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 $4/9 < 0.45$ 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 achieves a 0.483-approximation. 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 thus our work opens the possibility of a new class of algorithms for approximating CSPs.

翻译：我们给出一个$\widetilde{O}(\sqrt{n})$空间单趟$0.483$-近似流式算法，用于估计$n$个顶点有向图中的最大有向割规模（Max-DICUT）。该算法改进了Chou、Golovnev和Velusamy（FOCS 2020）提出的$O(\log n)$空间$4/9 < 0.45$近似算法，后者已知对$o(\sqrt{n})$空间算法是最优的。Max-DICUT是约束满足问题（CSP）的一个特例。在更广泛的背景下，我们首次证明存在一个CSP，使得$\widetilde{O}(\sqrt{n})$空间算法能够显著超越$o(\sqrt{n})$空间算法。我们工作的关键技术贡献在于提出了（有向）图的一阶快照及其估计量的概念。这些快照可用于模拟某些（非流式）Max-DICUT算法，包括Feige和Jozeph（Algorithmica, 2015）引入的“无意识”算法，他们证明其中一种算法能达到0.483近似。作者前期工作（SODA 2023）研究了有界度数图的受限情形，并观察到在此设定下，用$\ell_1$误差估计快照是直接的，且足以模拟无意识算法。但对于无界度数图，定义可实现且充分的估计量概念本身就具有微妙性。我们描述了新的快照估计概念，并通过精细的平滑技术证明其充分性，进而设计了一种算法——借助交织的顶点与边子抽样过程——来构建此类估计量的草图。在我们工作之前，针对一般实例上任意CSP的唯一流式算法均基于Max-DICUT的$O(\log n)$空间算法的推广，因此本研究开启了近似CSPs的新算法类别可能性。