5-Approximation for H-Treewidth Essentially as Fast as H-Deletion Parameterized by Solution Size

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.