A Polynomial Kernel for Funnel Arc Deletion Set

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 mathcal {C}-Arc Deletion Set (mathcal {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 mathcal {C}. In this work, we choose mathcal {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 kernels for this problem were known. Our main result is a kernel for Funnel-ADS with mathcal {O}(k^6) many vertices and mathcal {O}(k^7) many arcs, computable in mathcal {O}(nm) time, where n is the number of vertices and m the number of arcs in the input digraph.