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 packing of $k$ directed triangles in a tournament, and the {Induced 2-Path-Packing} (IPP) where we ask for a packing of $k$ 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 [{\sl Fomin, Le, Lokshtanov, Saurabh, Thomass\'e, Zehavi. ACM Trans. Algorithms, 2019}], where they gave a kernel of ${\cal O}(k^{3/2})$ vertices and ${\cal O}(k^{5/3})$ 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 $k^{1+\frac{{\cal O}(1)}{\sqrt{\log{k}}}}$ vertices and that IPP admits a kernel of ${\cal O}(k)$ vertices.