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⧵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 ̸ ⊆ coNP/poly).
@InProceedings{bentert_et_al:LIPIcs.IPEC.2024.14, author = {Bentert, Matthias and Fomin, Fedor V. and Hauser, Fanny and Saurabh, Saket}, title = {{The Parameterized Complexity Landscape of Two-Sets Cut-Uncut}}, booktitle = {19th International Symposium on Parameterized and Exact Computation (IPEC 2024)}, pages = {14:1--14:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-353-9}, ISSN = {1868-8969}, year = {2024}, volume = {321}, editor = {Bonnet, \'{E}douard and Rz\k{a}\.{z}ewski, Pawe{\l}}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2024.14}, URN = {urn:nbn:de:0030-drops-222400}, doi = {10.4230/LIPIcs.IPEC.2024.14}, annote = {Keywords: Fixed-parameter tractability, Polynomial Kernels, W\lbrack1\rbrack-hardness, XP, para-NP-Hardness} }
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