eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-08-30
66:1
66:16
10.4230/LIPIcs.ESA.2023.66
article
5-Approximation for ℋ-Treewidth Essentially as Fast as ℋ-Deletion Parameterized by Solution Size
Jansen, Bart M. P.
1
https://orcid.org/0000-0001-8204-1268
de Kroon, Jari J. H.
1
https://orcid.org/0000-0003-3328-9712
Włodarczyk, Michał
2
https://orcid.org/0000-0003-0968-8414
Eindhoven University of Technology, The Netherlands
University of Warsaw, Poland
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol274-esa2023/LIPIcs.ESA.2023.66/LIPIcs.ESA.2023.66.pdf
fixed-parameter tractability
treewidth
graph decompositions