Block Coordinate Descent Network Simplex Methods for Optimal Transport

TL;DR

The paper introduces BCDNS, a novel method combining network simplex and block coordinate descent for large-scale optimal transport problems, offering efficiency and exact solutions.

Contribution

It develops BCDNS, a new algorithm that decomposes OT problems into smaller subproblems, ensuring finite termination and improved computational efficiency.

Findings

BCDNS matches classical NS in accuracy

Reduces memory compared to Sinkhorn

Achieves up to tens of times speed-up

Abstract

We propose the Block Coordinate Descent Network Simplex (BCDNS) method for solving large-scale discrete Optimal Transport (OT) problems. BCDNS integrates the Network Simplex (NS) algorithm with a block coordinate descent (BCD) strategy, decomposing the full problem into smaller subproblems per iteration and reusing basis variables to ensure feasibility. We prove that BCDNS terminates in a finite number of iterations with an exact optimal solution, and we characterize its per-iteration complexity as O(s N), where s is a user-defined parameter in (0,1) and N is the total number of variables. Numerical experiments demonstrate that BCDNS matches the classical NS method in solution accuracy, reduces memory footprint compared to the Sinkhorn algorithm, achieves speed-ups of up to tens of times over the classical NS method, and exhibits runtime comparable to a high-precision Sinkhorn implementation.