# Problem 80: Merlin–Arthur Communication Complexity of Connectivity

We have a function $f:\mathcal{X}\times\mathcal{Y}\rightarrow\{0,1\}$. In the Merlin–Arthur communication model, Alice gets an $x\in\mathcal{X}$ and Bob gets a $y\in\mathcal{Y}$. Merlin is an all-knowing, all-powerful entity who sends them a proof at the beginning. Then Alice and Bob communicate to find $f(x,y)$. A protocol $\Pi$ solves $f$ if, for all $x,y$,
• $f(x,y) = 1 \implies \exists \text{ proof}: \Pr[\Pi(x,y,\text{ proof})=1] \ge 2/3$, and
• $f(x,y) = 0 \implies \forall \text{ proofs}: \Pr[\Pi(x,y,\text{ proof})=1] \le 1/3$.
We denote the communication complexity of $f$ in the above model as $\textrm{MA}^\rightarrow(f)$. The communication cost here includes the proof size. It is known that $\textrm{MA}^\rightarrow(\mathbf{Disj}) = \tilde{O}(\sqrt{n})$ [AaronsonW-09] and $\textrm{MA}^\rightarrow(\mathbf{InnerProd}) = \tilde{O}(\sqrt{n})$.
For $x,y\in\{0,1\}^{\binom{n}{2}}$, interpreting $x$ and $y$ as edges of an $n$ vertex graph, define $\mathbf{Connect}$ as follows. If $x\cup y$ is connected, $\mathbf{Connect}(x,y) = 1$, else $\mathbf{Connect}(x,y)=0$. Using the Ahn-Guha-McGregor [AhnGM-12] linear sketch for connectivity, we can show that $D^\rightarrow(\mathbf{Connect}) = \tilde{O}(n)$, where $D^\rightarrow$ denotes one-way communication complexity (Alice sends one message to Bob, and there is no Merlin).
Is $\textrm{MA}^\rightarrow(\mathbf{Connect}) = o(n)$?