# Packing Directed Cycles Efficiently

• Zeev Nutov
• Raphael Yuster
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3153)

## Abstract

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 $${\cal C}$$ of directed cycles in G to R  +  is a fractional dicycle packing of G if $$\sum_{e \in C \in {\cal C}} {\psi(C)} \leq w(e)$$ for each eE(G). The fractional dicycle packing number, denoted ν $$_{c}^{\rm *}$$(G,w), is the maximum value of $$\sum_{C \in {\cal C}} \psi(C)$$ taken over all fractional dicycle packings ψ. In case w ≡ 1 we denote the latter parameter by ν $$_{c}^{\rm *}$$(G).

Our main result is that ν $$_{c}^{\rm *}$$(G) – ν c (G)=o(n 2) 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(n 2) 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 $${\cal F}$$ be any fixed family of oriented graphs. For an oriented graph G, let $$\nu_{\cal F}(G)$$ denote the maximum number of arc-disjoint copies of elements of $${\cal F}$$ that can be found in G, and let $$\nu_{\cal F}^*(G)$$ denote the fractional relaxation. Then, $$\nu_{\cal F}^*(G) - \nu_{\cal F}(G)=o(n^2)$$. This lemma uses the recently discovered directed regularity lemma as its main tool.

It is well known that ν $$_{c}^{\rm *}$$(G,w) can be computed in polynomial time by considering the dual problem. However, it was an open problem whether an optimal fractional dicycle packing ψyielding ν $$_{c}^{\rm *}$$(G,w) can be generated in polynomial time. We prove that a maximum fractional dicycle packing yielding ν $$_{c}^{\rm *}$$(G,w) with at most O(n 2) dicycles receiving nonzero weight can be found in polynomial time.

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