Distributionally Robust Max Flows

Louis Chen, Will Ma, James B Orlin, David Simchi-Levi

Abstract: We study a distributionally robust max flow problem under the marginal distribution model, where the vector of arc capacities is random, with the marginals to the joint multivariate distribution being known, but the correlation being unknown. The goal is to compute the expected value of the max flow under the worst-case joint distribution of arc capacities. We provide a simple combinatorial proof that shows that for the case of finite-supported marginal distributions, this worst-case expectation can be efficiently computed, and moreover, the worst-case joint distribution can be explicitly constructed, despite being non-trivial in the sense that it is not a combination of monotonic or anti-monotonic couplings. Our technique is to use a related min-cost flow problem to generate a distribution over cuts in the graph, which in turn induces the worst-case joint distribution. It also provides an alternative interpretation of the problem as a zero-sum game between a capacity player and a cut player.