Hitting Minors on Bounded Treewidth Graphs. IV. An Optimal Algorithm

For a fixed finite collection of graphs F , the 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 F . by Courcelle’s Theorem, this problem can be solved in time fF (tw) · nO(1), where tw is the treewidth of G, for some function fF depending on F . In a recent series of articles, we have initiated the programme of optimizing asymptotically the function fF . Here we provide an algorithm showing that fF (tw) = 2O(tw · log tw) for every collection F . Prior to this work, the best known function fF was double-exponential in tw . In particular, our algorithm vastly extends the results of Jansen et al. [SODA 2014] for the particular case 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 F [Theor Comput Sci 2020] and general lower bounds [J Comput Syst Sci 2020], our algorithm yields the following complexity dichotomy when F = {H} contains a single connected graph H, assuming the Exponential Time Hypothesis: fH(tw) = 2Θ(tw) if H is a contraction of the chair or the banner, and fH(tw) = 2Θ(tw · log tw) otherwise.