A directed graph D is semicomplete if for every pair x, y of vertices of D, there is at least one arc between x and y. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph D = (V, A) and a pair of natural numbers k and `, we are to decide whether there is a subset X of V of size k such that the largest strong connectivity component in D − X has at most ` vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for ` = 1. We study parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters: k, `, ` + k and n − `. In particular, we prove that DCOC with parameter k on semicomplete digraphs can be solved in time O∗(216k) but not in time O∗(2o(k)) unless the Exponential Time Hypothesis (ETH) fails. The upper bound O∗(216k) implies the upper bound O∗(216(n−`)) for the parameter n− `. We complement the latter by showing that there is no algorithm of time complexity O∗(2o(n−`)) unless ETH fails. Finally, we improve (in dependency on `) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter ` + k on general digraphs from O∗(2O(k` log(k`))) to O∗(2O(k log(k`))). Note that Drange, Dregi and van’t Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time O∗(2o(k log `)) unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity O∗(2o(k log k)).
|Titel||15th International Symposium on Parameterized and Exact Computation, IPEC 2020|
|Redaktører||Yixin Cao, Marcin Pilipczuk|
|Forlag||Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing|
|Status||Udgivet - dec. 2020|
|Begivenhed||15th International Symposium on Parameterized and Exact Computation, IPEC 2020 - Virtual, Hong Kong, Kina|
Varighed: 14. dec. 2020 → 18. dec. 2020
|Konference||15th International Symposium on Parameterized and Exact Computation, IPEC 2020|
|By||Virtual, Hong Kong|
|Periode||14/12/2020 → 18/12/2020|
|Navn||Leibniz International Proceedings in Informatics, LIPIcs|
Bibliografisk noteFunding Information:
Funding Jørgen Bang-Jensen: Research supported by the Independent Research Fund Denmark under grant number DFF 7014-00037B. Gregory Gutin: Research supported by the Leverhulme Trust under grant number RPG-2018-161.
© Jørgen Bang-Jensen, Eduard Eiben, Gregory Gutin, Magnus Wahlström, and Anders Yeo;