expansions for reductions

Expansions for Reductions

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

Talk Outline

Talk Outline

q matchings, mutually disjoint in B

|N(S)| q|S|

Talk Outline

q matchings, mutually disjoint in B

|N(S)| q|S|

q-Expansion Lemma

Talk Outline

q-Expansion Lemma

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

Part 1A Tale of q Matchings

A

B

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

A

B

Assume that there are no isolated vertices in B.bleh

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

S

N(S)

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

A

B

en there exist two matchings saturatingA,bleh

A

B

en there exist two matchings saturatingA,bleh

A

B

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

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.

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.

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.

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

A

B

Part 2Two-Expansions and FVS

Ingredients

Ingredients

a high-degree vertex, v

Ingredients

a high-degree vertex, v

a small hitting set,sans v

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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...

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.

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.

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.

forest

hitting set that excludes v

v

forest

hitting set that excludes v

v

forest

hitting set that excludes v

v

forest

hitting set that excludes v

v

forest

hitting set that excludes v

v

forest

hitting set that excludes v

v

forest

hitting set that excludes v

v

forest

hitting set that excludes v

v

forest

hitting set that excludes v

v

forest

hitting set that excludes v

v

forest

hitting set that excludes v

v

by 2-expansion

lemma:

forest

hitting set that excludes v

v

by 2-expansion

lemma:

forest

hitting set that excludes v

v

by 2-expansion

lemma:

forest

hitting set that excludes v

v

by 2-expansion

lemma:

e Forward Direction

FVS6 k inG ⇒ FVS6 k inH

e Forward Direction

FVS6 k inG ⇒ FVS6 k inH

forest

hitting set that excludes v

v

by 2-expansion

lemma:

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

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

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.

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.

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.

e Reverse Direction

FVS6 k inG ⇐ FVS6 k inH

e Reverse Direction

FVS6 k inG ⇐ FVS6 k inH

forest

hitting set that excludes v

v

by 2-expansion

lemma:

e Only Danger

Cycles that:

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

¹neighbors inG, however

Part q

q-Expansions for Hittingθq-minor models

Ingredients

a high-degree vertex, v

a hitting set,sans v

Ingredients

a high-degree vertex, v

a hitting set,sans v

Ingredients

a high-degree vertex, v

an approximatehitting set

Ingredients

a high-degree vertex, v

a way of detecting(k+1)-flowers

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

theta-q-free area

hitting set that excludes v

v

theta-q-free area

hitting set that excludes v

v degree q into any

component?

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.

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.

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.

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.

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|.

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|.

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).

theta-q-free area

hitting set that excludes v

v

v

v

v

e Forward Direction

Solution6 k inG ⇒ Solution6 k inH

e Forward Direction

Solution6 k inG ⇒ Solution6 k inH

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

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

v

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.

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.

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.

e Reverse Direction

Solution6 k inG ⇐ Solution6 k inH

e Reverse Direction

Solution6 k inG ⇐ Solution6 k inH

v

e Only Danger

Minor models that:

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

²neighbors inG, however

e Only Danger

Minor models that:

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

²neighbors inG, however

v

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

v

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

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.

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.

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 .

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 .

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.

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.

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.

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.

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

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

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

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

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.

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.

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.

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.

ere are at most k such bags of constant size.

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

ere are at most k such bags of constant size.

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

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)}

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)}

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)}

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)}

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)}

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)}

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

And thus we are (finally!) done.

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

Dank u wel