# Difference between revisions of "Open Problems:80"

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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 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 $x,y$, | |

− | We have a function $f:\mathcal{X}\times\mathcal{Y}\rightarrow\{0,1\}$. In the | ||

* $f(x,y) = 1 \implies \exists \text{ proof}: \Pr[\Pi(x,y,\text{ proof})=1] \ge 2/3$, and | * $f(x,y) = 1 \implies \exists \text{ proof}: \Pr[\Pi(x,y,\text{ proof})=1] \ge 2/3$, and | ||

* $f(x,y) = 0 \implies \forall \text{ proofs}: \Pr[\Pi(x,y,\text{ proof})=1] \le 1/3$. | * $f(x,y) = 0 \implies \forall \text{ proofs}: \Pr[\Pi(x,y,\text{ proof})=1] \le 1/3$. | ||

+ | We denote the communication complexity of $f$ in the above model as $\textrm{MA}^\rightarrow(f)$. The communication cost here does not include the proof size. | ||

+ | It is known that $\textrm{MA}^\rightarrow(\mathbf{Disj}) = \tilde{O}(\sqrt{n})$ {{cite|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 {{cite|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). | |

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− | For $x,y\in\{0,1\}^{\binom{n}{2}}$, interpreting $x$ and $y$ as edges of an $n$ vertex graph, define $\ | ||

− | Is $\textrm{MA}^\rightarrow(\ | + | Is $\textrm{MA}^\rightarrow(\mathbf{Connect}) = o(n)$? |

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## Revision as of 03:44, 28 April 2017

Suggested by | Amit Chakrabarti |
---|---|

Source | Banff 2017 |

Short link | https://sublinear.info/80 |

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 does not include 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 $\textrm{MA}^\rightarrow(\mathbf{Connect}) = o(n)$?