Minors under Structural Parameterizations Bart M.P. Jansen and - PowerPoint PPT Presentation

Polynomial Kernels for Hitting Forbidden Minors under Structural Parameterizations Bart M.P. Jansen and Astrid Pieterse

Problem 𝐺 is a finite set of connected graphs 𝐺 -minor free deletion Given undirected graph 𝐻 and budget 𝑐 , can we remove 𝑐 vertices from 𝐻 such that it no longer has 𝐺 -minors? 𝐼 is a minor of 𝐻 G

Problem 𝐺 is a finite set of connected graphs 𝐺 -minor free deletion Given undirected graph 𝐻 and budget 𝑐 , can we remove 𝑐 vertices from 𝐻 such that it no longer has 𝐺 -minors? 𝐼 is a minor of 𝐻

Problem 𝐺 is a finite set of connected graphs 𝐺 -minor free deletion Given undirected graph 𝐻 and budget 𝑐 , can we remove 𝑐 vertices from 𝐻 such that it no longer has 𝐺 -minors? 𝐼 is a minor of 𝐻

Problem 𝐺 is a finite set of connected graphs 𝐺 -minor free deletion Given undirected graph 𝐻 and budget 𝑐 , can we remove 𝑐 vertices from 𝐻 such that it no longer has 𝐺 -minors? 𝐼 is a minor of 𝐻

Problem 𝐺 is a finite set of connected graphs 𝐺 -minor free deletion Given undirected graph 𝐻 and budget 𝑐 , can we remove 𝑐 vertices from 𝐻 such that it no longer has 𝐺 -minors? 𝐼 is a minor of 𝐻 H

𝐺 -minor free deletion Generalizes many known problems Vertex Cover for 𝐺 = {𝐿 2 } Can we remove 𝑐 vertices, such that 𝐻 becomes edgeless? Remove 3 vertices

𝐺 -minor free deletion Generalizes many known problems Vertex Cover for 𝐺 = {𝐿 2 } Can we remove 𝑐 vertices, such that 𝐻 becomes edgeless? Remove 3 vertices Feedback Vertex Set for 𝐺 = {𝐿 3 } Can we remove 𝑐 vertices, such that 𝐻 becomes acyclic? Remove 1 vertex

Kernelization 𝐺 -minor free deletion is NP-hard • Do preprocessing • Use an additional parameter 𝑙 to measure complexity 𝑜 bits 𝑞𝑝𝑚𝑧 𝑦 , 𝑙 𝑔(𝑙) bits time 𝑙 𝑙′ 𝑦 𝑦′ For which complexity measure, is good preprocessing possible? • 𝑔 𝑙 polynomial in 𝑙

Previous work General problem [Fomin, Jansen, Pilipczuk, J. Comput . Syst. Sci.’12] Let 𝑌 be a vertex cover of 𝐻 , there is a kernel of size 𝑞𝑝𝑚𝑧 𝑌 for 𝐺 -minor free deletion kernel 𝑐 𝑐′ |X|=3 General parameter [Bougeret , Sau, IPEC’17] modulator to treedepth 1 = vertex cover Let 𝑌 be a modulator to treedepth 𝜃 , there is a kernel of size 𝑞𝑝𝑚𝑧 𝑌 for vertex cover vertex cover = { 𝐿 2 }-minor free deletion

Main result We generalize both existing results, resolving an open question by Bougeret and Sau on FVS Theorem 𝐺 -minor free deletion parameterized by a modulator to treedepth 𝜃 has a polynomial kernel For more information & interesting proof techniques Come see the poster!