Conditioning and covariance on caterpillars.

Let X 1 , …, X n be joint {±1}-valued random variables. It is known that conditioning on a random subset of O(1/ϵ 2 ) of them reduces their average pairwise covariance to below ϵ (in expectation). We conjecture that O(1/ϵ 2 ) can be improved to O(1/ϵ). The motivation for the problem and our conjectured improvement comes from the theory of global correlation rounding for convex relaxation hierarchies. We suggest attempting the conjecture in the case that X 1 , …, X n are the leaves of an information flow tree. We prove the conjecture in the case that the information flow tree is a caterpillar graph (similar to a two-state hidden Markov model).