Min-max routing problems aim to minimize the maximum tour length among agents as they collaboratively visit all cities, i.e., the completion time. 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 a new deep-learning framework to solve large-scale min-max routing problems. We model the simultaneous decision-making of multiple agents as a sequential generation process, allowing the utilization of scalable deep-learning models for sequential decision-making. In the sequentially approximated problem, we propose a scalable contextual Transformer model, Equity-Transformer, which generates sequential actions considering an equitable workload among other agents. The effectiveness of Equity-Transformer is demonstrated through its superior performance in two representative min-max routing tasks: the min-max multiple traveling salesman problem (min-max mTSP) and the min-max multiple 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: https://github.com/kaist-silab/equity-transformer

翻译：最小最大路径规划问题旨在最小化多个智能体协同访问所有城市时的最大路径长度（即完成时间）。这类问题包含诸多具有实际应用价值但被公认为NP难的问题。现有方法在处理需要协调大量智能体覆盖数千个城市的大规模问题时面临挑战。本文提出一种新型深度学习框架用于求解大规模最小最大路径规划问题。我们将多智能体协同决策建模为序列生成过程，从而可利用可扩展的深度学习模型处理序列决策问题。在序列近似问题中，我们提出可扩展的上下文Transformer模型——Equity-Transformer，该模型在生成序列动作时考虑智能体间的公平工作负载分配。通过在两个代表性最小最大路径规划任务中的优越表现验证了Equity-Transformer的有效性：最小最大多旅行商问题（min-max mTSP）和最小最大多车辆取送货问题（min-max mPDP）。值得注意的是，在包含100台车辆和1000个城市的mTSP案例中，相比竞争性启发式算法（LKH3），我们的方法实现了约335倍的运行时间缩短和约53%的成本值降低。我们提供可复现的源代码：https://github.com/kaist-silab/equity-transformer