Concentration of Submodular Functions Under Negative Dependence

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, resolving an open problem raised in (citet{approx/QiuS22}). Previous work showed such concentration results for random variables that come from specific dependent-rounding algorithms (citet{focs/ChekuriVZ10,soda/HarveyO14}). We discuss some applications of our results to combinatorial optimization and beyond.

我们研究一个问题，即是否满足负面依赖的各种概念的随机变量是否满足了Chernoff样浓度的不平等。当随机变量满足负相关或负回归时，我们证明了下尾的浓度不平等，从而解决了在（citet {lot/qius22}）中提出的开放问题。先前的工作显示了来自特定依赖性符号算法的随机变量（citet {focs/chekurivz10，soda/harveyo14}）的这种浓度结果。我们讨论了我们的结果的一些应用，以组合优化。