Packing directed cycles efficiently

Abstract

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 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 νc*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 νc*(G). Our main result is that νc*(G) - νc(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 νc(G)-o(n2) in randomized polynomial time. Since computing νc(G) is an NP-Hard problem, and since almost all digraphs have νc(G) = Θ(n 2) 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 νℱ*(G) denote the fractional relaxation. Then, νℱ*(G) - νℱ(G) = o(n2). This lemma uses the recently discovered directed regularity lemma as its main tool. It is well known that νc*(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 νc*(G,w) can be generated in polynomial time. We prove that a maximum fractional dicycle packing yielding νc*(G, ω) with at most O(n2) dicycles receiving nonzero weight can be found in polynomial time. © Springer-Verlag 2004.