Difference between revisions of "Open Problems:45"

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* In random-order streams, $\Omega(\sqrt{n})$ space is needed to obtain a better than $1/2$ approximation {{cite|KapralovKS-15}}.
 
* In random-order streams, $\Omega(\sqrt{n})$ space is needed to obtain a better than $1/2$ approximation {{cite|KapralovKS-15}}.
 
* In adversarial streams, $\Omega(n)$ space is needed to obtain a better than $1/2$ approximation {{cite|KapralovK-19}}.
 
* In adversarial streams, $\Omega(n)$ space is needed to obtain a better than $1/2$ approximation {{cite|KapralovK-19}}.
 +
 +
The progress on general constraint satisfaction problems in the streaming setting:
 +
* For every $\varepsilon>0$, there is an $O(\log n)$ space $(4/9-\varepsilon)$-approximation sketching algorithm for MAX-DICUT {{cite|GuruswamiVV-17}}.
 +
* For every $\varepsilon>0$, there is an $O(\log n)$ space $(4/9-\varepsilon)$-approximation sketching algorithm for MAX-2AND {{cite|GuruswamiVV-17}}.
 +
* $\Omega(\sqrt{n})$ space is needed to obtain a better than $1/2$ approximation for MAX-DICUT (also MAX-2AND) {{cite|GuruswamiVV-17}}.
 +
* Dichotomy theorem for streaming approximation of all Boolean Max-2CSPs: For every Boolean Max-2CSP, there is an explicit constant

Revision as of 03:59, 27 June 2022

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

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].
  • In adversarial streams, $\Omega(n)$ space is needed to obtain a better than $1/2$ approximation [KapralovK-19].

The progress on general constraint satisfaction problems in the streaming setting:

  • For every $\varepsilon>0$, there is an $O(\log n)$ space $(4/9-\varepsilon)$-approximation sketching algorithm for MAX-DICUT [GuruswamiVV-17].
  • For every $\varepsilon>0$, there is an $O(\log n)$ space $(4/9-\varepsilon)$-approximation sketching algorithm for MAX-2AND [GuruswamiVV-17].
  • $\Omega(\sqrt{n})$ space is needed to obtain a better than $1/2$ approximation for MAX-DICUT (also MAX-2AND) [GuruswamiVV-17].
  • Dichotomy theorem for streaming approximation of all Boolean Max-2CSPs: For every Boolean Max-2CSP, there is an explicit constant