# Neuro CROSS exchange: Learning to CROSS exchange to solve realistic vehicle routing problems

arXiv: https://arxiv.org/abs/2206.02771 · HTML: https://mlsanity.com/paper/2206.02771/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 33 references that are in the map.
- First posted on arXiv 6 Jun 2022.

## Builds on (most cited first, 18 of 18)

- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Pointer Networks](https://mlsanity.com/paper/1506.03134/) (2015)
- [Neural Combinatorial Optimization with Reinforcement Learning](https://mlsanity.com/paper/1611.09940/) (2016)
- [Learning Combinatorial Optimization Algorithms over Graphs](https://mlsanity.com/paper/1704.01665/) (2017)
- [POMO: Policy Optimization with Multiple Optima for Reinforcement Learning](https://mlsanity.com/paper/2010.16011/) (2020)
- [Reinforcement Learning for Solving the Vehicle Routing Problem](https://mlsanity.com/paper/1802.04240/) (2018)
- [An Efficient Graph Convolutional Network Technique for the Travelling Salesman Problem](https://mlsanity.com/paper/1906.01227/) (2019)
- [Learning to Perform Local Rewriting for Combinatorial Optimization](https://mlsanity.com/paper/1810.00337/) (2018)
- [Combinatorial Optimization with Graph Convolutional Networks and Guided Tree Search](https://mlsanity.com/paper/1810.10659/) (2018)
- [Learning Improvement Heuristics for Solving Routing Problems](https://mlsanity.com/paper/1912.05784/) (2019)
- [Neural Large Neighborhood Search for the Capacitated Vehicle Routing Problem](https://mlsanity.com/paper/1911.09539/) (2019)
- [Learning to Delegate for Large-scale Vehicle Routing](https://mlsanity.com/paper/2107.04139/) (2021)
- [NeuroLKH: Combining Deep Learning Model with Lin-Kernighan-Helsgaun Heuristic for Solving the Traveling Salesman Problem](https://mlsanity.com/paper/2110.07983/) (2021)
- [Multi-Decoder Attention Model with Embedding Glimpse for Solving Vehicle Routing Problems](https://mlsanity.com/paper/2012.10638/) (2020)
- [Deep Policy Dynamic Programming for Vehicle Routing Problems](https://mlsanity.com/paper/2102.11756/) (2021)
- [Learning Collaborative Policies to Solve NP-hard Routing Problems](https://mlsanity.com/paper/2110.13987/) (2021)
- [ScheduleNet: Learn to solve multi-agent scheduling problems with reinforcement learning](https://mlsanity.com/paper/2106.03051/) (2021)
- [Learning to Solve Vehicle Routing Problems with Time Windows through Joint Attention](https://mlsanity.com/paper/2006.09100/) (2020)

## Built on by (most cited first, 2 of 2)

- [Ant Colony Sampling with GFlowNets for Combinatorial Optimization](https://mlsanity.com/paper/2403.07041/) (2024)
- [Heuristics for Vehicle Routing Problem: A Survey and Recent Advances](https://mlsanity.com/paper/2303.04147/) (2023)

## Abstract (arXiv)

CROSS exchange (CE), a meta-heuristic that solves various vehicle routing problems (VRPs), improves the solutions of VRPs by swapping the sub-tours of the vehicles. Inspired by CE, we propose Neuro CE (NCE), a fundamental operator of learned meta-heuristic, to solve various VRPs while overcoming the limitations of CE (i.e., the expensive O(n4) search cost). NCE employs a graph neural network to predict the cost-decrements (i.e., results of CE searches) and utilizes the predicted cost-decrements as guidance for search to decrease the search cost to O(n2). As the learning objective of NCE is to predict the cost-decrement, the training can be simply done in a supervised fashion, whose training samples can be prepared effortlessly. Despite the simplicity of NCE, numerical results show that the NCE trained with flexible multi-depot VRP (FMDVRP) outperforms the meta-heuristic baselines. More importantly, it significantly outperforms the neural baselines when solving distinctive special cases of FMDVRP (e.g., MDVRP, mTSP, CVRP) without additional training.