Packing directed cycles efficiently |
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Authors: | Zeev Nutov |
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Affiliation: | a Department of Computer Science, The Open University of Israel, Tel Aviv, Israel b Department of Mathematics, University of Haifa, Haifa 31905, Israel |
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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 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∈Cψ(C)?w(e) for each e∈E(G). The fractional dicycle packing number, denoted , is the maximum value of ∑C∈Cψ(C) taken over all fractional dicycle packings ψ. In case w≡1 we denote the latter parameter by .Our main result is that 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(n2) our result is a FPTAS for computing νc(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 νF(G) denote the maximum number of arc-disjoint copies of elements of F that can be found in G, and let denote the fractional relaxation. Then, . This lemma uses the recently discovered directed regularity lemma as its main tool.It is well known that can be 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. |
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Keywords: | Cycles Packing Digraph |
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