Submodular Dominance and Applications

In submodular optimization we often deal with the expected value of a submodular function $f$ on a distribution $\mathcal{D}$ over sets of elements. In this work we study such submodular expectations for negatively dependent distributions. We introduce a natural notion of negative dependence, which we call Weak Negative Regression (WNR), that generalizes both Negative Association and Negative Regression. We observe that WNR distributions satisfy Submodular Dominance, whereby the expected value of $f$ under $\mathcal{D}$ is at least the expected value of $f$ under a product distribution with the same element-marginals. Next, we give several applications of Submodular Dominance to submodular optimization. In particular, we improve the best known submodular prophet inequalities, we develop new rounding techniques for polytopes of set systems that admit negatively dependent distributions, and we prove existence of contention resolution schemes for WNR distributions.