Packing directed cycles efficiently

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 (n²) 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 (n²) in randomized polynomial time. Since computing ν_c (G) is an NP-Hard problem, and since almost all digraphs have ν_c (G) = Θ (n²) 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 (n²). 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 (n²) dicycles receiving nonzero weight can be found in polynomial time.