Multiparty Communication Complexity and Threshold Circuit Size of $\ensuremath{\sfAC}^0$
Paul W. Beame, Trinh Huynh · SIAM Journal on Computing · 2012
We prove an $n^{\Omega(1)}/4^k$ lower bound on the randomized k-party communication complexity of depth 4 $\ensuremath {\sf AC}^0$ functions in the number-on-forehead (NOF) model for up to $\Theta(\log n)$ players. These are the first nontrivial lower bounds for general NOF multiparty communication complexity for any $\ensuremath {\sf AC}^0$ function for $\omega(\log\log n)$ players. For nonconstant k the bounds are larger than all previous lower bounds for any $\ensuremath {\sf AC}^0$ function even for simultaneous communication complexity. Our lower bounds imply the first superpolynomial lower bounds for the simulation of $\ensuremath {\sf AC}^0$ by $\ensuremath {\sf MAJ\circ SYM\circ AND}$ circuits, showing that the well-known quasi-polynomial simulations of $\ensuremath {\sf AC}^0$ by such circuits due to Allender (1989) and Yao (1990) are qualitatively optimal,-1pt even for formulas of small constant depth. We also exhibit a depth 5 formula in ${\ensuremath {\sf NP}^{cc}_k}-{\ensuremath {\sf BPP}^{cc}_k}$ for k up to $\Theta(\log n)$ and derive $\Omega(2^{\sqrt{\log n}/\sqrt{k}})$ lower bound on the randomized k-party NOF communication complexity of set disjointness for up to $\Theta(\log^{1/3} n)$ players, which is significantly larger than the $O(\log\log n)$ players allowed in the best previous lower bounds for multiparty set disjointness. We prove other strong results for depth 3 and 4 $\ensuremath {\sf AC}^0$ functions.