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 ω. 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) ≤ ω(e) for each e ∈ E(G). The fractional dicycle packing number, denoted ν_c*l(G, ω), is the maximum value of ∑C∈C ψ(C) taken over all fractional dicycle packings ψ. In case ω = 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(n ²) our result is a FPTAS for computing ν_c(G) for almost all digraphs. The latter 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, ω) can be computed in polynomial time by considering the dual problem. However, it was an open problem whether an optimal fractional dicycle packing ψ yielding ν_c*(G,w) can be generated in polynomial time. We prove that a maximum fractional dicycle packing yielding ν_c*(G, ω) with at most O(n²) dicycles receiving nonzero weight can be found in polynomial time.