## Distributionally Robust Max Flows with Marginals

Published in *In preparation, to be submitted*, 2018

Consider an s-t flow network in which the arc capacities are distributed according to an unknown distribution, but with known marginals. We study the problem of finding a coupling of the marginals that yields the worst, expected Max-flow value. Towards solving this optimization problem, we identify a primal-dual formulation. As one might expect, the form relates to the well-known Max Flow, Min-Cut theorem. And this leads to the rather intriguing interpretation as a 2-player, zero-sum game wherein player 1 chooses what to set the arc capacities to and player 2 chooses an s-t cut. Our contributions in this talk are twofold: (1) we present a linear program (that is efficient with finite-supported marginals) that not only computes the worst-case expected value but is of a form amenable in a two-stage network design formulation, and (2) we show that the problem of finding the worst-case coupling of the stochastic arc capacities boils down to finding an optimal distribution over the set of s-t cuts, which we show can be found efficiently.

Recommended citation: Louis Chen, Will Ma, James Orlin, David Simchi-Levi. (2018). "Distributionally Robust Max Flows with Marginals." *In preparation, to be submitted*. 1(2). __https://www.dropbox.com/s/k6dsmnq7h803v8n/DROMaxFlows.pdf?dl=0__