# Equity-Transformer: Solving NP-hard Min-Max Routing Problems as Sequential Generation with Equity Context

arXiv: https://arxiv.org/abs/2306.02689 · HTML: https://mlsanity.com/paper/2306.02689/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 40 references that are in the map.
- First posted on arXiv 5 Jun 2023.

## Builds on (most cited first, 20 of 20)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Attention, Learn to Solve Routing Problems!](https://mlsanity.com/paper/1803.08475/) (2018)
- [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)
- [DIFUSCO: Graph-based Diffusion Solvers for Combinatorial Optimization](https://mlsanity.com/paper/2302.08224/) (2023)
- [Learning to Perform Local Rewriting for Combinatorial Optimization](https://mlsanity.com/paper/1810.00337/) (2018)
- [Learning Improvement Heuristics for Solving Routing Problems](https://mlsanity.com/paper/1912.05784/) (2019)
- [Generalize a Small Pre-trained Model to Arbitrarily Large TSP Instances](https://mlsanity.com/paper/2012.10658/) (2020)
- [Sym-NCO: Leveraging Symmetricity for Neural Combinatorial Optimization](https://mlsanity.com/paper/2205.13209/) (2022)
- [Learning to Delegate for Large-scale Vehicle Routing](https://mlsanity.com/paper/2107.04139/) (2021)
- [Learning 2-opt Heuristics for the Traveling Salesman Problem via Deep Reinforcement Learning](https://mlsanity.com/paper/2004.01608/) (2020)
- [Efficient Active Search for Combinatorial Optimization Problems](https://mlsanity.com/paper/2106.05126/) (2021)
- [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)
- [Simulation-guided Beam Search for Neural Combinatorial Optimization](https://mlsanity.com/paper/2207.06190/) (2022)
- [Learning Generalizable Models for Vehicle Routing Problems via Knowledge Distillation](https://mlsanity.com/paper/2210.07686/) (2022)
- [Heterogeneous Attentions for Solving Pickup and Delivery Problem via Deep Reinforcement Learning](https://mlsanity.com/paper/2110.02634/) (2021)
- [Efficient Neural Neighborhood Search for Pickup and Delivery Problems](https://mlsanity.com/paper/2204.11399/) (2022)
- [A Deep Reinforcement Learning Approach for Solving the Traveling Salesman Problem with Drone](https://mlsanity.com/paper/2112.12545/) (2021)
- [A Comprehensive Survey on the Multiple Travelling Salesman Problem: Applications, Approaches and Taxonomy](https://mlsanity.com/paper/2102.12772/) (2021)

## Built on by (most cited first, 16 of 16)

- [ReEvo: Large Language Models as Hyper-Heuristics with Reflective Evolution](https://mlsanity.com/paper/2402.01145/) (2024)
- [Ant Colony Sampling with GFlowNets for Combinatorial Optimization](https://mlsanity.com/paper/2403.07041/) (2024)
- [Neural Combinatorial Optimization Algorithms for Solving Vehicle Routing Problems: A Comprehensive Survey with Perspectives](https://mlsanity.com/paper/2406.00415/) (2024)
- [DPN: Decoupling Partition and Navigation for Neural Solvers of Min-max Vehicle Routing Problems](https://mlsanity.com/paper/2405.17272/) (2024)
- [PARCO: Parallel AutoRegressive Models for Multi-Agent Combinatorial Optimization](https://mlsanity.com/paper/2409.03811/) (2024)
- [Efficient Neural Combinatorial Optimization Solver for the Min-max Heterogeneous Capacitated Vehicle Routing Problem](https://mlsanity.com/paper/2507.21386/) (2025)
- [Learning to Solve the Min-Max Mixed-Shelves Picker-Routing Problem via Hierarchical and Parallel Decoding](https://mlsanity.com/paper/2502.10233/) (2025)
- [RL-SPH: Learning to Achieve Feasible Solutions for Integer Linear Programs](https://mlsanity.com/paper/2411.19517/) (2024)
- [USPR: Learning a Unified Solver for Profiled Routing](https://mlsanity.com/paper/2505.05119/) (2025)
- [Solving the Min-Max Multiple Traveling Salesmen Problem via Learning-Based Path Generation and Optimal Splitting](https://mlsanity.com/paper/2508.17087/) (2025)
- [End-to-end Deep Reinforcement Learning for Stochastic Multi-objective Optimization in C-VRPTW](https://mlsanity.com/paper/2512.01518/) (2025)
- [A Coalition Game for On-demand Multi-modal 3D Automated Delivery System](https://mlsanity.com/paper/2412.17252/) (2024)
- [Synergizing Reinforcement Learning and Genetic Algorithms for Neural Combinatorial Optimization](https://mlsanity.com/paper/2506.09404/) (2025)
- [Multi-Action Self-Improvement for Neural Combinatorial Optimization](https://mlsanity.com/paper/2510.12273/) (2025)
- [A Unified Knowledge Embedded Reinforcement Learning-based Framework for Generalized Capacitated Vehicle Routing Problems](https://mlsanity.com/paper/2605.14416/) (2026)
- [WeCon: An Efficient Weight-Conditioned Neural Solver for Multi-Objective Combinatorial Optimization Problems](https://mlsanity.com/paper/2605.22876/) (2026)

## Abstract (arXiv)

Min-max routing problems aim to minimize the maximum tour length among multiple agents by having agents conduct tasks in a cooperative manner. These problems include impactful real-world applications but are known as NP-hard. Existing methods are facing challenges, particularly in large-scale problems that require the coordination of numerous agents to cover thousands of cities. This paper proposes Equity-Transformer to solve large-scale min-max routing problems. First, we employ sequential planning approach to address min-max routing problems, allowing us to harness the powerful sequence generators (e.g., Transformer). Second, we propose key inductive biases that ensure equitable workload distribution among agents. The effectiveness of Equity-Transformer is demonstrated through its superior performance in two representative min-max routing tasks: the min-max multi-agent traveling salesman problem (min-max mTSP) and the min-max multi-agent pick-up and delivery problem (min-max mPDP). Notably, our method achieves significant reductions of runtime, approximately 335 times, and cost values of about 53% compared to a competitive heuristic (LKH3) in the case of 100 vehicles with 1,000 cities of mTSP. We provide reproducible source code: urlhttps://github.com/kaist-silab/equity-transformer.