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.

翻译：我们研究τ-有界密度边删除问题（τ-BDED），其定义为：给定一个无向图G，目标是删除尽可能少的边，从而得到一个图G'，使得G'中不存在密度超过τ的子图。图（子图）的密度定义为边数与顶点数之比。该问题最近被提出，并已证明对于τ∈{2/3, 3/4, 1 + 1/25}是NP难的，但对于τ∈{0,1/2,1}可在多项式时间内求解[Bazgan等人，JCSS 2025]。我们针对目标密度τ给出了完整的二分性结果：1. 若2τ∈ℕ（半整目标密度）或τ<2/3，则τ-BDED可在多项式时间内求解。2. 否则，τ-BDED是NP难的。我们进一步通过关于图G树宽的固定参数可处理性补充了NP难性结果。此外，对于整目标密度τ∈ℕ，我们证明了τ-BDED可在随机O(m^{1 + o(1)})时间内求解。我们的算法结果基于对受限网络上一种新型通用流问题的归约，该问题根据τ的不同，可通过最大s-t流或通用因子方法求解。我们认为这些流与匹配变体之间的联系具有独立的研究价值。