Abstract
We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time \(2^{o(t\log t)}\cdot n^{\mathcal {O}(1)}\) on general directed graphs, where t is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time \(2^{\mathcal {O}(t\log t)}\cdot n^{\mathcal {O}(1)}\). On the other hand, we show that if the input digraph is planar, then the running time can be improved to \(2^{\mathcal {O}(t)}\cdot n^{\mathcal {O}(1)}\).
Work supported by the National Science Centre of Poland, grant number 2013/11/D/ST6/03073 (MP, MW). The work of Ł. Kowalik is a part of the project TOTAL that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 677651). This research is a part of projects that have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreements No 714704 (AS). MP and MW are supported by the Foundation for Polish Science (FNP) via the START stipend programme. JN is supported by NWO Veni grant 639.021.438.
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Notes
- 1.
In general digraphs, DFVS and DFAS are well-known to be reducible to each other; see [5, Proposition 8.42 and Exercise 8.16]. These reductions, however, do not preserve planarity of the digraph in question.
- 2.
Throughout this paper, the treewidth of a directed graph is defined as the treewidth of its underlying undirected graph.
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Bonamy, M., Kowalik, Ł., Nederlof, J., Pilipczuk, M., Socała, A., Wrochna, M. (2018). On Directed Feedback Vertex Set Parameterized by Treewidth. In: Brandstädt, A., Köhler, E., Meer, K. (eds) Graph-Theoretic Concepts in Computer Science. WG 2018. Lecture Notes in Computer Science(), vol 11159. Springer, Cham. https://doi.org/10.1007/978-3-030-00256-5_6
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