  We introduce a new kernelization tool, called rainbow matching technique},
that is appropriate for the design of polynomial kernels for packing problems
and their hitting counterparts. Our technique capitalizes on the powerful
combinatorial results of [Graf, Harris, Haxell, SODA 2021]. We apply the
rainbow matching technique on four (di)graph packing or hitting problems,
namely the Triangle-Packing in Tournament problem (TPT), where we ask for a
packing of $k$ directed triangles in a tournament, Directed Feedback Vertex Set
in Tournament problem (FVST), where we ask for a (hitting) set of at most $k$
vertices which intersects all triangles of a tournament, the Induced
2-Path-Packing (IPP) where we ask for a packing of $k$ induced paths of length
two in a graph and Induced 2-Path Hitting Set problem (IPHS), where we ask for
a (hitting) set of at most $k$ vertices which intersects all 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
$O(k^{3/2})$ vertices for the two first problems and $O(k^{5/3})$ vertices for
the two last. 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 and FVST admit (almost linear)
kernels of $k^{1+\frac{O(1)}{\sqrt{\log{k}}}}$ vertices and that IPP and IPHS
admit kernels of $O(k)$ vertices.
