Kernelization for $H$-Packing Revisited
Data Structures and Algorithms
Summary
The authors study a problem where you want to find multiple copies of a small graph pattern inside a bigger graph without overlapping vertices. They improve existing general methods for a special set of graph patterns called subdivided stars by creating smaller, more efficient representations (kernels) of the problem. Their approach involves simplifying the graph into parts with limited connections and carefully keeping only important vertices. They also show that for some related patterns, it is provably harder to get such efficient simplifications, revealing unexpected complexity differences when slightly changing the pattern.
Authors
Tomohiro Koana, Soh Kumabe
Abstract
\textsc{$H$-Packing} asks whether a graph $G$ contains $k$ vertex-disjoint copies of a fixed pattern graph $H$. Via the standard reduction to \textsc{$d$-Set Packing}, one obtains generic kernels with $O(k^{|V(H)|-1})$ vertices and $O(k^{|V(H)|})$ edges. We revisit the question of beating these bounds for specific patterns $H$. Our main results concern subdivided stars. Let $S_{d_1,d_2}$ denote the subdivided star with $d_1$ branches of length $1$ and $d_2$ branches of length $2$. We obtain kernels with $O(k^2)$ vertices and $O(k^3)$ edges for $P_5=S_{0,2}$, for $S_{1,2}$, and for every $S_{d_1,1}$, kernels with $O(k^4)$ vertices and $O(k^6)$ edges for every fixed $S_{d_1,d_2}$ with $d_1\ge 1$, and a kernel with $O(k^2)$ vertices and $O(k^4)$ edges for the paw. Our proofs proceed in two steps. First, we reduce to instances in which all but a small part of the graph is independent, or in which the graph has a small vertex cover. Second, we reduce the independent side by keeping only a bounded number of witness vertices for each subset of the small part. On the negative side, we prove a lower bound for the line $S_{0,d}$. For every $d\ge 3$ and every $\varepsilon>0$, \textsc{$S_{0,d}$-Packing} does not admit a compression of size $O(k^{d-\varepsilon})$ unless $\NP\subseteq \coNP/\poly$. Thus, deleting a single vertex from the pattern may, surprisingly, make kernelization provably harder, showing that compressibility of \textsc{$H$-Packing} is not monotone under taking induced subgraphs.