The Parameterized Complexity Landscape of Two-Sets Cut-Uncut
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| Format: | Preprint |
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2024
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| _version_ | 1866929472790331392 |
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| author | Bentert, Matthias Fomin, Fedor V. Hauser, Fanny Saurabh, Saket |
| author_facet | Bentert, Matthias Fomin, Fedor V. Hauser, Fanny Saurabh, Saket |
| contents | In Two-Sets Cut-Uncut, we are given an undirected graph $G=(V,E)$ and two terminal sets $S$ and $T$. The task is to find a minimum cut $C$ in $G$ (if there is any) separating $S$ from $T$ under the following ``uncut'' condition. In the graph $(V,E \setminus C)$, the terminals in each terminal set remain in the same connected component. In spite of the superficial similarity to the classic problem Minimum $s$-$t$-Cut, Two-Sets Cut-Uncut is computationally challenging. In particular, even deciding whether such a cut of any size exists, is already NP-complete. We initiate a systematic study of Two-Sets Cut-Uncut within the context of parameterized complexity. By leveraging known relations between many well-studied graph parameters, we characterize the structural properties of input graphs that allow for polynomial kernels, fixed-parameter tractability (FPT), and slicewise polynomial algorithms (XP). Our main contribution is the near-complete establishment of the complexity of these algorithmic properties within the described hierarchy of graph parameters. On a technical level, our main results are fixed-parameter tractability for the (vertex-deletion) distance to cographs and an OR-cross composition excluding polynomial kernels for the vertex cover number of the input graph (under the standard complexity assumption NP is not contained in coNP/poly). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_13543 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Parameterized Complexity Landscape of Two-Sets Cut-Uncut Bentert, Matthias Fomin, Fedor V. Hauser, Fanny Saurabh, Saket Data Structures and Algorithms Discrete Mathematics In Two-Sets Cut-Uncut, we are given an undirected graph $G=(V,E)$ and two terminal sets $S$ and $T$. The task is to find a minimum cut $C$ in $G$ (if there is any) separating $S$ from $T$ under the following ``uncut'' condition. In the graph $(V,E \setminus C)$, the terminals in each terminal set remain in the same connected component. In spite of the superficial similarity to the classic problem Minimum $s$-$t$-Cut, Two-Sets Cut-Uncut is computationally challenging. In particular, even deciding whether such a cut of any size exists, is already NP-complete. We initiate a systematic study of Two-Sets Cut-Uncut within the context of parameterized complexity. By leveraging known relations between many well-studied graph parameters, we characterize the structural properties of input graphs that allow for polynomial kernels, fixed-parameter tractability (FPT), and slicewise polynomial algorithms (XP). Our main contribution is the near-complete establishment of the complexity of these algorithmic properties within the described hierarchy of graph parameters. On a technical level, our main results are fixed-parameter tractability for the (vertex-deletion) distance to cographs and an OR-cross composition excluding polynomial kernels for the vertex cover number of the input graph (under the standard complexity assumption NP is not contained in coNP/poly). |
| title | The Parameterized Complexity Landscape of Two-Sets Cut-Uncut |
| topic | Data Structures and Algorithms Discrete Mathematics |
| url | https://arxiv.org/abs/2408.13543 |