Expansions for Reductions

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Expansions for Reductions

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Page 1: Expansions for Reductions

Expansions for Reductions

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Expansions for Reductions

Joint work withFedor Fomin, Daniel Lokshtanov,

Geevarghese Philip, and Saket Saurabh

University of BergenUniversity of California, San DiegoThe Institute of Mathematical Sciences

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Talk Outline

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Talk Outline

q matchings, mutually disjoint in B

|N(S)| q|S|

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Talk Outline

q matchings, mutually disjoint in B

|N(S)| q|S|

q-Expansion Lemma

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Talk Outline

q-Expansion Lemma

with applications to Vertex Cover,Feedback Vertex set,and beyond.

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Part 1A Tale of q Matchings

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A

B

Consider a bipartite graph one of whose parts (say B) is atleast twice as big as the other (call thisA).

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A

B

Assume that there are no isolated vertices in B.bleh

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Suppose, further, that for every subset S inA,N(S) is atleast twice as large as |S|.

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S

N(S)

Suppose, further, that for every subset S inA,N(S) is atleast twice as large as |S|.

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A

B

en there exist two matchings saturatingA,bleh

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A

B

en there exist two matchings saturatingA,bleh

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A

B

en there exist two matchings saturatingA,and disjoint in B.

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Claim:

IfN(A) > q|A|, then there exists a subset S ofA such that:

there qmatchings saturating the subset Sthat are vertex-disjoint in B.

provided B does not have any isolated vertices.

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Claim:

IfN(A) > q|A|, then there exists a subset S ofA such that:

there qmatchings saturating the subset Sthat are vertex-disjoint in B.

provided B does not have any isolated vertices.

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Claim:

If |B| > q|A|, then there exists a subset S ofA such that:

there qmatchings saturating the subset Sthat are vertex-disjoint in B,

provided B does not have any isolated vertices.

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Crucially: it turns out that the endpoints of the matchingsin B (the larger set) do not have neighbors outside S.

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A

B

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Part 2Two-Expansions and FVS

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Ingredients

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Ingredients

a high-degree vertex, v

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Ingredients

a high-degree vertex, v

a small hitting set,sans v

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Given a high-degree vertex v, finding a small hitting set thatdoes not contain v.

Any hitting set whose size is a polynomial function of k.

When the largest collection of vertex disjoint paths fromN(v) toN(v) is small.

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Given a high-degree vertex v, finding a small hitting set thatdoes not contain v.

A subset whose removal makes the graph acyclic.

When the largest collection of vertex disjoint paths fromN(v) toN(v) is small.

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Given a high-degree vertex v, finding a small hitting set thatdoes not contain v.

A polynomial function of k.

When the largest collection of vertex disjoint paths fromN(v) toN(v) is small.

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Given a high-degree vertex v, finding a small hitting set thatdoes not contain v.

At least twice the size of the hitting set.

When the largest collection of vertex disjoint paths fromN(v) toN(v) is small.

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Given a high-degree vertex v, finding a small hitting set thatdoes not contain v.

Find an approximate hitting set T .

When the largest collection of vertex disjoint paths fromN(v) toN(v) is small.

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Given a high-degree vertex v, finding a small hitting set thatdoes not contain v.

If T does not contain v, we are done.

When the largest collection of vertex disjoint paths fromN(v) toN(v) is small.

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Given a high-degree vertex v, finding a small hitting set thatdoes not contain v.

Else: v ∈ T . Delete T \ v fromG.

When the largest collection of vertex disjoint paths fromN(v) toN(v) is small.

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Given a high-degree vertex v, finding a small hitting set thatdoes not contain v.

e only remaining cycles pass through v.

When the largest collection of vertex disjoint paths fromN(v) toN(v) is small.

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Given a high-degree vertex v, finding a small hitting set thatdoes not contain v.

Find an optimal cut set for paths fromN(v) toN(v).

When the largest collection of vertex disjoint paths fromN(v) toN(v) is small.

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Given a high-degree vertex v, finding a small hitting set thatdoes not contain v.

Why is this cut set small enough?

When the largest collection of vertex disjoint paths fromN(v) toN(v) is small.

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Given a high-degree vertex v, finding a small hitting set thatdoes not contain v.

When is this cut set small enough?

When the largest collection of vertex disjoint paths fromN(v) toN(v) is small.

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Given a high-degree vertex v, finding a small hitting set thatdoes not contain v.

When is this cut set small enough?

When the largest collection of vertex disjoint paths fromN(v) toN(v) is small.

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Given a high-degree vertex v, finding a small hitting set thatdoes not contain v.

When is this cut set small enough?

When the largest collection of vertex disjoint paths fromN(v) toN(v) is not small...

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Given a high-degree vertex v, finding a small hitting set thatdoes not contain v.

When is this cut set small enough?

When the largest collection of vertex disjoint paths fromN(v) toN(v) is not small... we get a reduction rule.

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Given a high-degree vertex v, finding a small hitting set thatdoes not contain v.

When is this cut set small enough?

More than k vertex-disjoint paths fromN(v) toN(v)→ v belongs to any hitting set of size at most k.

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Given a high-degree vertex v, finding a small hitting set thatdoes not contain v.

When is this cut set small enough?

So either v “forced”, or we have hitting set of suitable size.Notice that we arrive at either situation in polynomial time.

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forest

hitting set that excludes v

v

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forest

hitting set that excludes v

v

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forest

hitting set that excludes v

v

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forest

hitting set that excludes v

v

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forest

hitting set that excludes v

v

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forest

hitting set that excludes v

v

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forest

hitting set that excludes v

v

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forest

hitting set that excludes v

v

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forest

hitting set that excludes v

v

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forest

hitting set that excludes v

v

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forest

hitting set that excludes v

v

by 2-expansion

lemma:

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forest

hitting set that excludes v

v

by 2-expansion

lemma:

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forest

hitting set that excludes v

v

by 2-expansion

lemma:

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forest

hitting set that excludes v

v

by 2-expansion

lemma:

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e Forward Direction

FVS6 k inG ⇒ FVS6 k inH

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e Forward Direction

FVS6 k inG ⇒ FVS6 k inH

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forest

hitting set that excludes v

v

by 2-expansion

lemma:

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IfG has a FVS that either contains v or all of S, we are ingood shape.

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Consider now a FVS that:• Does not contain v,• and omits at least one vertex of S.

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Notice that this does not lead to a larger FVS:

For every vertex v in S that a FVS ofG leaves out,

it must pick a vertex u that kills no more than all of S.

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Notice that this does not lead to a larger FVS:

For every vertex v in S that a FVS ofG leaves out,

it must pick a vertex u that kills no more than all of S.

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Notice that this does not lead to a larger FVS:

For every vertex v in S that a FVS ofG leaves out,

it must pick a vertex u that kills no more than all of S.

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e Reverse Direction

FVS6 k inG ⇐ FVS6 k inH

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e Reverse Direction

FVS6 k inG ⇐ FVS6 k inH

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forest

hitting set that excludes v

v

by 2-expansion

lemma:

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e Only Danger

Cycles that:

• pass through v,• non-neighbors of v inH¹• and do not pass through S.

¹neighbors inG, however

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Part q

q-Expansions for Hittingθq-minor models

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Ingredients

a high-degree vertex, v

a hitting set,sans v

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Ingredients

a high-degree vertex, v

a hitting set,sans v

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Ingredients

a high-degree vertex, v

an approximatehitting set

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Ingredients

a high-degree vertex, v

a way of detecting(k+1)-flowers

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Assuming we have a small hitting set, and a vertex of largedegree...

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theta-q-free area

hitting set that excludes v

v

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theta-q-free area

hitting set that excludes v

v degree q into any

component?

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So the degree into every component is bounded by (q− 1),

but there could be lots of components!

If there are at leastq|HS|

components, we get a q-expansion!

If we can handle the expansion, note that we will end upwith a degree bound of:

(q− 1)q.

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So the degree into every component is bounded by (q− 1),but there could be lots of components!

If there are at leastq|HS|

components, we get a q-expansion!

If we can handle the expansion, note that we will end upwith a degree bound of:

(q− 1)q.

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So the degree into every component is bounded by (q− 1),but there could be lots of components!

If there are at leastq|HS|

components,

we get a q-expansion!

If we can handle the expansion, note that we will end upwith a degree bound of:

(q− 1)q.

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So the degree into every component is bounded by (q− 1),but there could be lots of components!

If there are at leastq|HS|

components, we get a q-expansion!

If we can handle the expansion, note that we will end upwith a degree bound of:

(q− 1)q.

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So the degree into every component is bounded by (q− 1),but there could be lots of components!

If there are at leastq|HS|

components, we get a q-expansion!

If we can handle the expansion, note that we will end upwith a degree bound of:

(q− 1)q|HS|.

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So the degree into every component is bounded by (q− 1),but there could be lots of components!

If there are at leastq|HS|

components, we get a q-expansion!

If we can handle the expansion, note that we will end upwith a degree bound of:

(q− 1)q|HS|.

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So the degree into every component is bounded by (q− 1),but there could be lots of components!

If there are at leastq|HS|

components, we get a q-expansion!

If we can handle the expansion, note that we will end upwith a degree bound of:

(q− 1)qpoly(k).

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theta-q-free area

hitting set that excludes v

v

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v

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v

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v

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e Forward Direction

Solution6 k inG ⇒ Solution6 k inH

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e Forward Direction

Solution6 k inG ⇒ Solution6 k inH

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IfG has a solution that either contains v or all of S, we are ingood shape.

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Consider now a solution that:• Does not contain v,• and omits at least one vertex of S.

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v

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Notice that this does not lead to a larger solution:

For every vertex v in S that a solution inG leaves out,

it must pick a vertex u that kills no more minor models thanall of S.

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Notice that this does not lead to a larger solution:

For every vertex v in S that a solution inG leaves out,

it must pick a vertex u that kills no more minor models thanall of S.

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Notice that this does not lead to a larger solution:

For every vertex v in S that a solution inG leaves out,

it must pick a vertex u that kills no more minor models thanall of S.

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e Reverse Direction

Solution6 k inG ⇐ Solution6 k inH

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e Reverse Direction

Solution6 k inG ⇐ Solution6 k inH

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v

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e Only Danger

Minor models that:

• pass through v,• non-neighbors of v inH²• and do not pass through S.

²neighbors inG, however

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e Only Danger

Minor models that:

• pass through v,• non-neighbors of v inH²• and do not pass through S.

²neighbors inG, however

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v

If there were minor models here, then vwould be acut-vertex. Since θq is a 2-connected graph: contradiction!

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v

If there were minor models here, then vwould be acut-vertex. Since θq is a 2-connected graph: contradiction!

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What’s left?Finding the small hitting set that avoids v.

• Assume we have an approximation algorithm.If v is not in the approximate solution, then we aredone.If v is in the approximate solution, again, delete all ofthe hitting set except v, so the remaining graph onlyhas minor models passing through v.

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What’s left?Finding the small hitting set that avoids v.

• Assume we have an approximation algorithm.If v is not in the approximate solution, then we aredone.If v is in the approximate solution, again, delete all ofthe hitting set except v, so the remaining graph onlyhas minor models passing through v.

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eoremLet F be a minor-closed family of graphs such that itsobstruction set contains a planar graph. Given a graphG, in polynomial time we can find a subset T ⊆ V(G)such that G[V \ T ] is an element of F and |T | 6(OPT · log3/2OPT). HereOPT is the size of smallestsuch set T .

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eoremLet F be a minor-closed family of graphs such that itsobstruction set contains a planar graph. Given a graphG, in polynomial time we can find a subset T ⊆ V(G)such that G[V \ T ] is an element of F and |T | 6(k · log3/2 k). HereOPT is the size of smallest such setT .

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What’s left?Finding the small hitting set that avoids v.

• So we have an approximate solution, T .• If v is not in the approximate solution, then we are

done.• If v is in the approximate solution, again, delete all of

the hitting set except v, so the remaining graph onlyhas minor models passing through v.

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What’s left?Finding the small hitting set that avoids v.

• So we have an approximate solution, T .

• If v is not in the approximate solution, then we aredone.

• If v is in the approximate solution, again, delete all ofthe hitting set except v, so the remaining graph onlyhas minor models passing through v.

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What’s left?Finding the small hitting set that avoids v.

• So we have an approximate solution, T .• If v is not in the approximate solution, then we are

done.

• If v is in the approximate solution, again, delete all ofthe hitting set except v, so the remaining graph onlyhas minor models passing through v.

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What’s left?Finding the small hitting set that avoids v.

• So we have an approximate solution, T .• If v is not in the approximate solution, then we are

done.• If v is in the approximate solution, again, delete all of

the hitting set except v, so the remaining graph onlyhas minor models passing through v.

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In this graph, all minor models of θq pass through v.

e graphG \ T

∪ {v}

has constant treewidth:

free of θc minor models + large treewidth=

(q× q) grid minor⇒ θc minor model

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In this graph, all minor models of θq pass through v.

e graphG \ T

∪ {v}

has constant treewidth:

free of θc minor models + large treewidth=

(q× q) grid minor⇒ θc minor model

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In this graph, all minor models of θq pass through v.

e graphG \ T

∪ {v}

has constant treewidth:

free of θc minor models + large treewidth=

(q× q) grid minor⇒ θc minor model

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In this graph, all minor models of θq pass through v.

e graphG \ T ∪ {v}

has constant treewidth.

free of θc minor models + large treewidth=

(q× q) grid minor⇒ θc minor model

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Promise: v is not at the center of a (k+ 1)-flower.

Stare at the tree decomposition ofG \ S.

At every bag, compute the number of vertex-disjoint minormodels that pass through v in the graphG \ S∪ {v}

Note the bags at which this measure flips.

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Promise: v is not at the center of a (k+ 1)-flower.

Stare at the tree decomposition ofG \ S.

At every bag, compute the number of vertex-disjoint minormodels that pass through v in the graphG \ S∪ {v}

Note the bags at which this measure flips.

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Promise: v is not at the center of a (k+ 1)-flower.

Stare at the tree decomposition ofG \ S.

At every bag, compute the number of vertex-disjoint minormodels that pass through v in the graphG \ S∪ {v}

Note the bags at which this measure flips.

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Promise: v is not at the center of a (k+ 1)-flower.

Stare at the tree decomposition ofG \ S.

At every bag, compute the number of vertex-disjoint minormodels that pass through v in the graphG \ S∪ {v}

Note the bags at which this measure flips.

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ere are at most k such bags of constant size.

Add all vertices in all these bags to the hitting set.

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ere are at most k such bags of constant size.

Add all vertices in all these bags to the hitting set.

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Checking if v is at the center of a (k+ 1) flower:

maximum number of vertex-disjoint minor models of θqpassing through v.

max |S|{∀p ∈ S ∃Xp ⊆ V,

∀p ∈ S, q ∈ S :p 6= q ⇒ Xp ∩ Xq = ∅

φq(Xp ∪ v)}

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Checking if v is at the center of a (k+ 1) flower:

maximum number of vertex-disjoint minor models of θqpassing through v.

max |S|{∀p ∈ S ∃Xp ⊆ V,

∀p ∈ S, q ∈ S :p 6= q ⇒ Xp ∩ Xq = ∅

φq(Xp ∪ v)}

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Checking if v is at the center of a (k+ 1) flower:

maximum number of vertex-disjoint minor models of θqpassing through v.

max |S|{∀r ∈ S ∃Xr ⊆ V,

∀p ∈ S, q ∈ S :p 6= q ⇒ Xp ∩ Xq = ∅

φq(Xp ∪ v)}

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Checking if v is at the center of a (k+ 1) flower:

maximum number of vertex-disjoint minor models of θqpassing through v.

max |S|{∀r ∈ S ∃Xr ⊆ V,

∀p ∈ S, q ∈ S :p 6= q ⇒ Xp ∩ Xq = ∅

φq(Xp∪v)}

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Checking if v is at the center of a (k+ 1) flower:

maximum number of vertex-disjoint minor models of θqpassing through v.

max |S|{∀r ∈ S ∃Xr ⊆ V,

∀p ∈ S, q ∈ S :p 6= q ⇒ Xp ∩ Xq = ∅

φq(Xp∪v)}

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Checking if v is at the center of a (k+ 1) flower:

maximum number of vertex-disjoint minor models of θqpassing through v.

max |S|{∀r ∈ S ∃Xr ⊆ V,

∀p ∈ S, q ∈ S :p 6= q ⇒ Xp ∩ Xq = ∅

φq(Xp∪v)}

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On a graph of constant treewidth, detecting large flowers ispolynomial time.

And thus we are (finally!) done.

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On a graph of constant treewidth, detecting large flowers ispolynomial time. And thus we are (finally!) done.

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Dank u wel