Open Problems:45

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

(Unless explicitly mentioned, the stream is adversarial)

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].
• $\Omega(n)$ 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 $(2/5-\varepsilon)$-approximation linear sketching algorithm for MAX-DICUT [GuruswamiVV-17].
• For every $\varepsilon>0$, there is an $O(\log n)$ space $(2/5-\varepsilon)$-approximation linear 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 $\alpha$ 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 [ChouGV-20].
• 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 [ChouGV-20].
• 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 [ChouGV-20].
• 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 [ChouGSV-21].
• 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 [ChouGSV-21a].

See Madhu Sudan's latest survey on streaming and sketching complexity of CSPs [Sudan-22].