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

TL;DR

This work resolves the complexity landscape of deleting edges to bound subgraph density (\(\tau\)-BDED) by proving a complete dichotomy: the problem is solvable in polynomial time when the target density is half-integral ($2\tau\in\mathbb{N}$) or strictly below \(2/3\), and NP-hard otherwise. The authors introduce a novel transshipment framework, \textit{rflow}, that reduces \(\tau\)-BDED to generalized flow problems, which in turn connect to General Factors; this yields a near-linear randomized algorithm for integral targets and fixed-parameter tractability in treewidth. A key technical contribution is showing how flow, matching, and factor problems interrelate in restricted networks, enabling efficient solutions in several cases and sharp hardness results in the rest. The results provide a principled, flow-based approach to a density-deletion problem with practical implications for graph modification and subgraph-density control, and open avenues for extensions to wider network classes and parameter regimes.

Abstract

We study $τ$-Bounded-Density Edge Deletion ($τ$-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 $τ$. 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 $τ\in \{2/3, 3/4, 1 + 1/25\}$, but polynomial-time solvable for $τ\in \{0,1/2,1\}$ [Bazgan et al., JCSS 2025]. We provide a complete dichotomy with respect to the target density $τ$: 1. If $2τ\in \mathbb{N}$ (half-integral target density) or $τ< 2/3$, then $τ$-BDED is polynomial-time solvable. 2. Otherwise, $τ$-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 $τ\in \mathbb{N}$, we show $τ$-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 $τ$, 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.

Figures (2)

• Figure 1: Example transformation of a tbded instance with $\tau \in\mathds{N}$ (on the left) to a flow instance (on the right). All arcs on the same layer of the flow network have the same capacity as indicated at the leftmost arc.
• Figure 2: A schematic illustration of the reduction in \ref{['thm:hardness_below_one']}. Given an instance of xlc with universe $X = \{x_1, x_2, \dots, x_{\ell q}\}$ and set collection $C = \{C_1, C_2, \dots, C_t\}$, we add a vertex $y_i$ for each element $x_i \in X$ and a vertex $c_j$ for each set $C_j \in C$. We add an edge $\{y_i, c_j\}$ whenever $x_i \in C_j$. We further add dummy stars with $\ell-1$ leaves and fully connect their centers to the vertices representing $C$. Any tbded solution then leaves $t$ stars with $\ell$ leaves and there is a a one-to-one mapping between (green) star centers and sets selected in an xlc solution.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Lemma 3
• Lemma 4: Cha00
• Theorem 5
• Lemma 6
• Lemma 7
• Lemma 9
• Lemma 10
• Theorem 11