# A Column Generation Approach to the Discrete Barycenter Problem

## TL;DR

This paper introduces two column generation methods to efficiently compute the discrete Wasserstein barycenter, significantly reducing memory usage and computation time compared to traditional linear programming approaches.

## Contribution

It develops and analyzes two novel column generation strategies, including a Dantzig-Wolfe decomposition, for solving large-scale discrete barycenter problems.

## Key findings

- Both strategies outperform full linear programming in speed.
- Memory requirements are dramatically reduced.
- Performance varies depending on input data.

## Abstract

The discrete Wasserstein barycenter problem is a minimum-cost mass transport problem for a set of discrete probability measures. Although an exact barycenter is computable through linear programming, the underlying linear program can be extremely large. For worst-case input, a best known linear programming formulation is exponential in the number of variables, but has a low number of constraints, making it an interesting candidate for column generation. In this paper, we devise and study two column generation strategies: a natural one based on a simplified computation of reduced costs, and one through a Dantzig-Wolfe decomposition. For the latter, we produce efficiently solvable subproblems, namely, a pricing problem in the form of a classical transportation problem. The two strategies begin with an efficient computation of an initial feasible solution. While the structure of the constraints leads to the computation of the reduced costs of all remaining variables for setup, both approaches may outperform a computation using the full program in speed, and dramatically so in memory requirement. In our computational experiments, we exhibit that, depending on the input, either strategy can become a best choice.