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作访问所有城市时的最大行程长度，即完成时间。这些问题包含重要的实际应用，但被公认为NP困难问题。现有方法面临挑战，特别是在需要协调大量智能体覆盖数千个城市的大规模问题中。本文提出一种新的深度学习框架用于解决大规模最小最大路径规划问题。我们将多个智能体的同步决策建模为序贯生成过程，从而能够利用可扩展的深度学习模型进行序贯决策。在序贯近似问题中，我们提出一种可扩展的上下文Transformer模型——公平性Transformer（Equity-Transformer），该模型在生成序贯动作时考虑智能体之间的工作负载公平性。通过两个代表性最小最大路径规划任务（最小最大多旅行商问题（min-max mTSP）和最小最大多取送货问题（min-max mPDP））的卓越性能，验证了公平性Transformer的有效性。值得注意的是，在mTSP的100辆车和1000个城市场景下，我们的方法与竞争性启发式算法（LKH3）相比，运行时间减少约335倍，成本值降低约53%。我们提供可复现的源代码：https://github.com/kaist-silab/equity-transformer