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• W2005943912 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 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 ) - 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 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 ν F ( 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 ) - ν 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 * ( 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. 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 ( n 2 ) dicycles receiving nonzero weight can be found in polynomial time." @default.
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• W2005943912 date "2007-01-01" @default.
• W2005943912 modified "2023-09-27" @default.
• W2005943912 title "Packing directed cycles efficiently" @default.
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• W2005943912 doi "https://doi.org/10.1016/j.dam.2006.04.033" @default.
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