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

TL;DR

The paper tackles F-Deletion and its Treewidth-$\eta$-Deletion specialization, addressing the nonuniform nature of classic kernel bounds by introducing lossy kernelization. It develops protrusion-based techniques to convert near-protrusions into true protrusion decompositions with controlled accuracy loss, and uses these decompositions to achieve uniform 2-approximate kernels of size polynomial in the parameter $k$ (not depending on $\mathcal{F}$ in the exponent). The authors further extend the framework to a $(1+\epsilon)$-lossy protocol with a bounded number of oracle calls, and derive linear kernels on sparse graph classes when $\mathcal{F}$ contains a planar graph, even for disconnected minors. The approach handles disconnected forbidden minors and topological-minor-free graph classes, with computable constants, and yields a unifying view that blends protrusion replacement, minor packing/covering, and boundaried graph composition. Overall, the work provides a substantial step toward uniform, lossy kernelization for broad minor-related deletion problems, with practical implications for kernel sizes on sparse graph families and for problems generalizing Vertex Cover and Feedback Vertex Set.

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.

Figures (3)

• Figure 1: Illustration to \ref{['lem:dichotomy:bounded']} for $F$ being a disjoint union $F_1 + F_1 + F_2 + F_3$ and $\mathcal{F} = \{F\}$. Top left: An $\mathcal{F}$-deletion set $S$ is marked with black discs. The gray areas correspond to those $P_i$ which belong to $I({F_1}), I({F_2})$. Their neighborhoods in $P_0$ are sketched as well. Each $P_i$ in the corresponding family contains a minor model of the graph drawn in brown and is disjoint from $S$. Top right: $\widehat{S}$ is obtained from $S$ by removing $S \cap P_1$ and adding the vertices marked with squares. A potential minor model $\Phi$ of $F$ in $G-\widehat{S}$ is drawn in blue. It places one copy of $F_1$ inside $P_1$. Bottom right: A new model $\Phi'$ of $F$ is obtained from $\Phi$ by moving the image of $F_1$ to an unused component in $I({F_1})$. But this model is also present in $G-S$ which leads to a contradiction.
• Figure 3: Illustration to \ref{['lem:partition-mod']}. The protocol starts by finding a $\beta$-approximate solution $S_1$ to the Treewidth-$\eta$-Deletion problem in $G[P_0]$. Then $Q_1$ is the set obtained after deleting $S_1$ from $P_0$; note that $\textnormal{tw}(G[Q_0]) \le \eta$. It then repeats the process of finding a solution to the Treewidth-$\eta$-Deletion problem inside the old solution $S_1$ and continues doing so until the obtained new solution is a large fraction of the previous solution. Afterwards, \ref{['lrr:three']} removes the final solution from the graph and compresses the remaining graph using \ref{['lem:bound-simplicial']}.
• Figure 4: Illustration to \ref{['thm:lossy-kernel']}. Left column: Top figure represents the protrusion decomposition; the oval is the set $X \cup Z$ and the different colored parts are $P_1,P_2,P_3,P_4$; the pink and blue parts are simplicial with pink and blue clique neighbourhoods in $X \cup Z$ respectively; the green and yellow parts are non-simplicial with green and yellow parts in $X \cup Z$ as their neighbourhoods respectively. The middle graph is obtained after removing the simplicial parts (pink and blue) and the last graph is obtained after reducing the simplicial parts into constant size using \ref{['lem:dichotomy:compress']}. Right column: The bottom-most part represents the tree decomposition of $G^3-S_3$, the middle part is the tree decomposition of $G^2-S_2$ and the top part is the tree decomposition of $G^1_{\textnormal{flow}} - S_2$ obtained by attaching the pink and blue tree decompositions of the pink and blue parts, after adding their pink and blue neighbourhoods to each bag of their tree decompositions, respectively.

Theorems & Definitions (49)

• Theorem 1: Uniform lossy kernel
• Theorem 2: Uniform lossy protocol
• Lemma 3: Handling disconnected forbidden minors
• Theorem 4: Linear kernel on sparse graphs
• Definition 5
• Lemma 6: fomin2019kernelization
• Definition 7
• Lemma 8
• Lemma 8: $\bigstar$
• Definition 9: Minor packing