Fixed-parameter tractability for subset feedback set problems with parity constraints

The subset feedback set problem, which is a generalization of the well-known feedback vertex set problem, is that we are given an undirected graph G with a vertex subset S and a positive integer k, and the goal is to find a vertex set X of size at most k such that G−X has no S-cycle, where an S-cycl...

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Published in:	Theoretical computer science Vol. 576; pp. 61 - 76
Main Authors:	Kakimura, Naonori, Kawarabayashi, Ken-ichi
Format:	Journal Article
Language:	English
Published:	Elsevier B.V 20.04.2015
Subjects:	Algorithms Feedback Fixed-parameter algorithm Graph minor theory Graphs Integers Parity Parity constraints Subset feedback set problem Fixed-parameter algorithm Subset feedback set problem Parity constraints Graph minor theory
ISSN:	0304-3975, 1879-2294
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Abstract	The subset feedback set problem, which is a generalization of the well-known feedback vertex set problem, is that we are given an undirected graph G with a vertex subset S and a positive integer k, and the goal is to find a vertex set X of size at most k such that G−X has no S-cycle, where an S-cycle is a cycle having at least one vertex of S. It was recently shown that this problem is fixed parameter tractable, where k is the parameter. In this paper, we further generalize this problem to one with the parity constraints, and show the fixed parameter tractability:1.For a parameter k, there exists a fixed-parameter algorithm that either finds a vertex set X of size k such that G−X has no S-cycle of even length, or concludes that such a vertex set does not exist.2.For a parameter k, there exists a fixed-parameter algorithm that either finds a vertex set X of size k such that G−X has no S-cycle of odd length, or concludes that such a vertex set does not exist.
AbstractList	The subset feedback set problem, which is a generalization of the well-known feedback vertex set problem, is that we are given an undirected graph G with a vertex subset S and a positive integer k, and the goal is to find a vertex set X of size at most k such that has no S-cycle, where an S-cycle is a cycle having at least one vertex of S. It was recently shown that this problem is fixed parameter tractable, where k is the parameter. In this paper, we further generalize this problem to one with the parity constraints, and show the fixed parameter tractability: For a parameter k, there exists a fixed-parameter algorithm that either finds a vertex set X of size k such that has no S-cycle of even length, or concludes that such a vertex set does not exist. For a parameter k, there exists a fixed-parameter algorithm that either finds a vertex set X of size k such that has no S-cycle of odd length, or concludes that such a vertex set does not exist. The subset feedback set problem, which is a generalization of the well-known feedback vertex set problem, is that we are given an undirected graph G with a vertex subset S and a positive integer k, and the goal is to find a vertex set X of size at most k such that G−X has no S-cycle, where an S-cycle is a cycle having at least one vertex of S. It was recently shown that this problem is fixed parameter tractable, where k is the parameter. In this paper, we further generalize this problem to one with the parity constraints, and show the fixed parameter tractability:1.For a parameter k, there exists a fixed-parameter algorithm that either finds a vertex set X of size k such that G−X has no S-cycle of even length, or concludes that such a vertex set does not exist.2.For a parameter k, there exists a fixed-parameter algorithm that either finds a vertex set X of size k such that G−X has no S-cycle of odd length, or concludes that such a vertex set does not exist.
Author	Kakimura, Naonori Kawarabayashi, Ken-ichi
Author_xml	– sequence: 1 givenname: Naonori surname: Kakimura fullname: Kakimura, Naonori email: kakimura@global.c.u-tokyo.ac.jp organization: Graduate School of Arts and Sciences, University of Tokyo, Tokyo 153-8902, Japan – sequence: 2 givenname: Ken-ichi surname: Kawarabayashi fullname: Kawarabayashi, Ken-ichi email: k_keniti@nii.ac.jp organization: National Institute of Informatics, 2-1-2, Hitotsubashi, Chiyoda-ku, Tokyo, Japan
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Keywords	Fixed-parameter algorithm Subset feedback set problem Parity constraints Graph minor theory
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Snippet	The subset feedback set problem, which is a generalization of the well-known feedback vertex set problem, is that we are given an undirected graph G with a...
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SubjectTerms	Algorithms Feedback Fixed-parameter algorithm Graph minor theory Graphs Integers Parity Parity constraints Subset feedback set problem
Title	Fixed-parameter tractability for subset feedback set problems with parity constraints
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