The multi-commodity flow (MCF) problem is a fundamental topic in network flow and combinatorial optimization, with broad applications in transportation, communication, and logistics, etc. Nowadays, the rapid expansion of allocation systems has posed challenges for existing optimization engines in balancing optimality and tractability. In this paper, we present Pram, the first ML-based method that leverages the reasoning power of multimodal language models (MLMs) for addressing the trade-off dilemma -- a great need of service providers. As part of our proposal, Pram (i) quickly computes high-quality allocations by dividing the original problem into local subproblems, which are then resolved by an MLM-powered "agent", and (ii) ensures global consistency by harmonizing these subproblems via a multi-agent reinforcement learning algorithm. Theoretically, we show that Pram, which learns to perform gradient descent in context, provably converges to the optimum within the family of MCF problems. Empirically, on real-world datasets and public topologies, Pram achieves performance comparable to, and in some cases even surpassing, linear programming solvers (very close to the optimal solution), and substantially lower runtimes (1 to 2 orders of magnitude faster). Moreover, Pram exhibits strong robustness (<10\% performance degradation under link failures or flow bursts), demonstrating MLM's generalization ability to unforeseen events. Pram is objective-agnostic and seamlessly integrates with mainstream allocation systems, providing a practical and scalable solution for future networks.

翻译：多商品流（MCF）问题是网络流与组合优化领域的一个基础课题，在交通、通信与物流等领域具有广泛应用。当前，分配系统的快速扩张对现有优化引擎在平衡最优性与可处理性方面提出了挑战。本文提出Pram，这是首个基于机器学习的方法，它利用多模态语言模型（MLMs）的推理能力来解决服务提供商迫切面临的权衡困境。作为我们方案的一部分，Pram（i）通过将原始问题分解为局部子问题，并由MLM驱动的“智能体”进行求解，从而快速计算高质量的分配方案；（ii）通过多智能体强化学习算法协调这些子问题，确保全局一致性。理论上，我们证明了Pram（其学习在上下文中执行梯度下降）可在MCF问题族内可证明地收敛至最优解。实证方面，在真实世界数据集与公共拓扑结构上，Pram实现了与线性规划求解器相当（非常接近最优解）甚至在某些情况下更优的性能，同时显著降低了运行时间（快1到2个数量级）。此外，Pram展现出强大的鲁棒性（在链路故障或流量突发情况下性能下降<10%），证明了MLM对未预见事件的泛化能力。Pram与目标无关，并可无缝集成到主流分配系统中，为未来网络提供了一个实用且可扩展的解决方案。