Generalize the CC–χ1 lower bound to relate the number of leaves L(f) to χ1(f)

Develop an analogue of the inequality CC(f) ≥ ⌈log χ1(f)⌉ + 1 that directly lower-bounds L(f), the minimum number of leaves in a deterministic protocol tree for a Boolean function f, in terms of χ1(f), the 1-partition number. Such a bound should be strong enough to transfer known hardness-of-approximation results for χ1(f) to L(f), noting that the simple conjecture L(f) ≥ 2·χ1(f) is false.

Background

Using a reduction from Vertex Cover, the paper (via prior work) shows that χ1(f) is hard to approximate. To leverage this for the protocol size measure L(f) (minimum number of leaves), one would need a structural inequality relating L(f) and χ1(f) analogous to the paper’s key observation CC(f) ≥ ⌈log χ1(f)⌉ + 1.

However, the straightforward conjecture L(f) ≥ 2·χ1(f) is false due to imbalanced protocols, and the authors explicitly note that they do not know how to generalize the CC–χ1 argument to L(f).