Problem 80: Merlin–Arthur Communication Complexity of Connectivity

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Suggested by Amit Chakrabarti
Source Banff 2017
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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 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 $\textrm{MA}^\rightarrow(f)$. Communication cost here does not include the proof size.

It is known that $\textrm{MA}^\rightarrow(\textrm{DISJ}) = \tilde{O}(\sqrt{n})$ [Aaronson-Wigderson-XX] and $\textrm{MA}^\rightarrow(\textrm{InnerProd}) = \tilde{O}(\sqrt{n})$.

For $x,y\in\{0,1\}^{\binom{n}{2}}$, interpreting $x$ and $y$ as edges of an $n$ vertex graph, define $\textrm{is-conn}$ as follows. If $x\cup y$ is connected, $\textrm{is-conn}(x,y) = 1$, else $\textrm{is-conn}(x,y)=0$. Using the Ahn-Guha-McGregor [AhnGM-XX] linear sketch for connectivity, we can show that $D^\rightarrow(\textrm{is-conn}) = \tilde{O}(n)$, where $D^\rightarrow$ denotes one-way communication complexity (Alice sends once message to Bob, and there is no Merlin).

Is $\textrm{MA}^\rightarrow(\textrm{is-conn}) = o(n)$?