Streaming Algorithms via Local Algorithms for Maximum Directed Cut
Raghuvansh R. Saxena, Noah G. Singer, Madhu Sudan, Santhoshini Velusamy
TL;DR
This paper develops sublinear-space streaming algorithms for Max-DICUT by adapting local, neighborhood-based procedures to streaming environments. The authors introduce neighborhood-type sampling and edge-type distributions, enabling accurate simulation of a constant-factor local algorithm in adversarial and random edge-order streams. They achieve a tight spectrum of results: a (1/2−ε)-approximation in o(n) space with one pass for bounded-degree graphs under adversarial order, a (1/2−ε)-approximation in O(log n) space for random order, and a multi-pass, logarithmic-space method approaching the 1/2 barrier for general graphs. The work connects local computation techniques to streaming CSPs, yields near-optimal trade-offs between space, passes, and approximation, and opens avenues for extending these approaches to broader CSP classes and quantum streaming variants.
Abstract
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 $O(n^{1-Ω(1)})$ (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 $Ω(\sqrt{n})$ 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.