Improved Streaming Algorithms for Maximum Directed Cut via Smoothed Snapshots

TL;DR

This paper introduces a new streaming algorithm with sublinear space complexity that achieves a better approximation ratio for the Max-DICUT problem in directed graphs, surpassing previous algorithms and opening new avenues for CSP approximations.

Contribution

The authors develop the first $ ilde{O}( oot{2} )$-space streaming algorithm for Max-DICUT that outperforms prior $o( oot{2} )$-space algorithms, introducing the concept of graph snapshots and their estimates.

Findings

Achieves a 0.483-approximation with $ ilde{O}( oot{2} )$ space.

First CSP for which $ ilde{O}( oot{2} )$-space algorithms outperform $o( oot{2} )$-space algorithms.

Introduces new notions of snapshot estimation and smoothing techniques.

Abstract

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.