Treffer: Packing directed cycles efficiently

Let G be a simple digraph. The dicycle packing number of G, denoted vc(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 & of directed cycles in G to R+ is a fractional dicycle packing of G if Σe∈C∈ψ(C)≤w(e) for each e e E(G). The fractional dicycle packing number, denoted v*c(G, w), is the maximum value of ΣC∈ψ(C) taken over all fractional dicycle packings ψ. In case w ≡ 1 we denote the latter parameter by v*c(G). Our main result is that v*c(G) - vc(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 vc(G)-o(n2) in randomized polynomial time. Since computing vc(G) is an NP-Hard problem, and since almost all digraphs have vc(G) = Θ(n2) our result is a FPTAS for computing vc(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 vF(G) denote the maximum number of arc-disjoint copies of elements of F that can be found in G, and let v*F(G) denote the fractional relaxation. Then, v*F(G) - vF(G) = o(n2). This lemma uses the recently discovered directed regularity lemma as its main tool. It is well known that v*c(G, w) can he 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.