Difference between revisions of "Open Problems:80"

From Open Problems in Sublinear Algorithms
Jump to: navigation, search
(Changed to the correct definition of MA model that states to ensure communication cost *includes* the proof size.)
 
(3 intermediate revisions by 2 users not shown)
Line 3: Line 3:
 
|who=Amit Chakrabarti
 
|who=Amit Chakrabarti
 
}}
 
}}
 
+
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 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
 
* $f(x,y) = 1 \implies \exists \text{ proof}: \Pr[\Pi(x,y,\text{ proof})=1] \ge 2/3$, 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 includes the proof size.
 +
It is known that $\textrm{MA}^\rightarrow(\mathbf{Disj}) = \tilde{O}(\sqrt{n})$ {{cite|AaronsonW-09}} and $\textrm{MA}^\rightarrow(\mathbf{InnerProd}) = \tilde{O}(\sqrt{n})$.
  
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.
+
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).
 
 
It is known that $\textrm{MA}^\rightarrow(\textrm{DISJ}) = \tilde{O}(\sqrt{n})$ {{cite|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 {{cite|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 $\textrm{MA}^\rightarrow(\textrm{is-conn}) = o(n)$?
+
Is $\textrm{MA}^\rightarrow(\mathbf{Connect}) = o(n)$?
<references />
 

Latest revision as of 16:34, 26 January 2019

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