Pram: A New Paradigm for Solving Multi-Commodity Flow Problems Using Multimodal Language Models

The ICLR 2026 accepted paper Pram proposes an innovative "divide-and-conquer" framework that uses multimodal large language models to solve complex multi-commodity flow optimization problems, achieving significant breakthroughs in computational efficiency and solution quality, and injecting an AI-driven new methodology into the field of operations research and optimization.

The Multi-Commodity Flow Problem (MCFP) is a classic challenge in operations research, widely used in scenarios such as logistics planning and traffic scheduling. Traditional methods like linear programming and heuristic algorithms face challenges of high computational complexity and slow convergence in large-scale instances. Large language models have great potential in reasoning and planning tasks, but how to apply them to structured optimization problems remains an open question.

The Pram framework adopts a three-step strategy of "Divide, Harmonize, Then Conquer":

1. Divide: Use multimodal visual understanding to convert network topology into graph representation and intelligently partition into independent sub-networks;
2. Harmonize: Coordinate dependencies through cross-subnetwork attention mechanisms in the harmony layer, and convert numerical objectives into semantic constraints;
3. Conquer: Assign to backend solvers (traditional or neural) for processing and integrate sub-problem solutions. Technical highlights include graph structure-aware multimodal encoding, progressive reasoning chains, and adaptive solver strategy selection.

Evaluation on standard MCFP benchmark datasets (synthetic networks + real logistics data) shows:

• Solution quality: Most instances are comparable to or better than commercial solvers, with obvious advantages in large-scale instances (nodes >1000);
• Computational efficiency: Convergence time for large instances is reduced from hours to minutes;
• Ablation experiments: Removing the visual encoding module leads to decreased decomposition quality, and removing the harmonization phase significantly reduces the feasibility of global solutions, verifying the framework's rationality.

The Pram methodology can be extended to combinatorial optimization problems such as the Traveling Salesman Problem and Vehicle Routing Problem; it provides a new paradigm for combining domain knowledge with AI in industry, lowering the modeling threshold; it represents the practice of AI for Science in operations research, proving that large language models can handle structured scientific problems. In the future, AI-driven intelligent optimization is expected to bring efficiency revolutions in industries such as logistics and manufacturing.