SPACE: Unifying Symmetric and Asymmetric Routing Problems for Generalist Neural Solver
Rongsheng Chen, Changliang Zhou, Canhong Yu, Yuanyao Chen, Yu Zhou, Zhuo Chen, Zhenkun Wang
Abstract
Generalist neural routing solvers have shown great potential in solving diverse vehicle routing problems (VRPs) with a unified model. However, existing solvers are typically limited to symmetric settings or degrade in performance when switching to asymmetric settings due to input inconsistencies or inherent structural differences, substantially limiting their practicality in real-world scenarios that encompass both scenarios. To address this limitation, we define the spatial position of each node based on the relative distances to a specific set of pivots and further propose a Spatial Pivot-Aligned Coordinate-free Embedding (SPACE) framework that unifies node representation and solution generation across symmetric and asymmetric VRPs. Specifically, we construct a bidirectional Frechet representation using a novel furthest pivot sampling strategy to enable invariant node representations across distinct problem settings. Furthermore, we introduce a weight-decomposed adaptive decoding mechanism that decouples geometric perception from problem representations, mitigating the overfitting of constraint decisions to a specific geometry setting. Extensive experiments on 110 VRP variants, comprising 55 symmetric problems and their asymmetric counterparts, demonstrate that SPACE achieves promising zero-shot generalization in both symmetric and asymmetric VRPs.
AI Impact Assessments
(1 models)Scientific Impact Assessment: SPACE
1. Core Contribution
SPACE addresses a genuine gap in neural combinatorial optimization: existing generalist VRP solvers either operate exclusively in symmetric settings (where node coordinates are available) or degrade significantly when applied to asymmetric settings (where only distance matrices exist). The paper proposes two key innovations:
Bidirectional Fréchet Representation (BFR): Grounded in Bourgain's Embedding Theorem, the method defines node positions via distances to a set of "pivot" nodes selected through furthest pivot sampling (FPS). By concatenating both outgoing and incoming distances to pivots, the representation captures directional information while maintaining a unified format across symmetric and asymmetric instances. This is theoretically motivated — the authors prove 1-Lipschitz continuity with respect to a symmetrized metric and provide conditional non-contraction bounds.
Weight-Decomposed Adaptive Decoding (WDAD): Inspired by DoRA, this mechanism decomposes decoder weights into a shared base (encoding general routing knowledge) and constraint-conditioned low-rank updates, preventing the coupling of geometric perception with problem-specific constraint logic.
2. Methodological Rigor
Theoretical foundations are reasonably well-established. The paper provides formal proofs for non-expansiveness (Proposition C.7), conditional non-contraction (C.8), coverage-induced separation (C.9), and necessity of bidirectionality (Corollary C.12). The authors are careful to distinguish between unconditional guarantees (non-expansiveness) and conditional ones (bounded distortion requires pivot separation). The connection from Bourgain's theorem to the practical BFR construction is clearly delineated, though the gap between the theoretical O(log n) distortion guarantee and the fixed M=8 pivots used in practice is not deeply analyzed.
Experimental evaluation is extensive: 110 VRP variants (55 symmetric + 55 asymmetric), comparisons against 7 neural baselines, ablation studies, scalability tests up to 7000 nodes on CVRPLIB benchmarks, and controlled comparisons with matched training sets. The ablation study systematically isolates contributions of each component. The inclusion of URS-Extended (retrained on the same 12 problems) in Appendix K is a commendable methodological choice that demonstrates gains come from architectural innovations rather than training set differences.
Potential concerns: (1) The paper adds ACVRPBTW to the training set compared to URS, which could confound comparisons in the main tables. While the controlled comparison addresses this, it appears only in the appendix. (2) Oracle solvers differ across variants (HGS-PyVRP, LKH3, OR-Tools with varying time limits), making cross-problem gap comparisons somewhat heterogeneous. (3) For asymmetric problems, some "negative gaps" (outperforming the oracle) suggest the oracle solver (OR-Tools with 20s limit) may not provide strong baselines for certain complex variants.
3. Potential Impact
Practical significance: Real-world logistics involves both symmetric (drone delivery) and asymmetric (urban one-way streets) routing. A unified solver eliminates the need to maintain separate models. The demonstrated scalability to 7000-node instances and zero-shot generalization to unseen constraint combinations are practically valuable.
Methodological influence: The pivot-based coordinate-free representation could influence broader graph learning beyond VRPs. The idea of defining node positions through distances to reference points, rather than explicit coordinates, has potential applications in any domain dealing with metric or quasimetric spaces (e.g., network optimization, molecular design). The WDAD mechanism provides a template for decoupling domain-invariant knowledge from task-specific adaptations in multi-task combinatorial optimization.
Limitations of impact scope: The method builds specifically on URS architecture and the autoregressive constructive paradigm. Transferability to improvement-based or non-autoregressive solvers is unclear. The paper also acknowledges that reduced reliance on constraint states can weaken performance on constraint-heavy variants seen during training.
4. Timeliness & Relevance
The paper addresses a timely need. The NCO community has been moving toward generalist solvers (MTPOMO, MVMoE, URS, GOAL), but the symmetric-asymmetric divide remains a significant barrier. With increasing attention to real-world routing applications (as evidenced by recent works on CVRPLIB and real-world benchmarks), bridging this gap is practically important. The work also connects to broader trends in foundation models and transfer learning for optimization.
5. Strengths & Limitations
Key Strengths:
Notable Weaknesses:
Reproducibility: Training requires ~5 days on a single RTX 4090, which is reasonable. All hyperparameters are specified. The paper references open-source code for baselines, though SPACE's own code availability is not mentioned.
Overall Assessment
SPACE makes a solid contribution by unifying symmetric and asymmetric VRP solving through a theoretically grounded coordinate-free representation. The experimental evidence is thorough and the improvements, particularly on asymmetric variants, are substantial. The work advances the state of generalist neural routing solvers toward greater practical applicability. The main limitations are the modest degradation on some symmetric variants and the reliance on weak oracle baselines for certain asymmetric problems.
Generated May 26, 2026
Comparison History (24)
Paper 1 offers a concrete, technically novel framework (pivot-based coordinate-free embedding + adaptive decoding) addressing a clear practical gap: unifying symmetric and asymmetric VRPs with demonstrated large-scale empirical validation across 110 variants and zero-shot generalization. This combination of methodological rigor, direct industrial applicability (logistics/routing), and timely relevance to generalist neural combinatorial solvers suggests strong near-term scientific and real-world impact. Paper 2 is broad and potentially influential conceptually, but appears more theory/synthesis-heavy with less clearly specified causal identification and intervention methodology, making impact less certain.
Paper 2 addresses a fundamental and economically critical algorithmic challenge (Vehicle Routing Problems) with a highly rigorous methodology, testing on 110 variants. By unifying symmetric and asymmetric routing, it offers profound implications for logistics and operations research. While Paper 1 provides a highly practical tool for 3D content creation, Paper 2 demonstrates broader algorithmic innovation and potential real-world impact across global supply chains and transportation networks.
While Paper 1 offers impressive engineering optimization for edge-AI deployment, Paper 2 tackles a fundamental limitation in neural combinatorial optimization. By creating a unified framework (SPACE) for both symmetric and asymmetric vehicle routing problems, it bridges a significant methodological gap. Its rigorous evaluation across 110 variants and theoretical novelty in coordinate-free embedding give it a deeper, long-lasting scientific impact in operations research and machine learning compared to the application-focused parameter reduction in Paper 1.
Paper 1 addresses a critical safety and reliability flaw in large reasoning models, which has broad implications across high-stakes fields like medicine. Its focus on LLM reasoning control offers higher timeliness, broader cross-disciplinary impact, and wider real-world applicability compared to the specialized neural combinatorial optimization focus of Paper 2.
Paper 2 presents a rigorous, well-defined technical contribution (SPACE framework) addressing a concrete limitation in neural combinatorial optimization with extensive experimental validation across 110 VRP variants. It offers clear methodological innovation with practical applications in logistics and operations research. Paper 1, while provocative, relies on auto-ethnographic methodology with a single subject, co-authored by the AI under study, raising serious methodological concerns about objectivity and reproducibility. Its claims about AI phenomenology and 'training strata' lack the empirical rigor needed for broad scientific impact.
Paper 2 likely has higher scientific impact: it introduces a broadly applicable representation/decoding framework unifying symmetric and asymmetric VRPs, a central class of combinatorial optimization problems with wide industrial relevance. The methodological contribution (pivot-based coordinate-free embedding, bidirectional Fréchet representation, adaptive decoding) is general and evaluated rigorously across 110 variants with zero-shot generalization—signals strong robustness and reusability. Paper 1 is valuable and timely for privacy-aware pediatric personalization, but its impact is more domain-specific and constrained by dataset scale/access, with broader uptake depending on community adoption of the benchmark.
Paper 1 addresses the rapidly expanding and highly relevant field of LLM agents, focusing on the critical challenges of personalization, proactiveness, and long-term memory. Benchmarks in this domain typically drive significant downstream research and have a broad cross-disciplinary impact. In contrast, while Paper 2 provides a strong methodological advance, its focus on vehicle routing problems is more specialized, resulting in a narrower potential impact compared to the widespread applicability of interactive AI agents.
Paper 1 addresses a broader and more impactful problem—unifying symmetric and asymmetric vehicle routing problems with a single neural solver across 110 VRP variants. Its novel SPACE framework with Fréchet representations and weight-decomposed decoding represents significant methodological innovation in neural combinatorial optimization. The breadth of impact across diverse routing problems and real-world applicability is substantial. Paper 2, while rigorous and practical in conformance checking, addresses a more niche problem within process mining with incremental improvements (combining LP with A*), limiting its cross-disciplinary impact.
Paper 2 (SPACE) likely has higher scientific impact due to broader applicability and cross-field relevance: a unified representation/decoder that generalizes zero-shot across symmetric and asymmetric VRPs targets a central, widely-used optimization class with direct logistics/transport applications. Methodologically, it proposes principled invariances (pivot-based coordinate-free embedding) and evaluates on a large suite (110 variants), suggesting robust generalization. Paper 1 is timely and useful for research integrity, but its impact is more domain-specific (peer-review workflows) and may be constrained by venue/process dependence despite providing a valuable benchmark.
Paper 2 proposes a fundamental methodological advancement by unifying symmetric and asymmetric vehicle routing problems, introducing novel representation and decoding mechanisms. Its extensive evaluation across 110 variants demonstrates high rigor and broad applicability in both machine learning and operations research. In contrast, Paper 1 primarily presents an engineering contribution by extending an existing method into a practical software library, which, while useful, offers less theoretical innovation and a narrower scope of fundamental scientific impact.
Paper 1 addresses a fundamental bottleneck in the rapidly expanding field of LLM-based agents: reliable and efficient planning. By setting a strategic research agenda and proposing a paradigm shift towards generating symbolic solvers, it has the potential to influence a broad range of AI and software engineering applications. While Paper 2 offers excellent methodological rigor and practical solutions for VRPs, Paper 1's conceptual framework impacts a much wider and highly active cross-disciplinary research area.
Paper 2 has higher estimated impact due to broader applicability and timeliness: a unified neural framework for both symmetric/asymmetric VRPs targets a widely used optimization family with immediate logistics/transport relevance. It proposes a novel representation (pivot-based coordinate-free embedding) plus decoding changes and is validated across 110 variants with zero-shot generalization, suggesting robustness and cross-domain utility within routing/ML/OR. Paper 1 offers strong rigor and large scaling gains for OOP/POMDP sensor selection, but the scope is more specialized and likely impacts a narrower community despite significant methodological advances.
Paper 2 (AVBench) likely has higher impact because benchmarks and evaluation protocols can become community standards, influencing many downstream model papers and enabling comparable progress across labs. Its automated, human-aligned, fine-grained metrics and learned evaluators address a timely bottleneck in audio-video generation, with clear applications in model selection, data filtering, and RLHF reward shaping. Methodologically, it proposes scalable supervision via controlled perturbations and probabilistic scoring tied to human judgment. Paper 1 is novel for VRP generalist solvers but targets a narrower domain.
Paper 1 addresses a fundamental limitation in neural combinatorial optimization by unifying symmetric and asymmetric routing problems, demonstrating results across 110 VRP variants. Its methodological contribution (coordinate-free embedding, weight-decomposed decoding) has broad applicability across operations research and AI. Paper 2 introduces a valuable but niche evaluation dataset for audio-based distress estimation in CBT, with impact primarily limited to mental health NLP. Paper 1's broader technical contribution, methodological novelty, and wider applicability across optimization and AI give it higher potential impact.
Paper 2 likely has higher impact: it targets the broadly urgent problem of reliable LLM deployment via selective prediction, proposing an inference-time prover–verifier protocol inspired by interactive proofs. The approach is timely, widely applicable across domains using LLMs (QA, agents, decision support), and offers a clear empirical evaluation framework (coverage–precision) plus robustness tests across model families and failure-mode analysis. Paper 1 is technically novel and rigorous for VRPs, with strong practical relevance in routing, but its impact is more domain-specific and narrower than reliability methods for foundation models.
Paper 1 focuses on automating symbolic regression, a foundational tool for scientific discovery. Improving the reliability of extracting physical laws from data has profound, cross-disciplinary implications for physics, biology, and chemistry. While Paper 2 offers a strong technical advancement for vehicle routing problems with high industrial utility, Paper 1's contribution directly accelerates fundamental scientific research itself, granting it a broader and more fundamental scientific impact.
Paper 1 addresses the critical and highly timely issue of LLM safety by innovatively bridging formal methods with AI testing. Its approach offers verifiable traceability and systematic guarantees, which are urgently needed as LLM integration expands across all sectors. While Paper 2 offers a strong contribution to optimization and routing, Paper 1's focus on AI safety grants it a much broader potential impact across multiple disciplines and real-world applications.
Paper 1 addresses a highly critical and timely challenge in artificial intelligence: optimizing the deployment and specialization of Large Language Models (LLMs) under strict resource constraints. Its modular approach (SkillWeave) has broad applicability across virtually all domains utilizing LLMs, offering significant real-world utility and performance gains (a 9B model beating a 32B model with 4x speedup). While Paper 2 presents a rigorous and innovative solution for Vehicle Routing Problems, its impact is largely confined to operations research and logistics, whereas Paper 1's contributions will likely influence a much wider spectrum of AI research and industry applications.
Paper 2 has higher potential scientific impact due to its extreme timeliness and broader implications for AI ecosystems. While Paper 1 offers a robust methodological advancement in combinatorial optimization for routing problems, Paper 2 provides the first large-scale empirical study of decentralized Agent-to-Agent networks. By exposing critical flaws in current A2A economies-such as easily manipulated scoring and lack of verifiable execution-Paper 2 directly informs the design, safety, and auditing of future autonomous AI agent platforms, an area currently experiencing massive cross-disciplinary growth and interest.
Paper 1 addresses a fundamental limitation in neural combinatorial optimization by unifying symmetric and asymmetric vehicle routing problems, demonstrating results across 110 VRP variants. Its novelty in coordinate-free embedding and broader applicability to real-world logistics gives it higher impact potential. Paper 2 presents a useful but incremental prompting technique for uncertainty detection in SLMs, limited to multiple-choice QA. Paper 1's methodological contributions (Fréchet representations, weight-decomposed decoding) and breadth of impact across combinatorial optimization are more substantial.