A Localized Method for the Multi-commodity Flow Problem

TL;DR

This paper presents a new theoretical framework and parallel algorithms for efficiently solving large-scale multicommodity flow problems by reformulating them as equilibrium searches with edge-separable convex optimization.

Contribution

It introduces a novel equilibrium-based formulation and develops the Potential Difference Reduction algorithms, enabling highly parallelized and scalable solutions for multicommodity flow feasibility.

Findings

Algorithms achieve high efficiency on benchmark problems

Parallelizable approach scales well with problem size

Heuristics provide computationally cheaper alternatives

Abstract

This paper introduces a novel theoretical framework and a suite of highly efficient, parallelizable algorithms for solving the large-scale multicommodity flow (MCF) feasibility problem. We reframe the classical constraint-satisfaction problem as an equilibrium search. By defining vertex-specific height functions and edge-specific congestion functions, we establish a new, intuitive optimality condition: a flow is feasible if and only if it corresponds to a zero-stable pseudo-flow, where all potential differences across the network are resolved. This condition gives rise to an edge-separable convex optimization problem, whose structure is inherently suited for massive parallelization. Based on this formulation, we develop a family of Potential Difference Reduction (PDR) algorithms. Our primary method, provably convergent, solves an exact quadratic programming subproblem for each edge in parallel. To address scenarios with a very large number of commodities, we propose two computationally cheaper heuristics based on adaptive gradient descent. Extensive numerical experiments on well-known benchmarks demonstrate the framework's remarkable performance. This work provides a powerful new approach for tackling large-scale MCF problems, while also identifying the formal analysis of the convergence rate as a promising direction for future research.