaining columns are all similarly small, so the attention distribution becomes nearly flat and the model forced make decisions under high uncertainty because there no clear preference signal left. Sinkhorn iteratively normalizes both rows and columns. This prevents any single column from absorbing too much global mass and redistributes attention more evenly across nodes. As a result, even after some nodes are selected, the remaining nodes still maintain meaningful relative attention differences instead collapsing to a uniform low baseline. practice, this makes the decoder's selection policy more stable and less dependent distribution-specife "hub" patterns.
G INFEASIBLE RATE OF HGS ON SYNTHETIC ACVRP
Since HGS can return infeasible solutions, we additionally report the infeasible rate to provide more complete picture of its practical reliability under each computational budget. The infeasible rate denotes the proportion of runs in which HGS fails construct solution that satisfies all vehicle capacity constraints. As shown in Table [4] the short-budget configuration (HGS-Shorl Short) exhibits a substantial fraction of infeasible robustness, driving the infeasible rate down to 2.5% on ACVRP100, below 5% on ACVRP500, and effectively zero for ACVRP200 and ACVRP1000. These results highlight that, while HGS can be made highly reliable given sufficient runtime, its solution quality under tight time budgets must be interpreted together with the corresponding infeasible rate.
Increasing the time budget HGS-Long substantially improves
H MORE SCENARIOS ON SVD-BASED INITIALIZATION
Our approach can apply to broader class of tasks where the instance be represented by relational matrix R (not necessarily distance matrix). encodes directed or asymmetric relations between entities, such costs, compatibiliries, precedences, processing times, or transition penalties.
23

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Published as a conference paper at ICLR 2026
Table 15: Results on FFSP with and without SVD-based initialization.
Method
FFSP20
FFSP50
FFSPI00
MS
Gap
Time
MS
Gap
Time
MS
Gap
Time
MatNet
28.0676
0.3
0.82m
52.5973
1.36m
93.0295
0.6
2.39m
MaD
27.9727
0.0
0.85m
52.3042
0.0
1.39m
92.4473
0.0
2.43m
By decomposing R. into asymmetric components at initialization and injecting these components into the attention mechanism, the model can recover and propagate relational signals throughout message passing, rather than relying solely on node-wise features.
representative example beyond routing is the Flexible Flow Shop Scheduling Problem (FFSP). In FFSP, multiple production stages each contain several parallel machines, and each job must be processed sequentially across stages. The decision is to assign jobs to machines and determine the processing order minimize makespan total completion time. In MatNet, FFSP instances are encoded as matrices of processing times between jobs and machines at each stage. These entries are relational features rather than geometric distances: they describe directed interactions such as "job i processed on machine m requires time pi.m""
To illustrate the generality of our method in this setting. applied the proposed asymmetric SVD- based relational-feane MatNet's original initialization with our decomposition- on instances of size N = 20. We then evaluated the resulting model in a zero-shot fashion on larger out-o--distribution sizes N 50 and N 100. The results (Table 15) show that the same decomposition atusstttention mechanism extends naturally to non-routing relational problems and yields strong generalization beyond the training size.
encoding to MatNet under the FFSP configuration. Specifically, we replaced
initialization and retrained the model
We report MS (makespan)as the primary objective, which measures the completion time of the last job in the schedule; lower MS indicates a better schedule with shorter overall production horizon. Under this greedy inference setting, replacing MatNet's original matrix encoder with our SVD-based relational-feature encoding yields consistent improvements across sizes.
I LARGE LANGUAGE MODELS (LLMs) USAGE
We employed LLMs limited, non-substantive role for (i) linguistic refinement and (ii) minor coding assistance.
Writing. All technical content and arguments were authored by the researchers. Drafts were subse- quently polished with LLMs to improve clarity, grammar, and stylistic consistency; no conceptual or methodological changes were introduced by LLM outputs.
Coding. Implementation work was primarily derived from Matnet wwo et al. 2021 and the design of our proposed SVD-based initialization and Sinkhorn normalization was developed independently by the authors.
Scope and provenance. L-s-iiisteed edits were performed under human supervision and verified for correctness. All ideas, algorithms, and results are attributable to the authors: LLMs were not used for data generation, evaluation, or decision making.
24


Reviews:
[{"review_id": "ak2zIFP6k1", "reviewer": "Reviewer_qwgM", "summary": "This paper addresses the challenge of solving asymmetric vehicle routing problems (AVRPs) with neural solvers. The authors propose RADAR, a framework that modifies existing neural solvers to handle asymmetric distance matrices. The contribution is twofold: (1) an SVD-based initialization to encode static asymmetry into compact node embedding, and (2) the use of Sinkhorn normalization in the attention mechanism. The authors validate RADAR on a wide range of synthetic and real-world asymmetric VRPs, showing superior performance and generalization compared to baselines.", "questions": "1. To verify the necessity of the proposed SVD initialization, could you provide results for two models that keep all other settings unchanged (including z-score normalization) and combine Sinkhorn normalization with forward-only distance features (i.e., top-10 nearest outgoing neighbors) and bidirectional distance features (i.e., concatenating top-10 outgoing and top-10 incoming neighbor distances), respectively?\n\n2. What is the number of iterations T used for Sinkhorn normalization?\n\nWhile the empirical performance of RADAR is promising, I have concerns regarding its technical novelty and the true attribution of improvement to claimed components.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The paper is well-written, and the methodology is explained clearly.\n\n2. The conceptual framing of the problem into \"static\" and \"dynamic\" asymmetry is interesting, providing a new perspective for addressing asymmetric routing problems.  \n\n3. The experimental evaluation is a major strength. It is comprehensive, rigorous, and includes a wide array of benchmarks\n\n4. The empirical evaluation showcases the strong performance of the RADAR compared to the SOTA baseline, especially in out-of-distribution generalization to larger problem sizes.", "weaknesses": "1. The paper's contribution lies in the clever application of existing techniques rather than the development of fundamentally new methods. Both SVD and Sinkhorn normalization are standard, well-established algorithms. While the engineering is solid, the work feels more incremental than transformative from a methodological standpoint.\n\n2. In Table 1, the authors state that all retrained baselines were evaluated using z-score normalization. However, the authors do not justify why this specific normalization was uniformly applied. Applying a non-native normalization may unfairly penalize baseline performance, making it unclear if the comparison was conducted under optimal conditions for all methods.\n\n3. The paper uses different normalizations for different datasets. However, the paper provides no ablation study to compare the effect of z-score versus min-max (or no normalization) on the same dataset. It remains unclear whether the impressive performance gains are fully attributable to the proposed SVD and Sinkhorn modules or are partially an artifact of a specific, un-analyzed normalization choice. \n\n4.  The ablation study in Table 6 severely weakens the case for the SVD-based initialization as a standalone contribution. The RADAR-SVD performs very poorly on larger-scale problems, achieving a gap of over 20% on ATSP1000. This poor generalization suggests that the strong performance is critically dependent on the Sinkhorn operation.\n\n5. Some critical hyperparameters' details are missing. For example, the number of iterations T for Sinkhorn normalization is not specified. It is unclear whether the increasing number of iterations significantly increases the computational load and performance of the model.", "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": "j3w8UzIeY8", "reviewer": "Reviewer_c4fR", "summary": "The paper addresses the problem of learning to solve vehicle routing problem,\nesp. with asymmetric distances. The authors propose a method with two \naspects:\n- i) to concatenate the left and right eigenvectors of the SVD of the distance\n  matrix as customer features (replacing customer coordinates of the symmetric\n  case), and\n- ii) to make the attention matrix doubly stochastic with a Sinkhorn transform\n  (instead of making it row stochastic by a softmax).\n\nIn experiments on synthetic and real TSP and CVRP problems they show\nthat their method outperforms existing neural routing methods, esp. \nfor larger out-of-distribution instances, e.g., for ACVRP100 only by 0.3%,\nbut for ACVRP1000 by 38%", "questions": "- q1. Can you separate the effects of your two contributions i) and ii) clearly in an ablation study?\n  Which of the baselines can be equipped with features i) or doubly stochastic attention scores ii)\n  and how do they perform then?\n- q2. How does HGS perform on problems in tab. 1 and 3? Do you really observe\n  a lift over OR heuristics?\n- q3. Your method performs better esp. in larger out-of-distribution settings. Can you explain or\n  provide some first analysis why this happens?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- s1. two simple ideas: using left and right eigenvectors for asymmetric distances and making attention\n  doubly stochastic.\n- s2. many experiments, also some on different problem variations.\n- s3. well written.", "weaknesses": "- w1. ablation study and thus the attribution of improvement to the two ideas is not fully clear.\n- w2. performance of OR heuristics is not fully clear.\n- w3. why the performance improvements are so much larger for larger problems or \n  out of distribution problems is not investigated.\n- w4. small methodological contribution: adding the eigenvectors is basically feature engineering the asymmetric\n  distance matrix characteristics of the problem.\n\nmore details:\n\nw1. ablation study and thus the attribution of improvement to the two ideas is not fully clear.\n- You could add eigenvector features basically to any method with customer features.\n- And you could use the Sinkhorn normalization with any method involving on attention.\n- Which of the two contributions, \n  - i) using eigenvectors of the distance matrix as customer features and \n  - ii) making the attention matrix douvly stochastic \n  is contributing how much to the success of the method?\n- The example you provide in fig. 5 is problematic:\n  - You compare on ATSP100, but on ATSP100 lifts of your method are very small.\n    It might be more convincing to also compare on ACVRP1000 where you have substantial lifts.\n  - You report for your method RADAR w/o sinkhorn an objective value of ~1.5925,\n    but in tab. 1 your closest baseline ReLD+ achieves an slightly even better objective:\n\t1.5900. So in this case, adding eigenvectors seems not to help?\n\nw2. performance of OR heuristics is not fully clear.\n- Why is the OR solver HGS not compared in tab. 1 and 3?\n- Why do we not see runtimes in tab. 2? W/o. runtimes comparing VRP solvers is not so meaningful.\n\nw3. why the performance improvements are so much larger for larger problems or \nout of distribution problems is not investigated.\n- This is an odd phenomenon: on the problems you train for, ACVRP100, you see only\n  a very small lift of 0.3%, but then on the larger out-of-distribution instances, e.g., \n  for ACVRP1000 the lift is a substantial 38%. Which aspect of your method \n  is responsible for this behavior?\n\nw4. small methodological contribution: \n- adding the eigenvectors is basically feature engineering the asymmetric\n  distance matrix characteristics of the problem.\n\nSmall points:\n- p1. It would be helpful to show in tab. 1 which of the baselines have access to which problem \n  information such as the asymmetric distance matrix, the demands etc.", "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": "ls04M1l8mK", "reviewer": "Reviewer_24SW", "summary": "The paper proposes RADAR, a neural framework for solving Vehicle Routing Problems (VRPs) with asymmetric distance matrices. RADAR addresses both static and dynamic asymmetries via two key components that learn asymmetry-aware embeddings. For static asymmetry, it applies Singular Value Decomposition (SVD) to the input distance matrix to obtain asymmetry-aware node embeddings. For dynamic asymmetry, i.e., asymmetries that emerge during encoder attention, RADAR replaces the standard softmax with Sinkhorn normalization within the attention mechanism.", "questions": "The proposed SVD-based embedding appears tailored to attention-based models. Although attention mechanisms are mainstream in many domains, it remains important to assess effectiveness under alternative network architectures. Such an analysis would clarify whether the observed gains are specific to certain architectures or are more generally applicable. Could the authors provide a discussion and, if possible, experiments on the choice of network architecture?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1.\tThe SVD-based embedding is simple yet empirically effective. The motivation is clearly articulated and technically convincing.\n\n2.\tThe paper presents extensive experiments across multiple datasets and VRP variants, providing solid empirical support for the method’s effectiveness.", "weaknesses": "While leveraging SVD to encode matrix asymmetry is a valuable idea, the current contribution is confined to routing problems. Asymmetric (Directed) edge structures widely exist in other graph-based combinatorial optimization tasks. This scope limitation reduces the broader impact and significance of the work. If the authors could extend the idea from routing distance asymmetry to general graph asymmetry and validate it on a more diverse set of problems, the contribution would be substantially strengthened.", "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": "canSoViWPT", "reviewer": "Reviewer_D1Sv", "summary": "This paper introduces RADAR, a scalable neural framework designed to address the limitation of existing neural solvers for Vehicle Routing Problems (VRPs) that primarily rely on symmetric Euclidean distances, which restricts their applicability to real-world scenarios with asymmetric distances (e.g., one-way streets, traffic congestion).  \nRADAR tackles asymmetry from both static and dynamic perspectives: for static asymmetry, it uses Truncated Singular Value Decomposition (TSVD) on the asymmetric distance matrix to initialize compact, generalizable node embeddings that capture each node’s role as a source (outbound costs) and destination (inbound costs); for dynamic asymmetry, it replaces standard row-wise softmax normalization with Sinkhorn normalization, which jointly normalizes rows and columns of the attention matrix to enforce balanced bidirectional flow and capture global distance information from both interacting nodes.  \nExtensive experiments on 17 synthetic VRP variants (e.g., ATSP, ACVRP) and 3 real-world datasets show that RADAR outperforms SOTA traditional and neural baselines.  \nI regard the asymmetric vehicle routing problem as highly significant yet relatively underexplored in current research. The approach proposed in this paper is both novel and interesting; however, its experimental design and results are not very solid. Specifically, the selected baseline algorithms are not specifically tailored for asymmetric vehicle routing problems, limiting the validity of comparative analyses. Additionally, the experimental evaluations only cover out-of-distribution scenarios—where models are trained on instances of size 100 and tested on instances of other sizes—while failing to include in-distribution scenarios, i.e., training on instances across all sizes and subsequent testing on instances of all sizes. This omission undermines the comprehensiveness of the model’s performance verification.", "questions": "I like the asymmetric vehicle routing problem and the idea of this article, but the current experimental results cannot fully demonstrate the effectiveness of this idea. If the author could provide additional explanations regarding the effectiveness of the idea, I would be happy to raise my rating score.\n1. It is suggested to conduct comparative experiments on algorithms specifically designed to solve the ACVRP problem(e.g. heuristics and meta-heuristics).\n2. The experimental design should incorporate in-distribution scenarios, specifically involving training the model on instances spanning all node sizes and subsequently evaluating its performance on test instances of all sizes.\n3. The results in Table 5 indicate that the existing baseline algorithms are designed for symmetric structure problems and its performance will deteriorate in the case of asymmetry. Therefore, maybe the authors could compare the experimental results of the algorithm proposed in this paper and the baseline algorithm in the symmetric structure scenario?\n4. Regarding the ATSP and ACVRP problems, maybe the authors could add the experimental results of the HGS algorithm?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. The asymmetric vehicle routing problem is not well studied yet, this paper addresses a critical real-world gap by explicitly modeling asymmetric distances (e.g., one-way streets, traffic in vehicle routing problems (VRPs).\n2. Features a dual-component design with strong theoretical grounding: Truncated Singular Value Decomposition (TSVD) initializes node embeddings to capture static asymmetry (distinguishing nodes as sources/destinations), while Sinkhorn normalization replaces softmax to model dynamic asymmetry via bidirectional global attention.\n3. Demonstrates robust generalization across scenarios: it outperforms both traditional solvers (e.g., LKH3, HGS) and SOTA neural baselines (e.g., ReLD, MatNet) in synthetic multi-task (16 VRP variants) datesets and real-world datasets.\n4. Well writing and easy to understand.", "weaknesses": "1. The ACVRP problem has been proposed for a long time, and there have been some dedicated efforts to solve this problem, including heuristic algorithms (e.g. Vigo D. A heuristic algorithm for the asymmetric capacitated vehicle routing problem[J]. European Journal of Operational Research, 1996, 89(1): 108-126.) and meta-heuristic algorithms (e.g. Leggieri V, Haouari M. A matheuristic for the asymmetric capacitated vehicle routing problem[J]. Discrete Applied Mathematics, 2018, 234: 139-150.). This paper does not discuss these previous works.\n2. The selected baseline algorithms are not specifically tailored for asymmetric vehicle routing problems, limiting the validity of comparative analyses. \n3. The experimental evaluations only cover out-of-distribution scenarios—where models are trained on instances of size 100 and tested on instances of other sizes—while failing to include in-distribution scenarios, i.e., training on instances across all sizes and subsequent testing on instances of all sizes, weakening result comparability.\n4. It lacks an online update mechanism to adapt to real-time dynamic changes (e.g., sudden traffic jams).\n5. The results of the ACVRPTW problem in Table 3. The method presented in this paper is not the optimal one, but it is highlighted in bold.", "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)