KRW Composition Theorems via Lifting

💡 Research Summary

The paper tackles the long‑standing Karchmer‑Raz‑Wigderson (KRW) composition conjecture, which asserts that for Boolean functions f and g the depth of a circuit computing the block‑composition f ⋄ g is essentially the sum of the depths of f and g. Proving this conjecture would imply P ⊄ NC¹, a major goal in circuit complexity. Prior work succeeded in handling any outer function f but only a few specific inner functions g (parity, universal relation, etc.). This work dramatically widens the class of admissible inner functions by exploiting lifting theorems that translate lower bounds from weak models (query complexity) to strong models (communication complexity).

Monotone KRW theorem.
The authors first consider the monotone setting, where both f and g are monotone and depth is measured by monotone circuit depth mD(·). The monotone Karchmer‑Wigderson relation mKW f is defined analogously to the classic KW relation, but the goal is to find a coordinate where x_i > y_i. It is known that mD(f) = CC(mKW f). The key observation is that many monotone functions g have their monotone depth lower‑bounded via a query‑to‑communication lifting theorem: for a suitable gadget g_d of size t, any search problem S satisfying Q(S) queries yields a lifted problem S ⊙ g_d with communication complexity Ω(Q(S)·t). By choosing S to be the search problem underlying mKW g, the authors obtain a lower bound on CC(mKW g) that is proportional to the query complexity of g. Using this bound, they prove that for every non‑constant monotone f and every monotone g that is “liftable” in this sense, the monotone KRW conjecture holds: \