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Pairwise independent correlation gap

Arjun Ramachandra, Karthik Natarajan

TL;DR

This paper introduces the pairwise independent correlation gap for nonnegative monotone submodular set functions, defined as $f^{+}(oldsymbol{x}) / f^{++}(oldsymbol{x})$, and contrasts it with the classical correlation gap under mutual independence. It proves a tight upper bound of $4/3$ in two regimes: $n=3$ with arbitrary marginals and general $n$ with all marginals small or all large, using LP duality and explicit pairwise distributions (Hunter–Worsley constructions). The results reveal a fundamental distinction between pairwise independence and mutual independence, and are shown to yield improved performance guarantees in distributionally robust optimization problems, including $k$-sum objectives under worst-case dependence. The work combines polyhedral analysis of the feasible function polytope with constructive probabilistic distributions and ends with a conjecture that the $4/3$ bound holds universally, inviting further verification or refutation.

Abstract

In this paper, we introduce the notion of a ``pairwise independent correlation gap'' for set functions with random elements. The pairwise independent correlation gap is defined as the ratio of the maximum expected value of a set function with arbitrary dependence among the elements with fixed marginal probabilities to the maximum expected value with pairwise independent elements with the same marginal probabilities. We show that for any nonnegative monotone submodular set function defined on $n$ elements, this ratio is upper bounded by $4/3$ in the following two cases: (a) $n = 3$ for all marginal probabilities and (b) all $n$ for small marginal probabilities (and similarly large marginal probabilities). This differs from the bound on the ``correlation gap'' which holds with mutual independence and showcases the fundamental difference between pairwise independence and mutual independence. We discuss the implication of the results with two examples and end the paper with a conjecture.

Pairwise independent correlation gap

TL;DR

This paper introduces the pairwise independent correlation gap for nonnegative monotone submodular set functions, defined as , and contrasts it with the classical correlation gap under mutual independence. It proves a tight upper bound of in two regimes: with arbitrary marginals and general with all marginals small or all large, using LP duality and explicit pairwise distributions (Hunter–Worsley constructions). The results reveal a fundamental distinction between pairwise independence and mutual independence, and are shown to yield improved performance guarantees in distributionally robust optimization problems, including -sum objectives under worst-case dependence. The work combines polyhedral analysis of the feasible function polytope with constructive probabilistic distributions and ends with a conjecture that the bound holds universally, inviting further verification or refutation.

Abstract

In this paper, we introduce the notion of a ``pairwise independent correlation gap'' for set functions with random elements. The pairwise independent correlation gap is defined as the ratio of the maximum expected value of a set function with arbitrary dependence among the elements with fixed marginal probabilities to the maximum expected value with pairwise independent elements with the same marginal probabilities. We show that for any nonnegative monotone submodular set function defined on elements, this ratio is upper bounded by in the following two cases: (a) for all marginal probabilities and (b) all for small marginal probabilities (and similarly large marginal probabilities). This differs from the bound on the ``correlation gap'' which holds with mutual independence and showcases the fundamental difference between pairwise independence and mutual independence. We discuss the implication of the results with two examples and end the paper with a conjecture.
Paper Structure (7 sections, 7 theorems, 28 equations, 3 tables)

This paper contains 7 sections, 7 theorems, 28 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Set Functions with Pairwise Independence
  4. A Pairwise Independent Continuous Extension
  5. Upper Bound for $n = 3$
  6. Upper Bound for General $n$ with All Small or All Large Values in $\hbox{\boldmath $x$}$
  7. Two Examples

Key Result

Theorem 1

Calinescu2Agrawal2012 For any $n$, any nonnegative monotone submodular function $f: 2^{N} \rightarrow \mathbb{R}_+$ and any $\hbox{\boldmath $x$} \in [0,1]^n$, ${f^{+}(\hbox{\boldmath $x$})}/{F(\hbox{\boldmath $x$})} \leq e/(e-1)$. This bound is sharp and attained when $f(S) =\min (|S|,1)$ and $\hbo

Theorems & Definitions (14)

  • Theorem 1
  • Definition 2
  • Lemma 3
  • Lemma 4
  • Theorem 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • Theorem 8
  • ...and 4 more