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

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 [Kapralov-Khanna-Sudan'15,Kogan-Krauthgamer'15].
• There exists a constant $\varepsilon_*>0$ such that obtaining a $(1-\varepsilon_*)$ approximation to MAX-CUT requires $\Omega(n)$ space [Kapralov-Khanna-Sudan-Velingker'17].
• In random-order streams, $\Omega(\sqrt{n})$ space is needed to obtain a better than $1/2$ approximation [Kapralov-Khanna-Sudan'15].