HITTING MINORS ON BOUNDED TREEWIDTH GRAPHS. IV. AN OPTIMAL ALGORITHM

For a fixed finite collection of graphs \scrF, the \scrF-M-Deletion problem is as follows: given an n-vertex input graph G, find the minimum number of vertices that intersect all minor models in G of the graphs in \scrF. by Courcelle's Theorem, this problem can be solved in time f_\scrF(\sanst\sansw) \cdot n^\scrO(1), where \sanst\sansw is the treewidth of G for some function f_\scrF depending on \scrF. In a recent series of articles, we have initiated the program of optimizing asymptotically the function f_\scrF. Here we provide an algorithm showing that f_\scrF(\sanst\sansw) = 2^\scrO(\sanst\sansw.\mathrm{l}\mathrm{o}\mathrm{g} \sanst\sansw⁾ for every collection \scrF. Prior to this work, the best known function f_\scrF was double-exponential in \sanst\sansw. In particular, our algorithm vastly extends the results of Jansen, Lokshtanov, and Saurabh [Proc. of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2014, pp. 1802-1811] for the particular case \scrF = \{K5,K3,3\} and of Kociumaka and Pilipczuk [Algorithmica, 81 (2019), pp. 3655-3691] for graphs of bounded genus, and answers an open problem posed by Cygan et al. [Inform. Comput., 256 (2017), pp. 62-82]. We combine several ingredients such as the machinery of boundaried graphs in dynamic programming via representatives, the Flat Wall Theorem, bidimensionality, the irrelevant vertex technique, treewidth modulators, and protrusion replacement. Together with our previous results providing single-exponential algorithms for particular collections \scrF [J. Baste, I. Sau, and D. M. Thilikos, Theoret. Comput. Sci., 814 (2020), pp. 135-152] and general lower bounds [J. Baste, I. Sau, and D. M. Thilikos, J. Comput. Syst. Sci., 109 (2020), pp. 56-77], our algorithm yields the following complexity dichotomy when \scrF = \{H\} contains a single connected graph H, assuming the Exponential Time Hypothesis: f_H(\sanst\sansw) = 2^\Theta(\sanst\sansw⁾ if H is a contraction of the \sansc\sansh\sansa\sansi\sansr or the \sansb\sansa\sansn\sansn\sanse\sansr, and f_H(\sanst\sansw) = 2^{\Theta(\sanst\sansw.}\mathrm{l}\mathrm{o}\mathrm{g} \sanst\sansw⁾ otherwise.