Tree-based formulation for the multi-commodity flow problem

2025-09-29

AI Summary

This paper addresses the scalability challenge in large-scale minimum-cost multicommodity flow problems where the number of commodities |K| vastly exceeds the number of sources |S|, rendering conventional linear programming (LP) formulations intractable due to exponential constraint growth. We propose a novel modeling framework based on source grouping and tree-structured flow representations. Specifically, flows originating from each source are expressed as convex combinations of spanning trees rooted at that source, with tree variables dynamically generated via column generation. To further reduce complexity, we introduce a source-based decomposition strategy, ensuring constraint count depends solely on |S|—not |K|. This approach drastically compresses model dimensionality, enabling scalable optimization on instances with up to one million commodities and one hundred thousand nodes. Empirical evaluation shows approximately 10× speedup over direct LP solving and consistent superiority over classical path-based column generation methods.

Technology Category

Constraint Satisfaction and Optimization: Distributed CSP/OptimizationSearch and Optimization: Combinatorial OptimizationMachine Learning: Scalability of ML Systems

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsEconomics, Online Markets and Human Computation: Cost models of using LLMs in production systemsSecurity and Privacy: Large-scale security measurements

Abstract

We introduce a tree-based formulation for the minimum-cost multi-commodity flow problem that addresses large-scale instances. The method decomposes the source-based model by representing flows as convex combinations of trees rooted at source nodes, and solves the resulting formulation with column generation. The number of demand constraints now depends on the number of sources $|S|$, not commodities $|K|$, yielding a compact master problem when $|S| ll |K|$. We conduct a computational study comparing tree-based decomposition against path-based column generation and direct LP solving. The results show speed-ups of up to one order of magnitude over direct LP solving, and improved scalability compared to path-based formulations. Tree-based decomposition enables solving instances with millions of commodities and hundreds of thousands of nodes. This makes it well-suited for applications in transportation and logistics networks where multiple demands often share common origins.

Problem

• Addresses large-scale multi-commodity flow problems efficiently
• Reduces problem complexity when sources are fewer than commodities
• Enables solving massive instances with millions of commodities

Innovation

• Tree-based decomposition for multi-commodity flow
• Column generation with source-rooted trees
• Compact formulation reducing constraint complexity