We explore the use of local algorithms in the design of streaming algorithms for the Maximum Directed Cut problem. Specifically, building on the local algorithm of Buchbinder et al. (FOCS'12) and Censor-Hillel et al. (ALGOSENSORS'17), we develop streaming algorithms for both adversarially and randomly ordered streams that approximate the value of maximum directed cut in bounded-degree graphs. In $n$-vertex graphs, for adversarially ordered streams, our algorithm uses $O(n^{1-\Omega(1)})$ (sub-linear) space and for randomly ordered streams, our algorithm uses logarithmic space. Moreover, both algorithms require only one pass over the input stream. With a constant number of passes, we give a logarithmic-space algorithm which works even on graphs with unbounded degree on adversarially ordered streams. Our algorithms achieve any fixed constant approximation factor less than $\frac12$. In the single-pass setting, this is tight: known lower bounds show that obtaining any constant approximation factor greater than $\frac12$ is impossible without using linear space in adversarially ordered streams (Kapralov and Krachun, STOC'19) and $\Omega(\sqrt{n})$ space in randomly ordered streams, even on bounded degree graphs (Kapralov, Khanna, and Sudan, SODA'15). In terms of techniques, our algorithms partition the vertices into a small number of different types based on the structure of their local neighborhood, ensuring that each type carries enough information about the structure to approximately simulate the local algorithm on a vertex with that type. We then develop tools to accurately estimate the frequency of each type. This allows us to simulate an execution of the local algorithm on all vertices, and thereby approximate the value of the maximum directed cut.

翻译：本文探讨了在最大有向割问题的流式算法设计中应用局部算法的方法。具体而言，基于Buchbinder等人（FOCS'12）和Censor-Hillel等人（ALGOSENSORS'17）的局部算法，我们针对对抗排序流和随机排序流，分别开发了用于近似有界度图中最大有向割值的流式算法。在$n$顶点图中，对于对抗排序流，我们的算法使用$O(n^{1-\Omega(1)})$（亚线性）空间；对于随机排序流，算法使用对数空间。此外，两种算法仅需对输入流进行一次扫描。通过常数次扫描，我们提出了一种对数空间算法，该算法甚至能处理对抗排序流中无度限制的图。我们的算法可实现任意小于$\frac12$的固定常数近似比。在单次扫描设置下，该界限是紧的：已知下界表明，在对抗排序流中（Kapralov和Krachun，STOC'19），若使用超过$\frac12$的常数近似比，则必须消耗线性空间；而在随机排序流中，即使对有界度图也需$\Omega(\sqrt{n})$空间（Kapralov、Khanna和Sudan，SODA'15）。在技术层面，我们的算法根据顶点局部邻域的结构将其划分为少量不同类型，确保每种类型携带足够的结构信息，从而能近似模拟该类型顶点上的局部算法。随后我们开发了精确估计各类出现频率的工具。这使得我们能够模拟在所有顶点上执行局部算法的过程，进而近似计算最大有向割的值。