A Cubic Kernel for Feedback Vertex Set and Loop Cutset

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 i...

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Published in:Theory of computing systems Vol. 46; no. 3; pp. 566 - 597
Main Authors: Bodlaender, Hans L., van Dijk, Thomas C.
Format: Journal Article
Language:English
Published: New York Springer-Verlag 01.04.2010
Springer Nature B.V
Subjects:
Algorithms
Approximation
Computer Science
Data analysis
Graph theory
Graphs
Linear programming
Polynomials
Studies
Theory of Computation
Algorithms
Preprocessing
Feedback vertex set
Kernelization algorithms
Loop cutset
Graphs
Polynomial kernels
Data reduction
Fixed parameter tractability
ISSN:1432-4350, 1433-0490
Online Access:Get full text
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Summary: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.
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ISSN:1432-4350
1433-0490
DOI:10.1007/s00224-009-9234-2