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Half-Approximating Maximum Dicut in the Streaming Setting

Amir Azarmehr, Soheil Behnezhad, Shane Ferrante, Mohammad Saneian

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'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 $n^{1-Ω_\varepsilon(1)}$ 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.

Half-Approximating Maximum Dicut in the Streaming Setting

Abstract

We study streaming algorithms for the maximum directed cut problem. The edges of an -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 space, a -approximation can be trivially obtained for any fixed 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., ) 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 -approximation, assuming that the graph is constant-degree (SODA'25). In this paper, we show that for any , a -approximation of maximum dicut value can be obtained with 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.
Paper Structure (23 sections, 12 theorems, 87 equations, 1 table, 7 algorithms)

This paper contains 23 sections, 12 theorems, 87 equations, 1 table, 7 algorithms.

Table of Contents

  1. Independent work:
  2. Introduction
  3. Further Related Work
  4. Technical Overview
  5. Background
  6. Our Techniques
  7. Mildly low-degree vertices.
  8. High-degree vertices.
  9. Preliminaries
  10. The problem.
  11. Streaming setting.
  12. Basic Setup and Initialization
  13. Constants
  14. Simplifying and Coloring the Input Graph
  15. Step 1. Reducing the number of edges.
  16. ...and 8 more sections

Key Result

Theorem 1

For any $\varepsilon > 0$, there is a randomized one-pass streaming algorithm that, with high probability, outputs a $(\tfrac{1}{2}-\varepsilon)$-approximation of $\mathsf{MaxDiCut}$ using $n^{1-\Omega_\varepsilon(1)}$ space.

Theorems & Definitions (34)

  • Theorem 1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Proposition 3.5
  • Definition 3.6
  • Lemma 4.1
  • proof
  • Lemma 5.1
  • ...and 24 more