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Cycle killer... qu'est-ce que c'est? On the comparative approximability of hybridization number and directed feedback vertex set

Abstract : We show that the problem of computing the hybridization number of two rooted binary phylogenetic trees on the same set of taxa X has a constant factor polynomial-time approximation if and only if the problem of computing a minimum-size feedback vertex set in a directed graph (DFVS) has a constant factor polynomial-time approximation. The latter problem, which asks for a minimum number of vertices to be removed from a directed graph to transform it into a directed acyclic graph, is one of the problems in Karp's seminal 1972 list of 21 NP-complete problems. However, despite considerable attention from the combinatorial optimization community it remains to this day unknown whether a constant factor polynomial-time approximation exists for DFVS. Our result thus places the (in)approximability of hybridization number in a much broader complexity context, and as a consequence we obtain that hybridization number inherits inapproximability results from the problem Vertex Cover. On the positive side, we use results from the DFVS literature to give an O(log r log log r) approximation for hybridization number, where r is the value of an optimal solution to the hybridization number problem.
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https://hal.inria.fr/hal-00765019
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Submitted on : Thursday, July 21, 2016 - 3:52:11 PM
Last modification on : Thursday, September 29, 2022 - 5:01:33 AM

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Steven Kelk, Leo van Iersel, Nela Lekic, Simone Linz, Celine Scornavacca, et al.. Cycle killer... qu'est-ce que c'est? On the comparative approximability of hybridization number and directed feedback vertex set. SIAM Journal on Discrete Mathematics, 2012, 26 (4), pp.1635-1656. ⟨10.1137/120864350⟩. ⟨hal-00765019⟩

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