Volume 6 (2010) Article 10 pp. 227-245
A Separation of NP and coNP in Multiparty Communication Complexity
We prove that $\mathsf{coNP} \not\subseteq \mathsf{MA}$ in the number-on-forehead model of multiparty communication complexity for up to $k=(1-\epsilon)\log n$ players, where $\epsilon>0$ is any constant. Specifically, we construct an explicit function $F:\left(\{0,1\}^n\right)^k\to\{0,1\}$ with co-nondeterministic complexity $O(\log n)$ and Merlin-Arthur complexity $n^{\Omega(1)}.$ The problem was open for $k\geq3.$ As a corollary, we obtain an explicit separation of $\mathsf{NP}$ and $\mathsf{coNP}$ for up to $k=(1-\epsilon)\log n$ players, complementing an independent result by Beame et al. (2010) who separate these classes nonconstructively for up to $k = 2^{(1-\epsilon)n}$ players.