Solving Large Multicommodity Network Flow Problems on GPUs
Fangzhao Zhang, Stephen Boyd
TL;DR
This paper tackles large-scale all-pairs multicommodity network flow by reformulating the problem with destination-based flow aggregation, reducing the number of variables. It develops a GPU-friendly solver using the primal-dual hybrid gradient (PDHG) method, with careful proximal operators and adaptive step sizes, enabling exact solutions to nonlinear convex MCF problems on GPUs. The approach achieves substantial speedups (up to 100–1000×) over CPU-based solvers and scales to problems with up to $nm=10^9$ variables, outperforming traditional methods and enabling larger, more complex networks. An open-source implementation and comprehensive experiments validate the practical impact for fast, scalable network optimization on modern hardware.
Abstract
We consider the all-pairs multicommodity network flow problem on a network with capacitated edges. The usual treatment keeps track of a separate flow for each source-destination pair on each edge; we rely on a more efficient formulation in which flows with the same destination are aggregated, reducing the number of variables by a factor equal to the size of the network. Problems with hundreds of nodes, with a total number of variables on the order of a million, can be solved using standard generic interior-point methods on CPUs; we focus on GPU-compatible algorithms that can solve such problems much faster, and in addition scale to much larger problems, with up to a billion variables. Our method relies on the primal-dual hybrid gradient algorithm, and exploits several specific features of the problem for efficient GPU computation. Numerical experiments show that our primal-dual multicommodity network flow method accelerates state of the art generic commercial solvers by $100\times$ to $1000\times$, and scales to problems that are much larger. We provide an open source implementation of our method.