## Abstract

Let G be a simple digraph. The dicycle packing number of G, denoted ν_{c} (G), is the maximum size of a set of arc-disjoint directed cycles in G. Let G be a digraph with a nonnegative arc-weight function w. A function ψ from the set C of directed cycles in G to R_{+} is a fractional dicycle packing of G if ∑_{e ∈ C ∈ C} ψ (C) ≤ w (e) for each e ∈ E (G). The fractional dicycle packing number, denoted ν_{c} ^{*} (G, w), is the maximum value of ∑_{C ∈ C} ψ (C) taken over all fractional dicycle packings ψ. In case w ≡ 1 we denote the latter parameter by ν_{c}^{*} (G). Our main result is that ν_{c}^{*} (G) - ν_{c} (G) = o (n^{2}) where n = | V (G) |. Our proof is algorithmic and generates a set of arc-disjoint directed cycles whose size is at least ν_{c} (G) - o (n^{2}) in randomized polynomial time. Since computing ν_{c} (G) is an NP-Hard problem, and since almost all digraphs have ν_{c} (G) = Θ (n^{2}) our result is a FPTAS for computing ν_{c} (G) for almost all digraphs. The result uses as its main lemma a much more general result. Let F be any fixed family of oriented graphs. For an oriented graph G, let ν_{F} (G) denote the maximum number of arc-disjoint copies of elements of F that can be found in G, and let ν_{F}^{*} (G) denote the fractional relaxation. Then, ν_{F}^{*} (G) - ν_{F} (G) = o (n^{2}). This lemma uses the recently discovered directed regularity lemma as its main tool. It is well known that ν_{c}^{*} (G, w) can be computed in polynomial time by considering the dual problem. We present a polynomial algorithm that finds an optimal fractional dicycle packing. Our algorithm consists of a solution to a simple linear program and some minor modifications, and avoids using the ellipsoid method. In fact, the algorithm shows that a maximum fractional dicycle packing with at most O (n^{2}) dicycles receiving nonzero weight can be found in polynomial time.

Original language | English |
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Pages (from-to) | 82-91 |

Number of pages | 10 |

Journal | Discrete Applied Mathematics |

Volume | 155 |

Issue number | 2 |

DOIs | |

State | Published - 15 Jan 2007 |

## Keywords

- Cycles
- Digraph
- Packing