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A polynomial kernel for funnel arc deletion set

Milani, Marcelo Garlet

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 kernels for this problem were 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 in the input digraph.
Published in: Algorithmica, 10.1007/s00453-022-00960-w, Springer Nature