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Theory of Computing Systems

, Volume 46, Issue 3, pp 566–597 | Cite as

A Cubic Kernel for Feedback Vertex Set and Loop Cutset

  • Hans L. BodlaenderEmail author
  • Thomas C. van Dijk
Open Access
Article

Abstract

The Feedback Vertex Set problem on unweighted, undirected graphs is considered. Improving upon a result by Burrage et al. (Proceedings 2nd International Workshop on Parameterized and Exact Computation, pp. 192–202, 2006), we show that this problem has a kernel with O(k 3) vertices, i.e., there is a polynomial time algorithm, that given a graph G and an integer k, finds a graph G′ with O(k 3) vertices and integer k′≤k, such that G has a feedback vertex set of size at most k, if and only if G′ has a feedback vertex set of size at most k′. Moreover, the algorithm can be made constructive: if the reduced instance G′ has a feedback vertex set of size k′, then we can easily transform a minimum size feedback vertex set of G′ into a minimum size feedback vertex set of G. This kernelization algorithm can be used as the first step of an FPT algorithm for Feedback Vertex Set, but also as a preprocessing heuristic for Feedback Vertex Set.

We also show that the related Loop Cutset problem also has a kernel of cubic size. The kernelization algorithms are experimentally evaluated, and we report on these experiments.

Keywords

Graphs Algorithms Kernelization algorithms Preprocessing Data reduction Feedback vertex set Loop cutset Polynomial kernels Fixed parameter tractability 

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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands

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