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The discrete Wasserstein barycenter problem is a minimum-cost mass transportproblem for a set of discrete probability measures. Although an exactbarycenter is computable through linear programming, the underlying linearprogram can be extremely large. For worst-case input, a best known linearprogramming formulation is exponential in the number of variables, but has alow number of constraints, making it an interesting candidate for columngeneration.  In this paper, we devise and study two column generation strategies: anatural one based on a simplified computation of reduced costs, and one througha Dantzig-Wolfe decomposition. For the latter, we produce efficiently solvablesubproblems, namely, a pricing problem in the form of a classicaltransportation problem. The two strategies begin with an efficient computationof an initial feasible solution. While the structure of the constraints leadsto 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, weexhibit that, depending on the input, either strategy can become a best choice.

A Column Generation Approach to the Discrete Barycenter Problem