A Kernel for Planar F-deletion: The Connected Case

Polynomial Kernels for Planar F-deletion


Fedor V. Fomin, Daniel Lokshtanov and Saket Saurabh


when F contains connected graphs

*

Fedor V. Fomin, Daniel Lokshtanov and Saket Saurabh

Outline

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qÜÉ=píçêó=pç=c~êPolynomial Kernels for Planar F-deletion

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|G| p(k)?


|G| p(k)?

Infer the existence of a protrusion

åç


|G| p(k)?


åç


Reject the instance

no protrusions?

|G| p(k)?


åç


Reject the instance

no protrusions?

Reduce the protrusion

|G| p(k)?


åç


Reject the instance

no protrusions?


Poly Kernel

|G| p(k)?


óÉëåç


Reject the instance

no protrusions?


Lemmas 17-23Poly Kernel

|G| p(k)?


óÉëåç


Reject the instance

no protrusions?


Theorem 2,3


|G| p(k)?


óÉëåç


Reject the instance

no protrusions?


Theorem 2,3


|G| p(k)?


óÉëåç

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Reject the instance

no protrusions?


Theorem 2,3

Poly Kernel

|G| p(k)?


óÉëåç


Reject the instance

no protrusions?


Poly Kernel

|G| p(k)?


óÉëåç


Reject the instance

no protrusions?


Poly Kernel

|G| p(k)?


óÉëåç


Infer near-protrusions

Reject the instance

no protrusions?


Poly Kernel

|G| p(k)?


óÉëåç


Infer near-protrusions

Reject the instance

no protrusions?

Reduce the protrusionIrrelevant Edges

X

G \ X

X

G \ X

Approximate F-deletion set

X

G \ X

constant treewidth zone


X

G \ X



X

G \ X



X

G \ X



constant boundary...?

k

k

k

k

The Space of all t-boundaried graphs




`çåíáåìÉÇKKKPolynomial Kernels for Planar F-deletion

X

G \ X




X

G \ X




X

G \ X




X

G \ X




X

G \ X




X

G \ X




For every guess we have a protrusion

But it may not be safe to reduce them!

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Suppose the protrusion gives us a way offinding irrelevant edges.


If every guess declares an edge to be irrelevant,then it is safe to remove it from G.


If every guess declares an edge to be irrelevant,then it is safe to remove it from G.


If there are more than poly(k) edges incident on a single vertex, then one of them is not relevant to any guess.

Deleting an edge never increases OPT.


G


G G\{e}


G\{e}G


G\{e}G

Determine why G is a no-instance,and don’t interfere with it.

Vertex Cover 2of sizeG does not have a






2of sizeG does not have a F-deletion Set

of sizeG does not have a F-deletion Set k

of sizeG does not have a F-deletion Set kG does not belong to [FDel]k


But [FDel]k is closed under minors, and hence has a finite

obstruction set S.


But [FDel]k is closed under minors, and hence has a finite

obstruction set S.

S “witnesses” the fact that G is a NO instance. Edges not involved in copies of S are... irrelevant!

We don’t know the obstruction sets.

We don’t know the obstruction sets.

Even if we did, minor models of graphs in S could be arbitrarily large.

X

G \ X



X

G \ X



X

G \ X



X

G \ X



X

G \ X



X

G \ X



Before

X

G \ X



Before After

X

G \ X


Approximate F-deletion setBefore After

X

G \ X



X

G \ X





Avoiding some obstruction to membership in FDelk



Large degree implies the existence of at least one irrelevant edge



Large degree implies the existence of at least one irrelevant edge

Use near-protrusions, cost vectors, finite index, CMSO expressibility.

X

G \ X

constant treewidth zone: tw=c


X

G \ X



X

G \ X



Case 1

There is a (k+c+1)-sized(u,v)-separator.

X

G \ X



Case 1


X

G \ X



Case 1


bounded by poly(k)

X

G \ X



bounded by poly(k)

Case 2

There is a (k+c+1) flowbetween u and v.

X

G \ X



bounded by poly(k)

Case 2



bounded by poly(k)

Case 2


G \ S


bounded by poly(k)

Case 2


G \ S


bounded by poly(k)

Case 2


G \ S


bounded by poly(k)

Case 2


G \ S


bounded by poly(k)

Case 2


G \ S

a separator of size (c+1)+k

PlanarPolynomial Kernels for F-deletion

qÜÉ=`ÉåëçêÉÇ=aÉí~áäëPlanarPolynomial Kernels for F-deletion

The presence of disconnected graphs in F opens up a can of worms, various details

need substantial tweaking.

The finite obstructions are implied by WQO of t-boundaried graphs of bounded treewidth with special minor operations.

Consequences of the kernel.Obstructions to [FDel]k are polynomially

bounded in k.

PlanarPolynomial Kernels for F-deletion

Polynomial Kernels for F-deletion

qç=_É=`çåíáåìÉÇKKKPolynomial Kernels for F-deletion

Daniel will answer your questions!

Daniel will answer your questions!

Education

A Kernel for Planar F-deletion: The Connected Case