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

TL;DR

Equity-Transformer reframes min-max multi-agent routing as sequential generation, generating one long action sequence that encodes $M$ agent tours with virtual order via multi-agent positional encoding and an equity context encoding. Trained with REINFORCE and a shared baseline, the model directly minimizes the maximum tour length $\mathcal{L}_{\text{cost}}$ by leveraging a Transformer encoder–decoder architecture that injects equity information at decoding time. Empirically, Equity-Transformer achieves orders-of-magnitude speedups and substantial cost reductions over classical heuristics (e.g., LKH3) and learning-based baselines on large-scale min-max mTSP and mPDP tasks, handling up to $N=5000$ cities and $M=500$ agents while maintaining a Pareto frontier between runtime and solution quality. The approach demonstrates robustness to distributional shifts and scales to real-world datasets, offering a practical, scalable framework for equitable, high-quality multi-agent routing. $L_{\text{cost}}(\bm{a};\bm{x})=\max_m L(\bm{a}^m;\bm{x})$, $x_1$ depot, and $x_{N+1}=\cdots=x_{N+M}=x_1$ are central to the formulation, while equity biases are implemented through $\text{MPE}$ and $\text{CE}$ within a Transformer.

Abstract

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: \url{https://github.com/kaist-silab/equity-transformer}.

Figures (6)

• Figure 1: Illustration of parallel planning and sequential planning on min-max VRPs. In sequential planning, when an action selects a depot index (gray colored $a_t$), the planning for currently active agent is terminated to start planning for the new agent that corresponds to the selected depot.
• Figure 2: Illustration of Equity-Transformer. The $L$ stands for the number of sequential layers, where we set $L=3$.
• Figure 3: Illustration of equity context encoding
• Figure 4: Time-performance trade-off graph for mTSP. The left and bottom indicate the Pareto frontier.
• Figure 5: Time-performance trade-off graph for mPDP. The left and bottom indicate the Pareto frontier.