Packing Directed Cycles Efficiently

Let G be a simple digraph. The dicycle packing number of G, denoted νc(G), is the maximum size of a set of arc-disjoint directed cycles in G. Let G be a digraph with a nonnegative arcweight 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) − νc(G) = o(n) where n = |V (G)|. Our proof is algorithmic and generates a set of arc-disjoint directed cycles whose size is at least νc(G) − o(n) in randomized polynomial time. Since computing νc(G) is an NP-Hard problem, and since almost all digraphs have νc(G) = Θ(n ) our result is a FPTAS for computing νc(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 νF(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) − νF(G) = o(n). 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 ψ yielding ν c (G,w). 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 yielding ν c (G,w) with at most O(n ) dicycles receiving nonzero weight can be found in polynomial time.