Submodular Dominance and Applications

TL;DR

This paper introduces Weak Negative Regression, a form of negative dependence in distributions, and demonstrates its implications for submodular optimization, including improved inequalities and new rounding techniques.

Contribution

It defines Weak Negative Regression, shows it satisfies Submodular Dominance, and applies this to enhance submodular optimization methods and bounds.

Findings

WNR distributions satisfy Submodular Dominance.

Improved submodular prophet inequalities.

New rounding techniques for negatively dependent distributions.

Abstract

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.