Protrusion Decompositions Revisited: Uniform Lossy Kernels for Reducing Treewidth and Linear Kernels for Hitting Disconnected Minors

TL;DR

This paper improves kernelization techniques for F-Deletion problems, achieving size bounds independent of the family F's complexity and extending linear kernels to broader sparse graph classes.

Contribution

It introduces a simple 2-approximate kernelization for Treewidth-d-Deletion with size g(d)*k^5 and shows how to approach optimal approximation factors with oracle calls, also generalizing linear kernels to more graph classes.

Findings

A 2-approximate kernelization with size g(d)*k^5.

Kernel size can be made arbitrarily close to optimal with oracle calls.

Linear kernels extended to graph classes excluding a topological minor.

Abstract

Let F be a finite family of graphs. In the F-Deletion problem, one is given a graph G and an integer k, and the goal is to find k vertices whose deletion results in a graph with no minor from the family F. This may be regarded as a far-reaching generalization of Vertex Cover and Feedback vertex Set. In their seminal work, Fomin, Lokshtanov, Misra & Saurabh [FOCS 2012] gave a polynomial kernel for this problem when the family F contains a planar graph. As the size of their kernel is g(F) * k^{f(F)}, a natural follow-up question was whether the dependence on F in the exponent of k can be avoided. The answer turned out to be negative: Giannapoulou, Jansen, Lokshtanov & Saurabh [TALG 2017] proved that this is already inevitable for the special case of the Treewidth-d-Deletion problem. In this work, we show that this non-uniformity can be avoided at the expense of a small loss. First, we present a simple 2-approximate kernelization algorithm for Treewidth-d-Deletion with kernel size g(d) * k^5. Next, we show that the approximation factor can be made arbitrarily close to 1, if we settle for a kernelization protocol with O(1) calls to an oracle that solves instances of size bounded by a uniform polynomial in k. We also obtain linear kernels on sparse graph classes when F contains a planar graph, whereas the previously known theorems required all graphs in F to be connected. Specifically, we generalize the kernelization algorithm by Kim, Langer, Paul, Reidl, Rossmanith, Sau & Sikdar [TALG 2015] on graph classes that exclude a topological minor.