1 research outputs found
Kernelization for Graph Packing Problems via Rainbow Matching
We introduce a new kernelization tool, called rainbow matching technique,
that is appropriate for the design of polynomial kernels for packing problems.
Our technique capitalizes on the powerful combinatorial results of [Graf,
Harris, Haxell, SODA 2021]. We apply the rainbow matching technique on two
(di)graph packing problems, namely the Triangle-Packing in Tournament problem
(\TPT), where we ask for a directed triangle packing in a tournament, and the
Induced 2-Path-Packing (\IPP) where we ask for a packing of induced paths
of length two in a graph. The existence of a sub-quadratic kernels for these
problems was proven for the first time in [Fomin, Le, Lokshtanov, Saurabh,
Thomass\'e, Zehavi. ACM Trans. Algorithms, 2019], where they gave a kernel of
vertices and vertices
respectively. In the same paper it was questioned whether these bounds can be
(optimally) improved to linear ones. Motivated by this question, we apply the
rainbow matching technique and prove that \TPT admits an (almost linear) kernel
of vertices and that \IPP admits
kernel of vertices