Erdocombining double acute accents-Pósa property and its algorithmic applications - Parity constraints, subset feedback set, and subset packing

Naonori Kakimura, Ken Ichi Kawarabayashi, Yusuke Kobayashi

Research output: Chapter in Book/Report/Conference proceedingConference contribution

21 Citations (Scopus)

Abstract

The well-known Erdocombining double acute accents-Pósa theorem says that for any integer k and any graph G, either G contains k vertex-disjoint cycles or a vertex set X of order at most c-k log k (for some constant c) such that G-X is a forest. Thomassen [39] extended this result to the even cycles, but on the other hand, it is well-known that this theorem is no longer true for the odd cycles. However, Reed [31] proved that this theorem still holds if we relax k vertex-disjoint odd cycles to k odd cycles with each vertex in at most two of them. These theorems initiate many researches in both graph theory and theoretical computer science. In the graph theory side, our problem setting is that we are given a graph and a vertex set S, and we want to extend all the above results to cycles that are required to go through a subset of S, i.e., each cycle contains at least one vertex in S (such a cycle is called an S-cycle). It was shown in [20] that the above Erdocombining double acute accents-Pósa theorem still holds for this subset version. In this paper, we extend both Thomassen's result and Reed's result in this way. In the theoretical computer science side, we investigate generalizations of the following well-known problems in the framework of parameterized complexity: the feedback set problem and the cycle packing problem. Our purpose here is to consider the following problems: the feedback set problem with respect to the S-cycles, and the S-cycle packing problem. We give the first fixed parameter algorithms for the two problems. Namely; 1. For fixed k, we can either find a vertex set X of size k such that G-X has no S-cycle, or conclude that such a vertex set does not exist in O(n 2m) time (independently obtained in [7]). 2. For fixed k, we can either find k vertex-disjoint S-cycles, or conclude that such k disjoint cycles do not exist in O(n 2m) time. We also extend the above results to those with the parity constraints as follows; 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 even S-cycle, 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 odd S-cycle, or concludes that such a vertex set does not exist. 3. For a parameter k, there exists a fixed parameter algorithm that either finds k vertex-disjoint even S-cycles, or concludes that such k disjoint cycles do not exist. 4. For a parameter k, there exists a fixed parameter algorithm that either finds k odd S-cycles with each vertex in at most two of them, or concludes that such k cycles do not exist.

Original languageEnglish
Title of host publicationProceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012
Pages1726-1736
Number of pages11
Publication statusPublished - 2012
Externally publishedYes
Event23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 - Kyoto, Japan
Duration: 2012 Jan 172012 Jan 19

Other

Other23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012
CountryJapan
CityKyoto
Period12/1/1712/1/19

Fingerprint

Acute
Parity
Packing
Feedback
Cycle
Subset
Graph theory
Vertex of a graph
Computer science
Fixed-parameter Algorithms
Disjoint
Odd Cycle
Theorem
Packing Problem
Computer Science
Odd
Parameterized Complexity
Graph in graph theory

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

Cite this

Kakimura, N., Kawarabayashi, K. I., & Kobayashi, Y. (2012). Erdocombining double acute accents-Pósa property and its algorithmic applications - Parity constraints, subset feedback set, and subset packing. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 (pp. 1726-1736)

Erdocombining double acute accents-Pósa property and its algorithmic applications - Parity constraints, subset feedback set, and subset packing. / Kakimura, Naonori; Kawarabayashi, Ken Ichi; Kobayashi, Yusuke.

Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012. 2012. p. 1726-1736.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Kakimura, N, Kawarabayashi, KI & Kobayashi, Y 2012, Erdocombining double acute accents-Pósa property and its algorithmic applications - Parity constraints, subset feedback set, and subset packing. in Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012. pp. 1726-1736, 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, 12/1/17.
Kakimura N, Kawarabayashi KI, Kobayashi Y. Erdocombining double acute accents-Pósa property and its algorithmic applications - Parity constraints, subset feedback set, and subset packing. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012. 2012. p. 1726-1736
Kakimura, Naonori ; Kawarabayashi, Ken Ichi ; Kobayashi, Yusuke. / Erdocombining double acute accents-Pósa property and its algorithmic applications - Parity constraints, subset feedback set, and subset packing. Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012. 2012. pp. 1726-1736
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abstract = "The well-known Erdocombining double acute accents-P{\'o}sa theorem says that for any integer k and any graph G, either G contains k vertex-disjoint cycles or a vertex set X of order at most c-k log k (for some constant c) such that G-X is a forest. Thomassen [39] extended this result to the even cycles, but on the other hand, it is well-known that this theorem is no longer true for the odd cycles. However, Reed [31] proved that this theorem still holds if we relax k vertex-disjoint odd cycles to k odd cycles with each vertex in at most two of them. These theorems initiate many researches in both graph theory and theoretical computer science. In the graph theory side, our problem setting is that we are given a graph and a vertex set S, and we want to extend all the above results to cycles that are required to go through a subset of S, i.e., each cycle contains at least one vertex in S (such a cycle is called an S-cycle). It was shown in [20] that the above Erdocombining double acute accents-P{\'o}sa theorem still holds for this subset version. In this paper, we extend both Thomassen's result and Reed's result in this way. In the theoretical computer science side, we investigate generalizations of the following well-known problems in the framework of parameterized complexity: the feedback set problem and the cycle packing problem. Our purpose here is to consider the following problems: the feedback set problem with respect to the S-cycles, and the S-cycle packing problem. We give the first fixed parameter algorithms for the two problems. Namely; 1. For fixed k, we can either find a vertex set X of size k such that G-X has no S-cycle, or conclude that such a vertex set does not exist in O(n 2m) time (independently obtained in [7]). 2. For fixed k, we can either find k vertex-disjoint S-cycles, or conclude that such k disjoint cycles do not exist in O(n 2m) time. We also extend the above results to those with the parity constraints as follows; 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 even S-cycle, 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 odd S-cycle, or concludes that such a vertex set does not exist. 3. For a parameter k, there exists a fixed parameter algorithm that either finds k vertex-disjoint even S-cycles, or concludes that such k disjoint cycles do not exist. 4. For a parameter k, there exists a fixed parameter algorithm that either finds k odd S-cycles with each vertex in at most two of them, or concludes that such k cycles do not exist.",
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