TY - GEN

T1 - Kernelization for Graph Packing Problems via Rainbow Matching

AU - Bessy, Stéphane

AU - Bougeret, Marin

AU - Thilikos, Dimitrios M.

AU - Wiederrecht, Sebastian

N1 - Publisher Copyright:
Copyright © 2023 by SIAM.

PY - 2023

Y1 - 2023

N2 - 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 (I2PP) 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 [Fomin, Le, Lokshtanov, Saurabh, Thomassé, Zehavi. ACM Trans. Algorithms, 2019], where they gave a kernel of O(k3/2) vertices and O(k5/3) 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 (Equation presented) vertices and that I2PP admits a kernel of O(k) vertices.

AB - 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 (I2PP) 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 [Fomin, Le, Lokshtanov, Saurabh, Thomassé, Zehavi. ACM Trans. Algorithms, 2019], where they gave a kernel of O(k3/2) vertices and O(k5/3) 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 (Equation presented) vertices and that I2PP admits a kernel of O(k) vertices.

KW - Kernelization

KW - Packing problems

KW - Parameterized algorithms

KW - Rainbow matching

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M3 - Conference contribution

AN - SCOPUS:85150333285

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 3654

EP - 3663

BT - 34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023

PB - Association for Computing Machinery

T2 - 34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023

Y2 - 22 January 2023 through 25 January 2023

ER -