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

2026-01-13

AI Summary

This work addresses the challenge of non-uniform kernel dependence on the graph family $\mathcal{F}$ in the $\mathcal{F}$-Deletion problem, particularly the inherent bottleneck in Treewidth-$d$-Deletion where exponential terms in kernel size appear unavoidable. To circumvent this limitation, the paper introduces an approximate kernelization framework that permits a $(1+\varepsilon)$-approximation loss and yields a unified polynomial kernel. By integrating protrusion decomposition, sparse graph structure theory, and oracle-based protocols, the authors present the first 2-approximate kernel for Treewidth-$d$-Deletion with size $g(d) \cdot k^5$. Moreover, they overcome connectivity constraints, extending the existence of linear kernels to graph families containing disconnected planar graphs, thereby broadening applicability to a wider class of graphs excluding topological minors.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

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.

Problem

• F-Deletion
• Treewidth
• Kernelization
• Graph Minors
• Planar Graph

Innovation

• lossy kernelization
• treewidth deletion
• uniform kernel
• disconnected minors
• sparse graph classes