# Learning with Foresight: Enhancing Neural Routing Policy via Multi-Node Lookahead Prediction

arXiv: https://arxiv.org/abs/2605.19975 · HTML: https://mlsanity.com/paper/2605.19975/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 48 references that are in the map.
- First posted on arXiv 19 May 2026.

## Builds on (most cited first, 28 of 28)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Pointer Networks](https://mlsanity.com/paper/1506.03134/) (2015)
- [Attention, Learn to Solve Routing Problems!](https://mlsanity.com/paper/1803.08475/) (2018)
- [Better & Faster Large Language Models via Multi-token Prediction](https://mlsanity.com/paper/2404.19737/) (2024)
- [POMO: Policy Optimization with Multiple Optima for Reinforcement Learning](https://mlsanity.com/paper/2010.16011/) (2020)
- [Blockwise Parallel Decoding for Deep Autoregressive Models](https://mlsanity.com/paper/1811.03115/) (2018)
- [ProphetNet: Predicting Future N-gram for Sequence-to-Sequence Pre-training](https://mlsanity.com/paper/2001.04063/) (2020)
- [Learning Improvement Heuristics for Solving Routing Problems](https://mlsanity.com/paper/1912.05784/) (2019)
- [The pitfalls of next-token prediction](https://mlsanity.com/paper/2403.06963/) (2024)
- [Neural Combinatorial Optimization with Heavy Decoder: Toward Large Scale Generalization](https://mlsanity.com/paper/2310.07985/) (2023)
- [Hybrid Genetic Search for the CVRP: Open-Source Implementation and SWAP* Neighborhood](https://mlsanity.com/paper/2012.10384/) (2020)
- [Efficient Active Search for Combinatorial Optimization Problems](https://mlsanity.com/paper/2106.05126/) (2021)
- [BQ-NCO: Bisimulation Quotienting for Efficient Neural Combinatorial Optimization](https://mlsanity.com/paper/2301.03313/) (2023)
- [Towards Omni-generalizable Neural Methods for Vehicle Routing Problems](https://mlsanity.com/paper/2305.19587/) (2023)
- [Towards Generalizable Neural Solvers for Vehicle Routing Problems via Ensemble with Transferrable Local Policy](https://mlsanity.com/paper/2308.14104/) (2023)
- [RouteFinder: Towards Foundation Models for Vehicle Routing Problems](https://mlsanity.com/paper/2406.15007/) (2024)
- [INViT: A Generalizable Routing Problem Solver with Invariant Nested View Transformer](https://mlsanity.com/paper/2402.02317/) (2024)
- [Rethinking Light Decoder-based Solvers for Vehicle Routing Problems](https://mlsanity.com/paper/2503.00753/) (2025)
- [PolyNet: Learning Diverse Solution Strategies for Neural Combinatorial Optimization](https://mlsanity.com/paper/2402.14048/) (2024)
- [Large Language Models as End-to-end Combinatorial Optimization Solvers](https://mlsanity.com/paper/2509.16865/) (2025)
- [ASP: Learn a Universal Neural Solver!](https://mlsanity.com/paper/2303.00466/) (2023)
- [Multi-Token Prediction Needs Registers](https://mlsanity.com/paper/2505.10518/) (2025)
- [Neural Deconstruction Search for Vehicle Routing Problems](https://mlsanity.com/paper/2501.03715/) (2025)
- [BOPO: Neural Combinatorial Optimization via Best-anchored and Objective-guided Preference Optimization](https://mlsanity.com/paper/2503.07580/) (2025)
- [Bridging Large Language Models and Optimization: A Unified Framework for Text-attributed Combinatorial Optimization](https://mlsanity.com/paper/2408.12214/) (2024)
- [CAMP: Collaborative Attention Model with Profiles for Vehicle Routing Problems](https://mlsanity.com/paper/2501.02977/) (2025)
- [Symmetric Replay Training: Enhancing Sample Efficiency in Deep Reinforcement Learning for Combinatorial Optimization](https://mlsanity.com/paper/2306.01276/) (2023)
- [Neural Genetic Search in Discrete Spaces](https://mlsanity.com/paper/2502.10433/) (2025)

## Built on by (most cited first, 1 of 1)

- [Depot-Closed Multi-Component Construction for Neural Vehicle Routing](https://mlsanity.com/paper/2609.35066/) (2026)

## Abstract (arXiv)

Neural policies have shown promise in solving vehicle routing problems due to their reduced reliance on handcrafted heuristics. However, current training paradigms suffer from a fundamental limitation: they primarily focus on next-node prediction for solution construction, resulting in myopic decision-making that undermines long-horizon planning capacity. To this end, we introduce Multi-node Lookahead Prediction (MnLP), a novel training strategy that extends the supervised learning paradigm to predict multiple future nodes simultaneously. We incorporate causal and discardable MnLP modules that operate exclusively during training, facilitating models to anticipate multi-step decisions while preserving inference-time efficiency. By incorporating multi-depth auxiliary supervision into the loss function, MnLP equips neural policies with the ability of long-range contextual understanding. Experimentally, MnLP outperforms existing training methods, improving the generalization capability of neural policies across various problem sizes, distributions, and real-world benchmarks. Moreover, MnLP can be seamlessly integrated into diverse neural architectures without introducing additional inference overhead.