Divide, Harmonize, Then Conquer It: Shooting Multi-Commodity Flow Problems with Multimodal Language Models

TL;DR

Pram is a novel ML-based approach using multimodal language models to efficiently solve multi-commodity flow problems by dividing, solving locally, and harmonizing globally, achieving near-optimal solutions with faster runtimes and robustness.

Contribution

This paper introduces Pram, the first ML-based method leveraging multimodal language models for multi-commodity flow problems, combining problem division, local solving, and global harmonization.

Findings

Achieves near-optimal solutions comparable to linear programming solvers.

Runs 10 to 100 times faster than traditional solvers.

Maintains robustness under network failures and flow bursts.

Abstract

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.