eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-03-02
7:1
7:13
10.4230/LIPIcs.IPEC.2017.7
article
Generalized Feedback Vertex Set Problems on Bounded-Treewidth Graphs: Chordality Is the Key to Single-Exponential Parameterized Algorithms
Bonnet, Édouard
Brettell, Nick
Kwon, O-joung
Marx, Dániel
It has long been known that Feedback Vertex Set can be solved in time 2^O(w log w)n^O(1) on graphs of treewidth w, but it was only recently that this running time was improved to 2^O(w)n^O(1), that is, to single-exponential parameterized by treewidth. We investigate which generalizations of Feedback Vertex Set can be solved in a similar running time. Formally, for a class of graphs P, Bounded P-Block Vertex Deletion asks, given a graph G on n vertices and positive integers k and d, whether G contains a set S of at most k vertices
such that each block of G-S has at most d vertices and is in P. Assuming that P is recognizable in polynomial time and satisfies a certain natural hereditary condition, we give a sharp characterization of when single-exponential parameterized algorithms are possible for fixed values of d:
- if P consists only of chordal graphs, then the problem can be solved in time 2^O(wd^2) n^{O}(1),
- if P contains a graph with an induced cycle of length ell>= 4, then the problem is not solvable in time 2^{o(w log w)} n^O(1)} even for fixed d=ell, unless the ETH fails.
We also study a similar problem, called Bounded P-Component Vertex Deletion, where the target graphs have connected components of small size instead of having blocks of small size, and present analogous results.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol089-ipec2017/LIPIcs.IPEC.2017.7/LIPIcs.IPEC.2017.7.pdf
fixed-parameter tractable algorithms
treewidth
feedback vertex set