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 = {𝐿2} 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 = {𝐿2} Can we remove 𝑐 vertices, such that 𝐻 becomes edgeless? Feedback Vertex Set for 𝐺 = {𝐿3} Can we remove 𝑐 vertices, such that 𝐻 becomes acyclic?

Remove 1 vertex Remove 3 vertices

Kernelization

𝐺-minor free deletion is NP-hard

• Do preprocessing
• Use an additional parameter 𝑙 to measure complexity

For which complexity measure, is good preprocessing possible?

• 𝑔 𝑙 polynomial in 𝑙

𝑦

𝑙

𝑦′

𝑙′

𝑜 bits 𝑔(𝑙) bits 𝑞𝑝𝑚𝑧 𝑦 , 𝑙 time

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 General parameter [Bougeret, Sau, IPEC’17] Let 𝑌 be a modulator to treedepth 𝜃, there is a kernel of size 𝑞𝑝𝑚𝑧 𝑌 for vertex cover

modulator to treedepth 1 = vertex cover vertex cover = {𝐿2}-minor free deletion kernel 𝑐 𝑐′

|X|=3

Main result

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