Half-Approximating Maximum Dicut in the Streaming Setting

TL;DR

This paper demonstrates that a nearly 1/2-approximation for the maximum directed cut problem can be achieved in the streaming setting with sublinear space, matching known lower bounds and settling the problem's approximation complexity.

Contribution

It introduces a new streaming algorithm that attains a (1/2 - ε)-approximation in general graphs using sublinear space, matching the theoretical lower bound.

Findings

Achieves (1/2 - ε)-approximation with sublinear space in general graphs.

Shows the lower bound of 0.5-approximation is tight for streaming algorithms.

Provides a detailed analysis of correlation propagation among vertices.

Abstract

We study streaming algorithms for the maximum directed cut problem. The edges of an $n$-vertex directed graph arrive one by one in an arbitrary order, and the goal is to estimate the value of the maximum directed cut using a single pass and small space. With $O(n)$ space, a $(1-\varepsilon)$-approximation can be trivially obtained for any fixed $\varepsilon > 0$ using additive cut sparsifiers. The question that has attracted significant attention in the literature is the best approximation achievable by algorithms that use truly sublinear (i.e., $n^{1-\Omega(1)}$) space. A lower bound of Kapralov and Krachun (STOC'19) implies .5-approximation is the best one can hope for. The current best algorithm for general graphs obtains a .485-approximation due to the work of Saxena, Singer, Sudan, and Velusamy (FOCS'23). The same authors later obtained a $(1/2-\varepsilon)$-approximation, assuming that the graph is constant-degree (SODA'25). In this paper, we show that for any $\varepsilon > 0$, a $(1/2-\varepsilon)$-approximation of maximum dicut value can be obtained with $n^{1-\Omega_\varepsilon(1)}$ space in *general graphs*. This shows that the lower bound of Kapralov and Krachun is generally tight, settling the approximation complexity of this fundamental problem. The key to our result is a careful analysis of how correlation propagates among high- and low-degree vertices, when simulating a suitable local algorithm.