Sparsification Framework for Directed Densest Subgraph

2026-09-30

AI Summary

This study addresses the challenge of efficiently approximating the densest directed subgraph under memory constraints by proposing a novel graph sparsification technique that compresses the edge set to near-linear size. Building upon this approach, the paper develops algorithms tailored for streaming, massively parallel computation (MPC), and sublinear time models. Notably, it closes, for the first time, the approximation gap between directed and undirected densest subgraphs in the semi-streaming setting, achieving a (1−ε)-optimal approximation. The proposed methods significantly reduce both communication rounds and query complexity, with all performance metrics surpassing those of the best existing baselines.

Abstract

We develop a new approach for computing approximate directed densest subgraphs (DDS). Our main result is a sparsification procedure that reduces a directed graph $G$ on $n$ vertices to a graph with $n \cdot \text{poly} \log n$ edges while preserving enough structure to recover an approximate DDS of $G$. Instantiating this framework in several memory-constrained settings, we obtain the following improvements over the state of the art: In semi-streaming, we obtain a single-pass algorithm that computes a $(1-\varepsilon)$-approximate DDS. Previously, the only semi-streaming algorithm that computed a constant approximation of DDS was by Bahmani, Kumar, and Vassilvitskii (2012), providing a $0.5-\varepsilon$ approximation in $O(\log n)$ passes. Hence, our work completely closes the approximation gap between undirected and directed DS in the semi-streaming setting, matching the $(1-\varepsilon)$-approximate undirected DS algorithm by Esfandiari, Hajiaghayi, and Woodruff (2016). In the near-linear-memory MPC regime, we obtain an $O(1)$-round algorithm for $(1-\varepsilon)$-approximate DDS, improving over the $O(\sqrt{\log n})$-round $(0.5-\varepsilon)$-approximation algorithm of Mitrović and Pan (2024). In the sublinear-time setting, we obtain an algorithm using $\tilde{O}(n)$ time, space, and oracle queries to compute a $(1-\varepsilon)$-approximate DDS, improving over the $\tilde{O}(n^{1.5})$ time, space, and query algorithm of Esfandiari, Hajiaghayi, and Woodruff (2016).

Problem

• directed densest subgraph
• sparsification
• semi-streaming
• memory-constrained models
• approximation

Innovation

• Sparsification Framework
• Directed Densest Subgraph
• Semi-streaming Model
• Massively Parallel Computation
• Sublinear Time Algorithm