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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. |
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− | == Updates == | + | == Update == |
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− | (''Unless explicitly mentioned, the stream is adversarial'')
| + | 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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− | The progress on the MAX-CUT problem in the streaming setting:
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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}}.
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− | * There exists a constant $\varepsilon_*>0$ such that obtaining a $(1-\varepsilon_*)$ approximation to MAX-CUT requires $\Omega(n)$ space {{cite|KapralovKSV-17}}.
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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}}.
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− | * $\Omega(n)$ space is needed to obtain a better than $1/2$ approximation {{cite|KapralovK-19}}.
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− | The progress on general constraint satisfaction problems in the streaming setting:
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− | * For every $\varepsilon>0$, there is an $O(\log n)$ space $(2/5-\varepsilon)$-approximation linear sketching algorithm for MAX-DICUT {{cite|GuruswamiVV-17}}.
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− | * For every $\varepsilon>0$, there is an $O(\log n)$ space $(2/5-\varepsilon)$-approximation linear sketching algorithm for MAX-2AND {{cite|GuruswamiVV-17}}.
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− | * $\Omega(\sqrt{n})$ space is needed to obtain a better than $1/2$ approximation for MAX-DICUT (also MAX-2AND) {{cite|GuruswamiVV-17}}.
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− | * Dichotomy theorem for streaming approximation of all Boolean Max-2CSPs: For every Boolean Max-2CSP, there is an explicit constant $\alpha$ such that for every $\varepsilon$, (i) there is an $O(\log n)$ space $(\alpha-\varepsilon)$-approximation linear sketching algorithm and (ii) $\Omega(\sqrt{n})$ space is needed to obtain a better than $\alpha$ streaming approximation {{cite|ChouGV-20}}.
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− | * For every $\varepsilon>0$, there is an $O(\log n)$ space $(4/9-\varepsilon)$-approximation linear sketching algorithm for MAX-DICUT; $\Omega(\sqrt{n})$ space is needed to obtain a better than $4/9$ streaming approximation {{cite|ChouGV-20}}.
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− | * For every $\varepsilon>0$, there is an $O(\log n)$ space $(1/\sqrt{2}-\varepsilon)$-approximation linear sketching algorithm for MAX-$k$SAT; $\Omega(\sqrt{n})$ space is needed to obtain a better than $1/\sqrt{2}$ streaming approximation {{cite|ChouGV-20}}.
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− | * Dichotomy theorem for sketching approximation of all Boolean Max-CSPs: For every Boolean Max-CSP, there is a constant $\alpha$ such that for every $\varepsilon$, (i) there is an $O(\log n)$ space $(\alpha-\varepsilon)$-approximation linear sketching algorithm and (ii) $\Omega(\sqrt{n})$ space is needed to obtain a better than $\alpha$ sketching approximation {{cite|ChouGSV-21}}.
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− | * Dichotomy theorem for sketching approximation of all finite Max-CSPs: For every finite Max-CSP, there is a constant $\alpha$ such that for every $\varepsilon$, (i) there is an $O(\log n)$ space $(\alpha-\varepsilon)$-approximation linear sketching algorithm and (ii) $\Omega(\sqrt{n})$ space is needed to obtain a better than $\alpha$ sketching approximation {{cite|ChouGSV-21a}}.
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− | See Madhu Sudan's latest survey on streaming and sketching complexity of CSPs {{cite|Sudan-22}}.
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