A Polynomial Kernel for Funnel Arc Deletion Set

TL;DR

This paper introduces a polynomial kernel for the Funnel-Arc Deletion Set problem, a variant of Directed Feedback Arc Set, providing a significant step in fixed-parameter tractability and kernelization for this NP-hard problem.

Contribution

It presents the first polynomial kernel with size bounds for Funnel-Arc Deletion Set, advancing the understanding of kernelization in directed graph problems.

Findings

Kernel size is O(k^6) vertices and O(k^7) arcs.

Kernelization algorithm runs in O(nm) time.

Problem is fixed-parameter tractable despite NP-hardness.

Abstract

In Directed Feeback 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-Arc Deletion Set 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-Arc Deletion Set 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.