Exact finite-dimensional gap for dimensions five and higher

Determine the precise worst-case pairwise independent correlation gap for monotone submodular functions in each finite dimension n satisfying 5 ≤ n < ∞, thereby narrowing the known interval between the lower bound 640/479 and the upper bound e/(e − 1).

Background

The paper establishes that the universal 4/3 upper bound holds tightly in dimension n = 4, while a prior counterexample shows that the bound fails for n ≥ 5. The paper also proves that the worst-case pairwise independent correlation gap approaches e/(e − 1) asymptotically as the dimension tends to infinity.

For finite dimensions 5 ≤ n < ∞, the authors report only a lower bound of 640/479 and the general upper bound e/(e − 1). The exact worst-case value in these dimensions is therefore explicitly left unresolved.

References

It also remains to be resolved whether pairwise-independent distributions approximate the maximum of a submodular function within a factor of $O(\sqrt{\log n})$ or whether a stronger lower bound exists.

— Universal set families for maximization of nonnegative submodular and XOS functions  (2609.19528 - Chekuri et al., 17 Sep 2026) in Section 5, Conclusions

For dimensions 5 ≤ n < ∞, the precise worst case pairwise independent correlation gap remains unknown and is bounded below by the current best known lower bound 640/479 > 4/3 and above by e/(e − 1).

— Dimension Dependent Correlation Gap Bounds under Restricted Independence  (2609.02659 - Ramachandra, 2 Sep 2026) in Section 5, Conclusion, p. 27