Polynomial Kernels for Deletion to Classes of Acyclic Digraphs

Authors Matthias Mnich, Erik Jan van Leeuwen

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Matthias Mnich
Erik Jan van Leeuwen

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Matthias Mnich and Erik Jan van Leeuwen. Polynomial Kernels for Deletion to Classes of Acyclic Digraphs. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 55:1-55:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


We consider the problem to find a set X of vertices (or arcs) with |X| <= k in a given digraph G such that D = G-X is an acyclic digraph. In its generality, this is DIRECTED FEEDBACK VERTEX SET or DIRECTED FEEDBACK ARC SET respectively. The existence of a polynomial kernel for these problems is a notorious open problem in the field of kernelization, and little progress has been made. In this paper, we consider both deletion problems with an additional restriction on D, namely that D must be an out-forest, an out-tree, or a (directed) pumpkin. Our main results show that for each of these three restrictions the vertex deletion problem remains NP-hard, but we can obtain a kernel with k^{O(1)} vertices on general digraphs G. We also show that, in contrast to the vertex deletion problem, the arc deletion problem with each of the above restrictions can be solved in polynomial time.
  • directed feedback vertex/arc set
  • parameterized algorithms
  • kernels


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