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

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.