Abstract
In Directed Feedback Arc Set (DFAS) we search for a set of at most k arcs which intersect every cycle in the input digraph. It is a well-known open problem in parameterized complexity to decide if DFAS admits a kernel of polynomial size. We consider 𝒞-Arc Deletion Set (𝒞-ADS), a variant of DFAS where we want to remove at most k arcs from the input digraph in order to turn it into a digraph of a class 𝒞. In this work, we choose 𝒞 to be the class of funnels. Funnel-ADS is NP-hard even if the input is a DAG, but is fixed-parameter tractable with respect to k. So far no polynomial kernel for this problem was known. Our main result is a kernel for Funnel-ADS with 𝒪(k⁶) many vertices and 𝒪(k⁷) many arcs, computable in 𝒪(nm) time, where n is the number of vertices and m the number of arcs of the input digraph.
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