We identify a sharp separation in the streaming space complexity of Maximum Cut when the algorithm must output an approximate cut (rather than only the approximate value). For dense graphs, we show that $O(n/\varepsilon^2)$ space is sufficient and that $Ω(n)$ space is necessary. In contrast, for graphs with $Θ(n/\varepsilon^2)$ edges, the situation is markedly different: we show that the problem requires $Ω(n \log(\varepsilon^2 n)/\varepsilon^2)$ space for any $\varepsilon=ω(1/\sqrt{n})$, which is tight for the full range of $\varepsilon$. We also give an $Ω(n \log n/\varepsilon^2)$-space lower bound against deterministic algorithms for outputting a $(1-\varepsilon)$ approximation to the value of the maximum cut. Using similar techniques we prove an analogous sharp separation in the streaming space complexity of Densest Subgraph and show that for every constant-arity CSP over a constant-size alphabet and the Similarity problem the space complexity in dense streams can be improved by shaving a logarithmic factor.

翻译：我们揭示了最大割问题在流式计算中，当算法必须输出近似割（而不仅仅是近似值）时，其空间复杂度存在显著分离。对于稠密图，$O(n/\varepsilon^2)$ 空间足够且 $Ω(n)$ 空间是必要的。相比之下，对于边数为 $Θ(n/\varepsilon^2)$ 的图，情况截然不同：对于任何 $\varepsilon=ω(1/\sqrt{n})$，该问题需要 $Ω(n \log(\varepsilon^2 n)/\varepsilon^2)$ 空间，且这一界对于 $\varepsilon$ 的全部取值范围是紧的。我们还给出了对于确定性算法输出最大割值的 $(1-\varepsilon)$ 近似所需的 $Ω(n \log n/\varepsilon^2)$ 空间下界。运用类似技术，我们证明了稠密子图问题在流式计算空间复杂度上也存在类似的显著分离，并表明对于任意具有恒定元数且字母表大小恒定的 CSP 问题以及相似性问题，稠密数据流中的空间复杂度可通过削去一个对数因子得到改进。