Difference between revisions of "Open Problems:101"

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{{Header
 
{{Header
|title=Vertex connectivity in the LOCAL model
 
 
|source=wola19
 
|source=wola19
 
|who=Sorrachai Yingchareonthawornchai
 
|who=Sorrachai Yingchareonthawornchai
 
}}
 
}}
 
 
In this question, the input is the underlying graph $G=(V,E)$, as well as parameters $\nu,k$ and vertex $v\in V$. The goal is to output either $\bot$ or a subset $S\subseteq V$, such that
 
In this question, the input is the underlying graph $G=(V,E)$, as well as parameters $\nu,k$ and vertex $v\in V$. The goal is to output either $\bot$ or a subset $S\subseteq V$, such that
  
- if $\bot$ is the output, there is no $S$ such that $v\in S$ with $|S| \leq \nu$ and $|N(S)| < k$;
+
* if $\bot$ is the output, there is no $S$ such that $v\in S$ with $|S| \leq \nu$ and $|N(S)| < k$;
  
- if the output is a set $S$, then $|N(S)| < k$.
+
* if the output is a set $S$, then $|N(S)| < k$.
  
 
It is known that this problem can be solved with $O(\nu k)$ queries, and either time $O(\nu^{3/2} k)$ (deterministic) or $O(\nu k^2)$ (randomized) {{Cite|NanongkaiSY-19a|NanongkaiSY-19b|ForsterY-19}}.
 
It is known that this problem can be solved with $O(\nu k)$ queries, and either time $O(\nu^{3/2} k)$ (deterministic) or $O(\nu k^2)$ (randomized) {{Cite|NanongkaiSY-19a|NanongkaiSY-19b|ForsterY-19}}.
  
 
'''Question:''' Can one achieve time $O(\nu k)$?
 
'''Question:''' Can one achieve time $O(\nu k)$?

Latest revision as of 18:36, 26 August 2019

Suggested by Sorrachai Yingchareonthawornchai
Source WOLA 2019
Short link https://sublinear.info/101

In this question, the input is the underlying graph $G=(V,E)$, as well as parameters $\nu,k$ and vertex $v\in V$. The goal is to output either $\bot$ or a subset $S\subseteq V$, such that

  • if $\bot$ is the output, there is no $S$ such that $v\in S$ with $|S| \leq \nu$ and $|N(S)| < k$;
  • if the output is a set $S$, then $|N(S)| < k$.

It is known that this problem can be solved with $O(\nu k)$ queries, and either time $O(\nu^{3/2} k)$ (deterministic) or $O(\nu k^2)$ (randomized) [NanongkaiSY-19a,NanongkaiSY-19b,ForsterY-19].

Question: Can one achieve time $O(\nu k)$?