When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2017.68
URN: urn:nbn:de:0030-drops-74301
URL: https://drops.dagstuhl.de/opus/volltexte/2017/7430/
 Go to the corresponding LIPIcs Volume Portal

### Linear-Time Kernelization for Feedback Vertex Set

 pdf-format:

### Abstract

In this paper, we give an algorithm that, given an undirected graph G of m edges and an integer k, computes a graph G' and an integer k' in O(k^4 m) time such that (1) the size of the graph G' is O(k^2), (2) k' \leq k, and (3) 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'. This is the first linear-time polynomial-size kernel for Feedback Vertex Set. The size of our kernel is 2k^2+k vertices and 4k^2 edges, which is smaller than the previous best of 4k^2 vertices and 8k^2 edges. Thus, we improve the size and the running time simultaneously. We note that under the assumption of NP \not\subseteq coNP/poly, Feedback Vertex Set does not admit an O(k^{2-\epsilon})-size kernel for any \epsilon>0.

Our kernel exploits k-submodular relaxation, which is a recently developed technique for obtaining efficient FPT algorithms for various problems. The dual of k-submodular relaxation of Feedback Vertex Set can be seen as a half-integral variant of A-path packing, and to obtain the linear-time complexity, we give an efficient augmenting-path algorithm for this problem. We believe that this combinatorial algorithm is of independent interest.

A solver based on the proposed method won first place in the 1st Parameterized Algorithms and Computational Experiments (PACE) challenge.

### BibTeX - Entry

@InProceedings{iwata:LIPIcs:2017:7430,
author =	{Yoichi Iwata},
title =	{{Linear-Time Kernelization for Feedback Vertex Set}},
booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
pages =	{68:1--68:14},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-041-5},
ISSN =	{1868-8969},
year =	{2017},
volume =	{80},
editor =	{Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},