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

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 (1). Previous work showed such concentration results for random variables that come from specific dependent-rounding algorithms (2, 3). We discuss some applications of our results to combinatorial optimization and beyond. We also show applications to the concentration of read- k families 4 under certain forms of negative dependence; we further show a simplified proof of the entropy-method approach of 4.