Density Matters: A Complexity Dichotomy of Deleting Edges to Bound Subgraph Density
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| Autores principales: | , , , , |
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| Formato: | Preprint |
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2026
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| _version_ | 1866911356993667072 |
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| author | Bentert, Matthias Breitkopf, Tom-Lukas Froese, Vincent Herrmann, Anton Nichterlein, André |
| author_facet | Bentert, Matthias Breitkopf, Tom-Lukas Froese, Vincent Herrmann, Anton Nichterlein, André |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03129 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Density Matters: A Complexity Dichotomy of Deleting Edges to Bound Subgraph Density Bentert, Matthias Breitkopf, Tom-Lukas Froese, Vincent Herrmann, Anton Nichterlein, André Data Structures and Algorithms Discrete Mathematics 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. |
| title | Density Matters: A Complexity Dichotomy of Deleting Edges to Bound Subgraph Density |
| topic | Data Structures and Algorithms Discrete Mathematics |
| url | https://arxiv.org/abs/2601.03129 |