Editing Open Problems:45
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== Updates == | == Updates == | ||
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The progress on the MAX-CUT problem in the streaming setting: | 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 {{cite|KapralovKS-15|KoganK-15}}. | * 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}}. | * 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}}. | * In random-order streams, $\Omega(\sqrt{n})$ space is needed to obtain a better than $1/2$ approximation {{cite|KapralovKS-15}}. | ||
− | * $\Omega(n)$ 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: | The progress on general constraint satisfaction problems in the streaming setting: | ||
− | * For every $\varepsilon>0$, there is an $O(\log n)$ space $( | + | * 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 $( | + | * 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}}. | * $\Omega(\sqrt{n})$ 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 | + | * Dichotomy theorem for streaming approximation of all Boolean Max-2CSPs: For every Boolean Max-2CSP, there is an explicit constant |
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