  We study a generalized distributionally robust chance-constrained set
covering problem (DRC) with a Wasserstein ambiguity set, where both decisions
and uncertainty are binary-valued. We establish the NP-hardness of DRC and
recast it as a two-stage stochastic program, which facilitates decomposition
algorithms. Furthermore, we derive two families of valid inequalities. The
first family targets the hypograph of a "shifted" submodular function, which is
associated with each scenario of the two-stage reformulation. We show that the
valid inequalities give a complete description of the convex hull of the
hypograph. The second family mixes inequalities across multiple scenarios and
gains further strength via lifting. Our numerical experiments demonstrate the
reliability of the DRC model and the effectiveness of our proposed
reformulation and valid inequalities.
