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.

翻译：我们引入了一种新的核化工具，称为"彩虹匹配技术"，该技术适用于设计打包问题及其对偶击中问题的多项式核。该技术利用了[Graf, Harris, Haxell, SODA 2021]中强有力的组合结果。我们将彩虹匹配技术应用于四个（有向）图打包或击中问题，即锦标赛中的三角形打包问题（TPT），其中要求在一个锦标赛中打包$k$个有向三角形；锦标赛中的有向反馈顶点集问题（FVST），其中要求至多选取$k$个顶点以击中所有三角形；诱导2路径打包问题（IPP），其中要求在图中打包$k$条长度为2的诱导路径；以及诱导2路径击中集问题（IPHS），其中要求至多选取$k$个顶点以击中所有长度为2的诱导路径。这些问题是否存在次二次核首次在[Fomin, Le, Lokshtanov, Saurabh, Thomassé, Zehavi. ACM Trans. Algorithms, 2019]中得到证明，该文给出了前两个问题的$O(k^{3/2})$顶点核和后两个问题的$O(k^{5/3})$顶点核。同一篇论文中提出了这些界能否（最优地）改进为线性核的问题。受此问题启发，我们应用彩虹匹配技术证明了TPT和FVST具有（几乎线性的）$k^{1+\frac{O(1)}{\sqrt{\log{k}}}}$顶点核，而IPP和IPHS具有$O(k)$顶点核。