5-Approximation for ℋ-Treewidth Essentially as Fast as ℋ-Deletion Parameterized by Solution Size
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The notion of ℋ-treewidth, where ℋ is a hereditary graph class, was recently introduced as a generalization of the treewidth of an undirected graph. Roughly speaking, a graph of ℋ-treewidth at most k can be decomposed into (arbitrarily large) ℋ-subgraphs which interact only through vertex sets of size O(k) which can be organized in a tree-like fashion. ℋ-treewidth can be used as a hybrid parameterization to develop fixed-parameter tractable algorithms for ℋ-deletion problems, which ask to find a minimum vertex set whose removal from a given graph G turns it into a member of ℋ. The bottleneck in the current parameterized algorithms lies in the computation of suitable tree ℋ-decompositions. We present FPT approximation algorithms to compute tree ℋ-decompositions for hereditary and union-closed graph classes ℋ. Given a graph of ℋ-treewidth k, we can compute a 5-approximate tree ℋ-decomposition in time f(O(k)) · n^O(1) whenever ℋ-deletion parameterized by solution size can be solved in time f(k) · n^O(1) for some function f(k) ≥ 2^k. The current-best algorithms either achieve an approximation factor of k^O(1) or construct optimal decompositions while suffering from non-uniformity with unknown parameter dependence. Using these decompositions, we obtain algorithms solving Odd Cycle Transversal in time 2^O(k)· n^O(1) parameterized by 𝖻𝗂𝗉𝖺𝗋𝗍𝗂𝗍𝖾-treewidth and Vertex Planarization in time 2^O(k log k)· n^O(1) parameterized by 𝗉𝗅𝖺𝗇𝖺𝗋-treewidth, showing that these can be as fast as the solution-size parameterizations and giving the first ETH-tight algorithms for parameterizations by hybrid width measures.
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