Hitting minors on bounded treewidth graphs. IV. An optimal algorithm

For a fixed finite collection of graphs cal F, the cal F-M-DELETION problem asks, given an n-vertex input graph G, for the minimum number of vertices that intersect all minor models in G of the graphs in cal F. by Courcelle Theorem, this problem can be solved in time f_cal F(tw)· nO(1), where tw is the treewidth of G, for some function f_cal F depending on cal F In a recent series of articles, we have initiated the programme of optimizing asymptotically the function f_cal F. Here we provide an algorithm showing that f_cal F(tw) = 2O(tw· log tw) for every collection cal F. Prior to this work, the best known function f_cal F was double-exponential in tw. In particular, our algorithm vastly extends the results of Jansen et al. [SODA 2014] for the particular case cal F=K5,K3,3 and of Kociumaka and Pilipczuk [Algorithmica 2019] for graphs of bounded genus, and answers an open problem posed by Cygan et al. [Inf Comput 2017]. 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 cal F [Theor Comput Sci 2020] and general lower bounds [J Comput Syst Sci 2020], our algorithm yields the following complexity dichotomy when cal F = H contains a single connected graph H, assuming the Exponential Time Hypothesis: fH(tw)=2Theta(tw) if H is a contraction of the chair or the banner, and fH(tw)=2Theta(tw· log tw) otherwise.