At IPEC 2020, Bergougnoux, Bonnet, Brettell, and Kwon (Close Relatives of Feedback Vertex Set Without Single-Exponential Algorithms Parameterized by Treewidth, IPEC 2020, LIPIcs vol. 180, pp. 3:1-3:17) showed that a number of problems related to the classic Feedback Vertex Set (FVS) problem do not admit a 2^{o(k log k)} ⋅ n^{𝒪(1)}-time algorithm on graphs of treewidth at most k, assuming the Exponential Time Hypothesis. This contrasts with the 3^{k} ⋅ k^{𝒪(1)} ⋅ n-time algorithm for FVS using the Cut&Count technique. During their live talk at IPEC 2020, Bergougnoux et al. posed a number of open questions, which we answer in this work. - Subset Even Cycle Transversal, Subset Odd Cycle Transversal, Subset Feedback Vertex Set can be solved in time 2^{𝒪(k log k)} ⋅ n in graphs of treewidth at most k. This matches a lower bound for Even Cycle Transversal of Bergougnoux et al. and improves the polynomial factor in some of their upper bounds. - Subset Feedback Vertex Set and Node Multiway Cut can be solved in time 2^{𝒪(k log k)} ⋅ n, if the input graph is given as a cliquewidth expression of size n and width k. - Odd Cycle Transversal can be solved in time 4^k ⋅ k^{𝒪(1)} ⋅ n if the input graph is given as a cliquewidth expression of size n and width k. Furthermore, the existence of a constant ε > 0 and an algorithm performing this task in time (4-ε)^k ⋅ n^{𝒪(1)} would contradict the Strong Exponential Time Hypothesis. A common theme of the first two algorithmic results is to represent connectivity properties of the current graph in a state of a dynamic programming algorithm as an auxiliary forest with 𝒪(k) nodes. This results in a 2^{𝒪(k log k)} bound on the number of states for one node of the tree decomposition or cliquewidth expression and allows to compare two states in k^{𝒪(1)} time, resulting in linear time dependency on the size of the graph or the input cliquewidth expression.
@InProceedings{jacob_et_al:LIPIcs.IPEC.2021.21, author = {Jacob, Hugo and Bellitto, Thomas and Defrain, Oscar and Pilipczuk, Marcin}, title = {{Close Relatives (Of Feedback Vertex Set), Revisited}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {21:1--21:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.21}, URN = {urn:nbn:de:0030-drops-154049}, doi = {10.4230/LIPIcs.IPEC.2021.21}, annote = {Keywords: feedback vertex set, treewidth, cliquewidth} }
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