Difference between revisions of "Open Problems:45"

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'''Question:''' What about general constraint satisfaction problems with fixed clause-length and alphabet-size? In this case it is even not known how to obtain $O(n \operatorname{polylog} n)$ space bound.
 
'''Question:''' What about general constraint satisfaction problems with fixed clause-length and alphabet-size? In this case it is even not known how to obtain $O(n \operatorname{polylog} n)$ space bound.
  
== Update ==
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== Updates ==
 
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The progress on the MAX-CUT problem in the streaming setting:
It was shown in {{cite|KapralovKS-15|KoganK-15}} that estimating the maximum cut to within a factor of $(1-\varepsilon)$ requires $n^{1-O(\varepsilon)}$ space in graph streams. This was further improved in {{cite|KapralovKSV-17}} who showed that there exists some fixed constant $\varepsilon_*$ for which obtaining a $(1-\varepsilon_*)$ approximation to MAX-CUT requires $\Omega(n)$ space. Moreover, {{cite|KapralovKS-15}} proved that even in random-ordered streams, $\Omega(\sqrt{n})$ space is needed to obtain a better than $1/2$ approximation.
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* Estimating the maximum cut to within a factor of $(1-\varepsilon)$ requires $n^{1-O(\varepsilon)}$ space {{cite|KapralovKS-15|KoganK-15}}.
 +
* There exists a constant $\varepsilon_*>0$ such that obtaining a $(1-\varepsilon_*)$ approximation to MAX-CUT requires $\Omega(n)$ space {{cite|KapralovKSV-17}}.
 +
* In random-order streams, $\Omega(\sqrt{n})$ space is needed to obtain a better than $1/2$ approximation {{cite|KapralovKS-15}}.

Latest revision as of 04:28, 28 April 2017

Suggested by Robert Krauthgamer
Source Bertinoro 2011
Short link https://sublinear.info/45

The problem is defined as follows: given a stream of edges of an $n$-node graph $G$, estimate the value of the maximum cut in $G$.

Question: Is there an algorithm with an approximation factor strictly better than $1/2$ that uses $o(n)$ space?

Background: Note that $1/2$ is achievable using random assignment argument. Moreover, using sparsification arguments [Trevisan-09,AhnG-09], one can obtain a better approximation ratio using $O(n \operatorname{polylog} n)$ space. Woodruff and Bhattacharyya (private communication) noted that subsampling $O(n/\epsilon^2)$ edges gives, with high probability, an $\epsilon$-additive approximation for all cuts, and thus $1+\epsilon$ multiplicative approximation for MAX-CUT.

Question: What about general constraint satisfaction problems with fixed clause-length and alphabet-size? In this case it is even not known how to obtain $O(n \operatorname{polylog} n)$ space bound.

Updates[edit]

The progress on the MAX-CUT problem in the streaming setting:

  • Estimating the maximum cut to within a factor of $(1-\varepsilon)$ requires $n^{1-O(\varepsilon)}$ space [KapralovKS-15,KoganK-15].
  • There exists a constant $\varepsilon_*>0$ such that obtaining a $(1-\varepsilon_*)$ approximation to MAX-CUT requires $\Omega(n)$ space [KapralovKSV-17].
  • In random-order streams, $\Omega(\sqrt{n})$ space is needed to obtain a better than $1/2$ approximation [KapralovKS-15].