We study learning-augmented streaming algorithms for estimating the value of MAX-CUT in a graph. In the classical streaming model, while a $1/2$-approximation for estimating the value of MAX-CUT can be trivially achieved with $O(1)$ words of space, Kapralov and Krachun [STOC'19] showed that this is essentially the best possible: for any $\epsilon > 0$, any (randomized) single-pass streaming algorithm that achieves an approximation ratio of at least $1/2 + \epsilon$ requires $\Omega(n / 2^{\text{poly}(1/\epsilon)})$ space. We show that it is possible to surpass the $1/2$-approximation barrier using just $O(1)$ words of space by leveraging a (machine learned) oracle. Specifically, we consider streaming algorithms that are equipped with an $\epsilon$-accurate oracle that for each vertex in the graph, returns its correct label in $\{-1, +1\}$, corresponding to an optimal MAX-CUT solution in the graph, with some probability $1/2 + \epsilon$, and the incorrect label otherwise. Within this framework, we present a single-pass algorithm that approximates the value of MAX-CUT to within a factor of $1/2 + \Omega(\epsilon^2)$ with probability at least $2/3$ for insertion-only streams, using only $\text{poly}(1/\epsilon)$ words of space. We also extend our algorithm to fully dynamic streams while maintaining a space complexity of $\text{poly}(1/\epsilon,\log n)$ words.

翻译：我们研究了用于估计图中最大割值的学习增强型流算法。在经典流模型中，虽然使用$O(1)$字空间即可平凡地实现估计最大割值的$1/2$近似，但Kapralov与Krachun [STOC'19]证明这本质上是最优结果：对于任意$\epsilon > 0$，任何达到至少$1/2 + \epsilon$近似比的（随机化）单遍流算法都需要$\Omega(n / 2^{\text{poly}(1/\epsilon)})$空间。我们证明，通过利用（机器学习）预言机，仅用$O(1)$字空间即可突破$1/2$近似界限。具体而言，我们考虑配备$\epsilon$精度预言机的流算法，该预言机对图中每个顶点，以$1/2 + \epsilon$的概率返回其在对应图最优最大割解中的正确标签$\{-1, +1\}$，否则返回错误标签。在此框架下，我们提出一种单遍算法，对于仅插入流，以至少$2/3$的概率在$\text{poly}(1/\epsilon)$字空间内实现$1/2 + \Omega(\epsilon^2)$因子的最大割值近似。我们还将算法扩展至全动态流场景，同时保持$\text{poly}(1/\epsilon,\log n)$字的空间复杂度。