Density Matters: A Complexity Dichotomy of Deleting Edges to Bound Subgraph Density

2026-01-06

arXiv.org

AI Summary

This study addresses the τ-bounded density edge deletion problem (τ-BDED), which seeks to remove the minimum number of edges from a graph so that the density of every subgraph does not exceed a given threshold τ. The authors establish the first complete complexity dichotomy for this problem with respect to τ: it is polynomial-time solvable when 2τ is an integer or τ < 2/3, and NP-hard otherwise. For integer τ, they design a randomized algorithm running in near-linear time, O(m^{1+o(1)}). Moreover, the problem is shown to be fixed-parameter tractable when parameterized by treewidth. Their approach integrates tools from graph theory, parameterized algorithms, max s-t flow, and general factor theory, revealing a novel connection between flow and matching variants.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We study $\tau$-Bounded-Density Edge Deletion ($\tau$-BDED), where given an undirected graph $G$, the task is to remove as few edges as possible to obtain a graph $G'$ where no subgraph of $G'$ has density more than $\tau$. The density of a (sub)graph is the number of edges divided by the number of vertices. This problem was recently introduced and shown to be NP-hard for $\tau \in \{2/3, 3/4, 1 + 1/25\}$, but polynomial-time solvable for $\tau \in \{0,1/2,1\}$ [Bazgan et al., JCSS 2025]. We provide a complete dichotomy with respect to the target density $\tau$: 1. If $2\tau \in \mathbb{N}$ (half-integral target density) or $\tau<2/3$, then $\tau$-BDED is polynomial-time solvable. 2. Otherwise, $\tau$-BDED is NP-hard. We complement the NP-hardness with fixed-parameter tractability with respect to the treewidth of $G$. Moreover, for integral target density $\tau \in \mathbb{N}$, we show $\tau$-BDED to be solvable in randomized $O(m^{1 + o(1)})$ time. Our algorithmic results are based on a reduction to a new general flow problem on restricted networks that, depending on $\tau$, can be solved via Maximum s-t-Flow or General Factors. We believe this connection between these variants of flow and matching to be of independent interest.

Problem

• edge deletion
• subgraph density
• density bound
• graph modification
• NP-hardness

Innovation

• complexity dichotomy
• edge deletion
• graph density
• fixed-parameter tractability
• generalized flow