In the Max-Cut problem in the streaming model, an algorithm is given the edges of an unknown graph $G = (V,E)$ in some fixed order, and its goal is to approximate the size of the largest cut in $G$. Improving upon an earlier result of Kapralov, Khanna and Sudan, it was shown by Kapralov and Krachun that for all $\varepsilon>0$, no $o(n)$ memory streaming algorithm can achieve a $(1/2+\varepsilon)$-approximation for Max-Cut. Their result holds for single-pass streams, i.e.~the setting in which the algorithm only views the stream once, and it was open whether multi-pass access may help. The state-of-the-art result along these lines, due to Assadi and N, rules out arbitrarily good approximation algorithms with constantly many passes and $n^{1-\delta}$ space for any $\delta>0$. We improve upon this state-of-the-art result, showing that any non-trivial approximation algorithm for Max-Cut requires either polynomially many passes or polynomially large space. More specifically, we show that for all $\varepsilon>0$, a $k$-pass streaming $(1/2+\varepsilon)$-approximation algorithm for Max-Cut requires $\Omega_{\varepsilon}\left(n^{1/3}/k\right)$ space. This result leads to a similar lower bound for the Maximum Directed Cut problem, showing the near optimality of the algorithm of [Saxena, Singer, Sudan, Velusamy, SODA 2025]. Our lower bounds proceed by showing a communication complexity lower bound for the Distributional Implicit Hidden Partition (DIHP) Problem, introduced by Kapralov and Krachun. While a naive application of the discrepancy method fails, we identify a property of protocols called ``globalness'', and show that (1) any protocol for DIHP can be turned into a global protocol, (2) the discrepancy of a global protocol must be small. The second step is the more technically involved step in the argument, and therein we use global hypercontractive inequalities.

翻译：在流式模型的最大割问题中，算法按固定顺序接收未知图$G = (V,E)$的边，其目标是近似$G$中最大割的规模。Kapralov和Krachun改进了Kapralov、Khanna与Sudan的早期结果，证明了对于任意$\varepsilon>0$，不存在使用$o(n)$内存的流式算法能实现最大割的$(1/2+\varepsilon)$近似。该结论适用于单轮流式场景（即算法仅遍历数据流一次），而多轮访问是否有助于提升性能则悬而未决。沿此方向的最新成果由Assadi和N给出，他们排除了任何固定轮数且使用$n^{1-\delta}$空间（$\delta>0$）的算法获得任意好近似解的可能性。我们改进了这一前沿结果，证明任何非平凡的最大割近似算法必须消耗多项式轮数或多项式量级的空间。具体而言，我们证明对于任意$\varepsilon>0$，$k$轮流式$(1/2+\varepsilon)$近似最大割算法需要$\Omega_{\varepsilon}\left(n^{1/3}/k\right)$空间。该结论可推导出最大有向割问题的类似下界，从而表明[Saxena, Singer, Sudan, Velusamy, SODA 2025]所提算法近乎最优。我们的下界证明通过建立Kapralov与Krachun提出的分布隐式隐藏划分问题的通信复杂度下界实现。虽然直接应用差异法会失败，但我们定义了协议的一种“全局性”特质，并证明：（1）任何DIHP协议可转化为全局协议；（2）全局协议的差异度必然很小。第二步是论证中技术更复杂的环节，我们在该步骤中运用了全局超压缩不等式。