Dimension Dependent Correlation Gap Bounds under Restricted Independence

Abstract

The pairwise independent correlation gap is the ratio of the maximum expected value of a set function under arbitrary dependence to that under pairwise independence, measuring the loss from this independence restriction. Under mutual independence, this gap is universally bounded by $e/(e-1)$ for monotone submodular functions. With pairwise independence, a tighter $4/3$ upper bound was established for several special cases, including $n=3$, and conjectured to hold universally. A recent AI-assisted counterexample disproved this conjecture for $n=5$, leaving the validity of the $n=4$ bound and the tight worst case bound open. We resolve both questions. First, for $n=4$, we establish that the $4/3$ bound holds universally and is tight using an AI-assisted proof combining theoretical analysis and computational verification. The proof combines a structural characterization of optimal numerator vertices, permutation symmetry, cone certificate systems, Bernstein polynomial representations, recursive simplex subdivision, and verification of $2,745$ Bernstein coefficient systems. Second, we show that the worst case pairwise independent correlation gap attains $e/(e-1)$ asymptotically by constructing an instance with identical marginal probabilities and a monotone submodular union coverage function on a ground set partitioned into $m$ blocks. The number of blocks grows sublinearly with the ground set size. The result follows by constructing a feasible solution to a scaled asymptotic reduced dual of the pairwise independent linear program and immediately extends to $t$-wise independent random elements ($t\ge2$), since $t$-wise independence implies pairwise independence. Thus, pairwise independence, despite being the least restrictive form of independence in the $t$-wise independence hierarchy, can be as restrictive as mutual independence in the worst case.