Improved Streaming Algorithms for Maximum Directed Cut via Smoothed Snapshots

We give an $\widetilde{O}(\sqrt{n})$-space single-pass 0.483-approximation streaming algorithm for estimating the maximum directed cut size (Max-DICUT) in a directed graph on n vertices. This improves over an $O(\log n)$-space $4 / 9\lt 0.45$ approximation algorithm due to Chou, Golovnev, and Velusamy (FOCS 2020), which was known to be optimal for $o(\sqrt{n})$-space algorithms. Max-DICUT is a special case of a constraint satisfaction problem (CSP). In this broader context, we give the first CSP for which algorithms with $\widetilde{O}(\sqrt{n})$- space can provably outperform $o(\sqrt{n})$- space algorithms. The key technical contribution of our work is development of the notions of a first-order snapshot of a (directed) graph and of estimates of such snapshots. These snapshots can be used to simulate certain (non-streaming) Max-DICUT algorithms, including the “oblivious” algorithms introduced by Feige and Jozeph (Algorithmica, 2015), who showed that one such algorithm Previous work of the authors (SODA 2023) studied the restricted case of bounded-degree graphs, and observed that in this setting, it is straightforward to estimate the snapshot with $\ell_{1}$ errors and this suffices to simulate oblivious algorithms. But for unbounded-degree graphs, even defining an achievable and sufficient notion of estimation is subtle. We describe a new notion of snapshot estimation and prove its sufficiency using careful smoothing techniques, and then develop an algorithm which sketches such an estimate via a delicate process of intertwined vertex- and edge-subsampling. Prior to our work, the only streaming algorithms for any CSP on general instances were based on generalizations of the $O(\log n)$-space algorithm for Max-DICUT, and can roughly be characterized as based on “zeroth” order snapshots. Our work thus opens the possibility of a new class of algorithms for approximating CSPs by demonstrating that more sophisticated snapshots can outperform cruder ones in the case of Max-DICUT.