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，适用于设计打包问题及其击中问题的多项式核。该技术利用了[Graf, Harris, Haxell, SODA 2021]中强大的组合结果。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），其中我们询问在锦标赛中是否存在$k$个有向三角形的打包；锦标赛中的有向反馈顶点集问题（FVST），其中我们询问是否存在一个最多包含$k$个顶点的（击中）集，该集与锦标赛中的所有三角形相交；诱导2-路径打包问题（IPP），其中我们询问在图中是否存在$k$条长度为2的诱导路径的打包；以及诱导2-路径击中集问题（IPHS），其中我们询问是否存在一个最多包含$k$个顶点的（击中）集，该集与图中所有长度为2的诱导路径相交。这些问题在[Fomin, Le, Lokshtanov, Saurabh, Thomassé, Zehavi. ACM Trans. Algorithms, 2019]中首次证明了次二次核的存在性，其中针对前两个问题给出了$O(k^{3/2})$个顶点的核，针对后两个问题给出了$O(k^{5/3})$个顶点的核。同一篇论文中提出了这些界是否可以（最优地）改进为线性界的问题。受此问题启发，我们应用彩虹匹配技术，证明了TPT和FVST承认（近似线性的）核，顶点数为$k^{1+\frac{O(1)}{\sqrt{\log{k}}}}$，而IPP和IPHS承认$O(k)$个顶点的核。