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Generalized feedback vertex set problems on bounded-treewidth graphs: chordality is the key to single-exponential parameterized algorithms

Abstract : 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, the Bounded P-Block Vertex Deletion problem asks, given a graph G on n vertices and positive integers k and d, whether there is a set S of at most k vertices of G 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 4, then the problem is not solvable in time 2 o(w log w) n O(1) even for fixed d = , unless the ETH fails. As a warm up, we consider the analogous Bounded P-Component Vertex Deletion problem requiring each connected component to be a member of P, and show that chordality is also the key to single-exponential parameterized algorithms for this problem.
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Submitted on : Friday, January 25, 2019 - 2:16:27 PM
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Long-term archiving on: : Friday, April 26, 2019 - 3:17:29 PM

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Edouard Bonnet, Nick Brettell, O-Joung Kwon, Dániel Marx. Generalized feedback vertex set problems on bounded-treewidth graphs: chordality is the key to single-exponential parameterized algorithms. 12th International Symposium on Parameterized and Exact Computation (IPEC 2017), Sep 2017, Vienne, Austria. pp.7. ⟨hal-01994406⟩

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