## Abstract

A key result in the field of kernelization, a subfield of parameterized complexity, states that the classic Disjoint Cycle Packing problem, i.e. finding k vertex disjoint cycles in a given graph G, admits no polynomial kernel unless NP ⊆ coNP/poly. However, very little is known about this problem beyond the aforementioned kernelization lower bound (within the parameterized complexity framework). In the hope of clarifying the picture and better understanding the types of "constraints" that separate "kernelizable" from "non-kernelizable" variants of Disjoint Cycle Packing, we investigate two relaxations of the problem. The first variant, which we call Almost Disjoint Cycle Packing, introduces a "global" relaxation parameter t. That is, given a graph G and integers k and t, the goal is to find at least k distinct cycles such that every vertex of G appears in at most t of the cycles. The second variant, Pairwise Disjoint Cycle Packing, introduces a "local" relaxation parameter and we seek at least k distinct cycles such that every two cycles intersect in at most t vertices. While the Pairwise Disjoint Cycle Packing problem admits a polynomial kernel for all t ≥ 1, the kernelization complexity of Almost Disjoint Cycle Packing reveals an interesting spectrum of upper and lower bounds. In particular, for t = k/c, where c could be a function of k, we obtain a kernel of size O(2^{c2} k^{7+c} log^{3} k) whenever c ϵ o( √ k). Thus the kernel size varies from being sub-exponential when c ϵ o( √ k), to quasipolynomial when c ϵ o(logℓ k), ℓ ϵ ℝ+, and polynomial when c ϵ O(1). We complement these results for Almost Disjoint Cycle Packing by showing that the problem does not admit a polynomial kernel whenever t ϵ O(kϵ), for any 0 ≤ ϵ < 1.

Original language | English |
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Title of host publication | 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016 |

Editors | Yuval Rabani, Ioannis Chatzigiannakis, Davide Sangiorgi, Michael Mitzenmacher |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770132 |

DOIs | |

State | Published - 1 Aug 2016 |

Externally published | Yes |

Event | 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016 - Rome, Italy Duration: 12 Jul 2016 → 15 Jul 2016 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 55 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016 |
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Country/Territory | Italy |

City | Rome |

Period | 12/07/16 → 15/07/16 |

## Keywords

- Cycle packing
- Kernelization
- Parameterized complexity
- Relaxation

## ASJC Scopus subject areas

- Software