Concentration of Submodular Functions and Read-k Families Under Negative Dependence

TL;DR

This paper establishes Chernoff-like concentration inequalities for submodular functions of negatively dependent random variables, advancing understanding of their probabilistic behavior and applications in combinatorial optimization.

Contribution

It proves a new concentration inequality for negatively associated or negatively regressed variables, partially resolving an open problem and simplifying existing proofs.

Findings

Proves concentration bounds for negatively dependent variables.

Applies results to read-k families under negative dependence.

Provides simplified proof of existing entropy-method results.

Abstract

We study the question of whether submodular functions of random variables satisfying various notions of negative dependence satisfy Chernoff-like concentration inequalities. We prove such a concentration inequality for the lower tail when the random variables satisfy negative association or negative regression, partially resolving an open problem raised in (Qiu and Singla [QS22]). Previous work showed such concentration results for random variables that come from specific dependent-rounding algorithms (Chekuri, Vondrak, and Zenklusen [CVZ10] and Harvey and Olver [HO14]). We discuss some applications of our results to combinatorial optimization and beyond. We also show applications to the concentration of read-k families [Gav+15] under certain forms of negative dependence; we further show a simplified proof of the entropy-method approach of [Gav+15].