Half-Approximating Maximum Dicut in the Streaming Setting

2025-12-27

AI Summary

This paper studies the sublinear-space approximation of Max-Dicut in the streaming model: can a $(1/2 - varepsilon)$-approximation be achieved in a single pass using $n^{1-Omega_varepsilon(1)}$ space? Prior work by Kapralov and Krachun established $1/2$ as a lower bound on the approximability in sublinear space, but tightness remained open. We present the first $(1/2 - varepsilon)$-approximation algorithm for general directed graphs under this space constraint, thereby matching the lower bound and fully characterizing the streaming approximation complexity of Max-Dicut. Technically, our approach combines local algorithm simulation with a refined analysis of correlation propagation between high- and low-degree vertices—overcoming the previous best $0.485$-approximation (FOCS’23) and establishing the theoretical optimum for general graphs.

Technology Category

• Search and Optimization: Local Search
• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Data Stream Mining

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

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., $n^{1-Ω(1)}$) space. A lower bound of Kapralov and Krachun (STOC'20) 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 $n^{1-Ω_varepsilon(1)}$ 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.

Problem

• Estimating maximum directed cut value in streaming graph algorithms
• Achieving near-0.5 approximation with sublinear space in general graphs
• Settling tight approximation complexity for maximum dicut problem

Innovation

• Half-approximation algorithm for maximum dicut
• Sublinear space streaming using correlation analysis
• Simulating local algorithm for general graphs