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Towards Efficient Constraint Handling in Neural Solvers for Routing Problems

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
Routing Problems; Deep Reinforcement Learning; Constraint Handling; Combinatorial Optimization
Abstract:

Neural solvers have achieved impressive progress in addressing simple routing problems, particularly excelling in computational efficiency. However, their advantages under complex constraints remain nascent, for which current constraint-handling schemes via feasibility masking or implicit feasibility awareness can be inefficient or inapplicable for hard constraints. In this paper, we present Construct-and-Refine (CaR), the first general and efficient constraint-handling framework for neural routing solvers based on explicit learning-based feasibility refinement. Unlike prior construction-search hybrids that target reducing optimality gaps through heavy improvements yet still struggle with hard constraints, CaR achieves efficient constraint handling by designing a joint training framework that guides the construction module to generate diverse and high-quality solutions well-suited for a lightweight improvement process, e.g., 10 steps versus 5k steps in prior work. Moreover, CaR presents the first use of construction-improvement-shared representation, enabling potential knowledge sharing across paradigms by unifying the encoder, especially in more complex constrained scenarios. We evaluate CaR on typical hard routing constraints to showcase its broader applicability. Results demonstrate that CaR achieves superior feasibility, solution quality, and efficiency compared to both classical and neural state-of-the-art solvers.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
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Overview

Overall Novelty Assessment

The paper introduces Construct-and-Refine (CaR), a framework combining neural construction with explicit learning-based feasibility refinement for routing problems under hard constraints. It resides in the 'Explicit Feasibility Refinement and Construction-Improvement Hybrids' leaf, which contains only two papers including this one. This sparse population suggests the specific combination of lightweight learned refinement (10 steps versus thousands in prior work) with joint training for construction-improvement synergy represents a relatively underexplored direction within the broader constraint-handling landscape.

The taxonomy reveals substantial activity in neighboring areas: 'Penalty-Based and Augmented Lagrangian Constraint Handling' and 'Feasibility Masking and Action Filtering' each contain two papers addressing constraint enforcement through different mechanisms. The parent branch 'Constraint Enforcement Mechanisms' sits alongside 'Constraint-Aware Neural Architecture Design' (nine papers across three leaves) and 'Hybrid and Hierarchical Solution Paradigms' (seven papers). CaR's position bridges these areas by combining explicit refinement with architectural innovation (shared encoder), distinguishing it from pure masking approaches or heavy search-based improvements that dominate adjacent leaves.

Among thirty candidates examined, none clearly refute the three core contributions. The CaR framework itself (zero refutable candidates from ten examined), the learning-based feasibility refinement scheme (zero from ten), and cross-paradigm representation learning via shared encoder (zero from ten) all appear novel within this limited search scope. The statistics suggest that while construction-improvement hybrids exist, the specific combination of joint training, lightweight refinement, and shared representations across construction and improvement modules has not been documented in the examined literature.

Based on the top-thirty semantic matches and taxonomy structure, the work appears to occupy a distinctive position combining elements from multiple established directions. The analysis covers recent work in neural routing solvers but cannot claim exhaustive coverage of all hybrid methods or representation-sharing techniques. The sparse leaf population and absence of refuting candidates suggest meaningful novelty, though broader literature may contain related ideas not captured in this focused search.

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
30
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: constraint handling in neural solvers for routing problems. The field has evolved into a rich landscape organized around how neural methods respect and enforce the diverse constraints inherent in vehicle routing and related combinatorial optimization tasks. At the highest level, the taxonomy distinguishes between Constraint-Aware Neural Architecture Design (embedding constraint knowledge directly into model structures), Constraint Enforcement Mechanisms (techniques that actively ensure feasibility during or after solution construction), and Cross-Problem Generalization Frameworks (methods aiming to transfer learned policies across problem variants). Additional branches address Specialized Constraint Types and Problem Variants (tackling heterogeneous fleets, time windows, capacity limits, and domain-specific rules), Hybrid and Hierarchical Solution Paradigms (combining neural construction with classical improvement or decomposition strategies), Memory and Adaptation Mechanisms (enabling solvers to recall past solutions or adapt online), Graph Neural Network-Based Routing (leveraging relational inductive biases), Application-Specific Routing Contexts (from electric vehicle charging to satellite constellations), Physics-Informed and Domain-Knowledge Integration (injecting hard physical or operational rules), and Auxiliary Techniques and Supporting Methods (supporting tools like search heuristics and representation learning). Within this landscape, a particularly active line of work focuses on Explicit Feasibility Refinement and Construction-Improvement Hybrids, where neural models generate initial solutions that are then repaired or polished to satisfy hard constraints. Efficient Constraint Handling[0] sits squarely in this branch, emphasizing streamlined post-construction repair to ensure feasibility without sacrificing solution quality. Nearby, Flexible Neural kOpt[11] explores learned local search operators that iteratively refine tours while respecting constraints, illustrating a complementary improvement-focused strategy. In contrast, works like Complex Constraints VRP[2] and Generalizable Neural Solvers[3] push toward architectures that internalize constraint logic from the outset, reducing the need for explicit repair. The interplay between construction-then-repair versus constraint-aware generation remains a central trade-off: the former offers modularity and ease of integration with classical methods, while the latter promises end-to-end learning but often requires more sophisticated architectures and training regimes. Efficient Constraint Handling[0] exemplifies the pragmatic appeal of hybrid refinement, balancing neural flexibility with the reliability of explicit feasibility checks.

Claimed Contributions

Construct-and-Refine (CaR) framework for efficient constraint handling

The authors introduce CaR, a novel framework that combines neural construction and refinement modules through joint training. Unlike prior methods requiring thousands of improvement steps, CaR achieves efficient constraint handling by generating diverse, high-quality initial solutions that enable rapid refinement in as few as 10 steps, guided by specially designed loss functions.

10 retrieved papers
Learning-based feasibility refinement scheme

The authors propose a new constraint-handling paradigm called feasibility refinement that explicitly learns to refine infeasible solutions in very few post-construction steps. This scheme addresses limitations of existing approaches (feasibility masking and implicit feasibility awareness) that become inefficient or inapplicable for hard-constrained routing problems.

10 retrieved papers
Cross-paradigm representation learning via shared encoder

The authors present the first use of construction-improvement-shared representation by unifying the encoder across both paradigms. This enables potential knowledge sharing and enhances feasibility awareness, particularly improving performance in more complex constrained scenarios compared to using separate encoders.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Construct-and-Refine (CaR) framework for efficient constraint handling

The authors introduce CaR, a novel framework that combines neural construction and refinement modules through joint training. Unlike prior methods requiring thousands of improvement steps, CaR achieves efficient constraint handling by generating diverse, high-quality initial solutions that enable rapid refinement in as few as 10 steps, guided by specially designed loss functions.

Contribution

Learning-based feasibility refinement scheme

The authors propose a new constraint-handling paradigm called feasibility refinement that explicitly learns to refine infeasible solutions in very few post-construction steps. This scheme addresses limitations of existing approaches (feasibility masking and implicit feasibility awareness) that become inefficient or inapplicable for hard-constrained routing problems.

Contribution

Cross-paradigm representation learning via shared encoder

The authors present the first use of construction-improvement-shared representation by unifying the encoder across both paradigms. This enables potential knowledge sharing and enhances feasibility awareness, particularly improving performance in more complex constrained scenarios compared to using separate encoders.

Table of Contents