Simultaneous Multiparty Communication Complexity of Composed Functions
In the Number On the Forehead (NOF) multiparty communication model, k players want to evaluate a function F : X_1 ×...× X_k→ Y on some input (x_1,...,x_k) by broadcasting bits according to a predetermined protocol. The input is distributed in such a way that each player i sees all of it except x_i. In the simultaneous setting, the players cannot speak to each other but instead send information to a referee. The referee does not know the players' input, and cannot give any information back. At
... e end, the referee must be able to recover F(x_1,...,x_k) from what she obtained. A central open question, called the n barrier, is to find a function which is hard to compute for polylog(n) or more players (where the x_i's have size poly(n)) in the simultaneous NOF model. This has important applications in circuit complexity, as it could help to separate ACC^0 from other complexity classes. One of the candidates belongs to the family of composed functions. The input to these functions is represented by a k× (t· n) boolean matrix M, whose row i is the input x_i and t is a block-width parameter. A symmetric composed function acting on M is specified by two symmetric n- and kt-variate functions f and g, that output f∘ g(M)=f(g(B_1),...,g(B_n)) where B_j is the j-th block of width t of M. As the majority function MAJ is conjectured to be outside of ACC^0, Babai et. al. suggested to study MAJ∘ MAJ_t, with t large enough. So far, it was only known that t=1 is not enough for MAJ∘ MAJ_t to break the n barrier in the simultaneous deterministic NOF model. In this paper, we extend this result to any constant block-width t>1, by giving a protocol of cost 2^O(2^t)^2^t+1(n) for any symmetric composed function when there are 2^Ω(2^t) n players.