Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Sharmila Duppala, George Z. Li, Juan Luque, Aravind Srinivasan, Renata Valieva

TL;DR

This paper addresses Chernoff-like concentration for submodular functions of negatively dependent random variables, introducing 1-negative association (1-NA) as a practical, intermediate form of dependence. It develops an exponential-moments framework showing that 1-NA (and, for binary variables, weak negative regression) suffices to transfer independent-case submodular concentration bounds to dependent settings, with the baseline given by the submodular multilinear extension $F$ evaluated at the marginals ($\mu_0=F(x)$). The results yield a concrete lower-tail bound $\Pr[f(X) \le (1-\delta)\mu_0] \le \exp(-\mu_0\delta^2/2)$ and extend to read-$k$ families, providing a unified, principled route to concentration for a broad class of negatively dependent models and dependent rounding schemes. Applications include near-linear-time, group-fair maximum-coverage-type problems, and simplification of entropy-method proofs for read-$k$ supermodular functions, with implications for rounding algorithms, online settings, and strongly Rayleigh measures.

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].

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

TL;DR

This paper addresses Chernoff-like concentration for submodular functions of negatively dependent random variables, introducing 1-negative association (1-NA) as a practical, intermediate form of dependence. It develops an exponential-moments framework showing that 1-NA (and, for binary variables, weak negative regression) suffices to transfer independent-case submodular concentration bounds to dependent settings, with the baseline given by the submodular multilinear extension evaluated at the marginals (). The results yield a concrete lower-tail bound and extend to read- families, providing a unified, principled route to concentration for a broad class of negatively dependent models and dependent rounding schemes. Applications include near-linear-time, group-fair maximum-coverage-type problems, and simplification of entropy-method proofs for read- supermodular functions, with implications for rounding algorithms, online settings, and strongly Rayleigh measures.

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].
Paper Structure (19 sections, 9 theorems, 67 equations)

This paper contains 19 sections, 9 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Techniques.
  3. Applications.
  4. Related Work.
  5. Preliminaries
  6. Notions of Negative Dependence
  7. Negative Cylinder Dependence.
  8. Negative Association.
  9. Negative Regression.
  10. Strong Rayleigh.
  11. Submodular Functions
  12. Submodular Functions.
  13. Supermodular Functions.
  14. Mutlilinear Extension.
  15. Submodular Chernoff Bounds
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $X_1,\ldots,X_n$ be binary random variables with mean $x_1,\ldots,x_n$ satisfying negative association (or negative regression). Let $f$ be a non-negative monotone submodular function with marginal values in $[0,1]$ and let $F$ be the multilinear extension of $f$. If we let $\mu_0=F(x_1,\ldots,x

Theorems & Definitions (24)

  • Definition 1.1
  • Theorem 1.2
  • Definition 3.1
  • Definition 3.2
  • Claim 3.3
  • proof
  • Claim 3.4
  • proof
  • Claim 3.5
  • proof
  • ...and 14 more