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Page 1: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Introduction to the Regularity Lemma

Speaker: Joseph, Chuang-Chieh Lin

Advisor: Professor Maw-Shang Chang

Computation Theory LaboratoryDept. Computer Science and Information Engineering

National Chung Cheng University, Taiwan

July 8, 2008

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 2: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Outline

1 Introduction

2 Regular pairs and their properties

3 Szemeredi’s Regularity Lemma

4 A simple application

5 Conclusion and remarks

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 3: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Outline

1 Introduction

2 Regular pairs and their properties

3 Szemeredi’s Regularity Lemma

4 A simple application

5 Conclusion and remarks

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 4: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Introduction

Theorem 1.1 (Szemeredi’s Theorom)

Let k be a positive integer and let 0 < δ < 1. Then there

exists a positive integer N = N(k, δ), such that for every

A ⊂ {1, 2, ...,N}, |A| ≥ δN, A contains an arithmetic

progression of length k.

A branch of Ramsey theory (see also Van der Waerden’s

theorem).

How about N(3, 1/2)?

{1, 2, 3, 4, 5, 6, 7, 8}{1, 2, 3, 4, 5, 6, 7, 8, 9}

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 5: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Introduction

Theorem 1.1 (Szemeredi’s Theorom)

Let k be a positive integer and let 0 < δ < 1. Then there

exists a positive integer N = N(k, δ), such that for every

A ⊂ {1, 2, ...,N}, |A| ≥ δN, A contains an arithmetic

progression of length k.

A branch of Ramsey theory (see also Van der Waerden’s

theorem).

How about N(3, 1/2)?

{1, 2, 3, 4, 5, 6, 7, 8}{1, 2, 3, 4, 5, 6, 7, 8, 9}

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 6: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Introduction

Theorem 1.1 (Szemeredi’s Theorom)

Let k be a positive integer and let 0 < δ < 1. Then there

exists a positive integer N = N(k, δ), such that for every

A ⊂ {1, 2, ...,N}, |A| ≥ δN, A contains an arithmetic

progression of length k.

A branch of Ramsey theory (see also Van der Waerden’s

theorem).

How about N(3, 1/2)?

{1, 2, 3, 4, 5, 6, 7, 8}{1, 2, 3, 4, 5, 6, 7, 8, 9}

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 7: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Introduction

Theorem 1.1 (Szemeredi’s Theorom)

Let k be a positive integer and let 0 < δ < 1. Then there

exists a positive integer N = N(k, δ), such that for every

A ⊂ {1, 2, ...,N}, |A| ≥ δN, A contains an arithmetic

progression of length k.

A branch of Ramsey theory (see also Van der Waerden’s

theorem).

How about N(3, 1/2)?

{1, 2, 3, 4, 5, 6, 7, 8}{1, 2, 3, 4, 5, 6, 7, 8, 9}

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 8: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Related to a famous result...

Theorem

The primes contain arbitrarily long

arithmetic progressions.

(Terence Tao and Ben J. Green, 2004)

Terence Tao (2006 Fields Medal)

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 9: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Introduction (contd.)

The best-known bounds for N(k, δ):

C log(1/δ)k−1

≤ N(k , δ) ≤ 22δ−22

k+9

.

The Regularity Lemma (Szemeredi 1978) was invented as anauxiliary lemma in the proof of Szemeredi’s Theorom.

Roughly speaking, every graph (large enough) can, in somesense, be approximated by (pseudo-)random graphs.

Helpful in proving theorems for arbitrary graphs whenever thecorresponding result is easy for random graphs.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 10: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Introduction (contd.)

The best-known bounds for N(k, δ):

C log(1/δ)k−1

≤ N(k , δ) ≤ 22δ−22

k+9

.

The Regularity Lemma (Szemeredi 1978) was invented as anauxiliary lemma in the proof of Szemeredi’s Theorom.

Roughly speaking, every graph (large enough) can, in somesense, be approximated by (pseudo-)random graphs.

Helpful in proving theorems for arbitrary graphs whenever thecorresponding result is easy for random graphs.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 11: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Introduction (contd.)

The best-known bounds for N(k, δ):

C log(1/δ)k−1

≤ N(k , δ) ≤ 22δ−22

k+9

.

The Regularity Lemma (Szemeredi 1978) was invented as anauxiliary lemma in the proof of Szemeredi’s Theorom.

Roughly speaking, every graph (large enough) can, in somesense, be approximated by (pseudo-)random graphs.

Helpful in proving theorems for arbitrary graphs whenever thecorresponding result is easy for random graphs.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 12: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Introduction (contd.)

The best-known bounds for N(k, δ):

C log(1/δ)k−1

≤ N(k , δ) ≤ 22δ−22

k+9

.

The Regularity Lemma (Szemeredi 1978) was invented as anauxiliary lemma in the proof of Szemeredi’s Theorom.

Roughly speaking, every graph (large enough) can, in somesense, be approximated by (pseudo-)random graphs.

Helpful in proving theorems for arbitrary graphs whenever thecorresponding result is easy for random graphs.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 13: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Introduction (contd.)

Endre Szemerei

About 15 years later, its power was noted and plenty results ingraph theory and theoretical computer science have beenworked out.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 14: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Outline

1 Introduction

2 Regular pairs and their properties

3 Szemeredi’s Regularity Lemma

4 A simple application

5 Conclusion and remarks

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 15: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Density of bipartite graphs

Definition 2.1

Given a bipartite graph G = (A,B ,E ), E ⊂ A × B . Thedensity of G is defined to be

d(A,B) =e(A,B)

|A| · |B |,

where e(A,B) is the number of edges between A, B .

A perfect matching of G has density 1/n if |A| = |B | = n.

d(A,B) = 1 If G is complete bipartite.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 16: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Density of bipartite graphs

Definition 2.1

Given a bipartite graph G = (A,B ,E ), E ⊂ A × B . Thedensity of G is defined to be

d(A,B) =e(A,B)

|A| · |B |,

where e(A,B) is the number of edges between A, B .

A perfect matching of G has density 1/n if |A| = |B | = n.

d(A,B) = 1 If G is complete bipartite.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 17: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

ǫ-regular pair

Definition 2.2

Let ǫ > 0. Given a graph G and two disjoint vertex sets A ⊂ V ,B ⊂ V , we say that the pair (A,B) is ǫ-regular if for every X ⊂ A

and Y ⊂ B satisfying

|X | ≥ ǫ|A| and |Y | ≥ ǫ|B |,

we have|d(X ,Y ) − d(A,B)| < ǫ.

If G = (A,B ,E ) is a complete bipartite graph, then (A,B) isǫ-regular for every ǫ > 0.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 18: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

ǫ-regular pair (contd.)

1/2-regular 1/2-irregular 1/2-irregular

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 19: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Regularity is preserved when moving to subsets

Fact 2.3

Assume that

(A,B) is a ǫ-regular and d(A,B) = d, and

A′ ⊂ A and B ′ ⊂ B satisfy |A′| ≥ γ|A| and |B ′| ≥ γ|B | for

some γ ≥ ǫ,

then

(A′,B ′) is a max{2ǫ, γ−1ǫ}-regular and

d(A′,B ′) ≥ d − ǫ or d(A′,B ′) ≤ d + ǫ.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 20: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of Fact 2.3

Consider A′′ ⊂ A′ and B ′′ ⊂ B ′′, s.t. |A′′| ≥ ǫγ· γ|A′| ≥ ǫ|A|

and |B ′′| ≥ ǫγ· γ|B ′| ≥ ǫ|B |.

|d(A′′, B ′′) − d(A, B)| < ǫ.Hence |d(A′, B ′) − d(A′′, B ′′)| < 2ǫ.

Furthermore, since |d(A′,B ′) − d(A,B)| < ǫ,

d − ǫ < d(A′, B ′) < d + ǫ.|d(A′, B ′) − d(A′′, B ′′)| < 2ǫ.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 21: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of Fact 2.3

Consider A′′ ⊂ A′ and B ′′ ⊂ B ′′, s.t. |A′′| ≥ ǫγ· γ|A′| ≥ ǫ|A|

and |B ′′| ≥ ǫγ· γ|B ′| ≥ ǫ|B |.

|d(A′′, B ′′) − d(A, B)| < ǫ.Hence |d(A′, B ′) − d(A′′, B ′′)| < 2ǫ.

Furthermore, since |d(A′,B ′) − d(A,B)| < ǫ,

d − ǫ < d(A′, B ′) < d + ǫ.|d(A′, B ′) − d(A′′, B ′′)| < 2ǫ.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 22: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of Fact 2.3

Consider A′′ ⊂ A′ and B ′′ ⊂ B ′′, s.t. |A′′| ≥ ǫγ· γ|A′| ≥ ǫ|A|

and |B ′′| ≥ ǫγ· γ|B ′| ≥ ǫ|B |.

|d(A′′, B ′′) − d(A, B)| < ǫ.Hence |d(A′, B ′) − d(A′′, B ′′)| < 2ǫ.

Furthermore, since |d(A′,B ′) − d(A,B)| < ǫ,

d − ǫ < d(A′, B ′) < d + ǫ.|d(A′, B ′) − d(A′′, B ′′)| < 2ǫ.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 23: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of Fact 2.3

Consider A′′ ⊂ A′ and B ′′ ⊂ B ′′, s.t. |A′′| ≥ ǫγ· γ|A′| ≥ ǫ|A|

and |B ′′| ≥ ǫγ· γ|B ′| ≥ ǫ|B |.

|d(A′′, B ′′) − d(A, B)| < ǫ.Hence |d(A′, B ′) − d(A′′, B ′′)| < 2ǫ.

Furthermore, since |d(A′,B ′) − d(A,B)| < ǫ,

d − ǫ < d(A′, B ′) < d + ǫ.|d(A′, B ′) − d(A′′, B ′′)| < 2ǫ.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 24: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Most degrees into a large set are large

Fact 2.4

Let (A,B) be an ǫ-regular pair and d(A,B) = d. Then for any

Y ⊂ B, |Y | > ǫ|B | we have

#{x ∈ A | deg(x ,Y ) ≤ (d − ǫ)|Y |} ≤ ǫ|A|,

where deg(x ,Y ) is the number of neighbors of x in Y .

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 25: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of Fact 2.4

Let δ > ǫ be a constant.

Let X = {x ∈ A | deg(x ,Y ) ≤ (d − ǫ)|Y |}.

Assume |X | = δ|A| > ǫ|A|.

Clearly d(X ,Y ) ≤ δ|A|·(d−ǫ)|Y |δ|A||Y | ≤ d − ǫ.

But d − ǫ < d(X ,Y ) by the regularity of (A,B) and|Y | > ǫ|B |.

A contradiction occurs.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 26: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Outline

1 Introduction

2 Regular pairs and their properties

3 Szemeredi’s Regularity Lemma

4 A simple application

5 Conclusion and remarks

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 27: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

The famous Regularity Lemma

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 28: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

The famous Regularity Lemma (contd.)

Theorem 3.1 (Szemeredi’s Regularity Lemma, 1978)

For every ǫ > 0 and positive integer t, there exists two integers

M(ǫ, t) and N(ǫ, t) such that

For every graph G (V ,E ) with at least N(ǫ, t) vertices, thereis a partition (V0,V1,V2, . . . ,Vk) of V with:

t ≤ k ≤ M(ǫ, t),|V0| ≤ ǫn, and

|V1| = |V2| = . . . = |Vk |

such that at least (1 − ǫ)(

k2

)

of pairs (Vi ,Vj) are ǫ-regular.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 29: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

One of the proofs...

A combinatorial proof:

k sets ⇒ refine to k · 2k−1 sets → refine to (k2k−1) · 2k2k−1−1

⇒ . . . .

A tower of 2’s of height O(1/ǫ5) (since O(1/ǫ5) refinementsrequired).

e.g., 222222

: a tower of 2’s of height 5.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 30: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

One of the proofs...

A combinatorial proof:

k sets ⇒ refine to k · 2k−1 sets → refine to (k2k−1) · 2k2k−1−1

⇒ . . . .

A tower of 2’s of height O(1/ǫ5) (since O(1/ǫ5) refinementsrequired).

e.g., 222222

: a tower of 2’s of height 5.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 31: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Lower bound of M(ǫ, t) (k has to be in the worst case)

The tower dependence on 1/ǫ is necessary (by TimothyGowers [4]).

Constructive proof by Alon et al. [2]

M(n) = O(n2.2376) time (matrix multiplication).

“Deciding if a given partition of an input graph satisfies theproperty guaranteed by the regularity lemma” isco-NP-complete [2].

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 32: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Lower bound of M(ǫ, t) (k has to be in the worst case)

The tower dependence on 1/ǫ is necessary (by TimothyGowers [4]).

Constructive proof by Alon et al. [2]

M(n) = O(n2.2376) time (matrix multiplication).

“Deciding if a given partition of an input graph satisfies theproperty guaranteed by the regularity lemma” isco-NP-complete [2].

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 33: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Lower bound of M(ǫ, t) (k has to be in the worst case)

The tower dependence on 1/ǫ is necessary (by TimothyGowers [4]).

Constructive proof by Alon et al. [2]

M(n) = O(n2.2376) time (matrix multiplication).

“Deciding if a given partition of an input graph satisfies theproperty guaranteed by the regularity lemma” isco-NP-complete [2].

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 34: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Outline

1 Introduction

2 Regular pairs and their properties

3 Szemeredi’s Regularity Lemma

4 A simple application

5 Conclusion and remarks

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 35: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Triangle Removal Lemma

Lemma 4.1 (Triangle Removal Lemma)

For all 0 < δ < 1, there exists ǫ = ǫ(δ), such that for every

n-vertex graph G, at least one of the following is true:

1. G can be made triangle-free by removing < δn2 edges.

2. G has ≥ ǫn3 triangles.

We show this lemma by making use of the Regularity Lemma.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 36: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Triangle Removal Lemma

Lemma 4.1 (Triangle Removal Lemma)

For all 0 < δ < 1, there exists ǫ = ǫ(δ), such that for every

n-vertex graph G, at least one of the following is true:

1. G can be made triangle-free by removing < δn2 edges.

2. G has ≥ ǫn3 triangles.

We show this lemma by making use of the Regularity Lemma.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 37: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of the Triangle Removal Lemma

The regularity Lemma

For every ǫ > 0 and positive integer t, there exists two integers M(ǫ, t) andN(ǫ, t) such that

For every graph G(V , E) with at least N(ǫ, t) vertices, there is a partition

(V0, V1, V2, . . . , Vk) of V with:

t ≤ k ≤ M(ǫ, t),|V0| ≤ ǫn, and|V1| = |V2| = . . . = |Vk |

such that at least (1 − ǫ)`

k

2

´

of pairs (Vi , Vj) are ǫ-regular.

Let ǫ = δ10 and t = 10

δ.

Star with an arbitrary graph G (n ≥ N(ǫ, t)).

Find a δ10 -regular partition into k = k( δ

10 , 10δ

) blocks.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 38: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of the Triangle Removal Lemma

The regularity Lemma

For every ǫ > 0 and positive integer t, there exists two integers M(ǫ, t) andN(ǫ, t) such that

For every graph G(V , E) with at least N(ǫ, t) vertices, there is a partition

(V0, V1, V2, . . . , Vk) of V with:

t ≤ k ≤ M(ǫ, t),|V0| ≤ ǫn, and|V1| = |V2| = . . . = |Vk |

such that at least (1 − ǫ)`

k

2

´

of pairs (Vi , Vj) are ǫ-regular.

Let ǫ = δ10 and t = 10

δ.

Star with an arbitrary graph G (n ≥ N(ǫ, t)).

Find a δ10 -regular partition into k = k( δ

10 , 10δ

) blocks.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 39: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of the Triangle Removal Lemma (contd.)

Using the partition we justobtained, we define a reducedgraph G ′ as follows:

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 40: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of the Triangle Removal Lemma (contd.)

I: Remove all edges betweennon-regular pairs (at most δ

10n2

edges).

≤ δ10

(

k2

)

irregular pairs, and atmost ( n

k)2 edges between

each pair.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 41: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of the Triangle Removal Lemma (contd.)

II: Remove all edges inside blocks(at most δ

10n2 edges).

k blocks, and each containsat most

(

n/k2

)

edges,t ≤ k

≤ n2

k≤ δ

10n2 edges areremoved.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 42: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of the Triangle Removal Lemma (contd.)

III: Remove all edges between pairsof density < δ

2 (at most δ2n2

edges).

≤ δ2 ( n

k)2 edges between a pair

of density < δ2 , and at most

(

k

2

)

such pairs.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 43: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of the Triangle Removal Lemma (contd.)

Totally at most (δ/10 + δ/10 + δ/2)n2 < δn2 edges areremoved.

Thus if G ′ contains no triangle, the first condition of thelemma is satisfied.

Hence we suppose that G ′ contains a triangle and continue tosee the second condition of the lemma.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 44: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of the Triangle Removal Lemma (contd.)

Totally at most (δ/10 + δ/10 + δ/2)n2 < δn2 edges areremoved.

Thus if G ′ contains no triangle, the first condition of thelemma is satisfied.

Hence we suppose that G ′ contains a triangle and continue tosee the second condition of the lemma.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 45: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of the Triangle Removal Lemma (contd.)

By some technical reasons, we may assume V0 = ∅ and letm = n/k be the size of the blocks (V1,V2, . . . ,Vk).

A triangle in G ′ must go between three different blocks, sayA, B , and C .

If there is an edge between A and B ⇒ there must be manyedges (by Step III).

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 46: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of the Triangle Removal Lemma (contd.)

By some technical reasons, we may assume V0 = ∅ and letm = n/k be the size of the blocks (V1,V2, . . . ,Vk).

A triangle in G ′ must go between three different blocks, sayA, B , and C .

If there is an edge between A and B ⇒ there must be manyedges (by Step III).

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 47: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of the Triangle Removal Lemma (contd.)

By some technical reasons, we may assume V0 = ∅ and letm = n/k be the size of the blocks (V1,V2, . . . ,Vk).

A triangle in G ′ must go between three different blocks, sayA, B , and C .

If there is an edge between A and B ⇒ there must be manyedges (by Step III).

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 48: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of the Triangle Removal Lemma (contd.)

Since “most degrees into a largeset are large”

≤ m/4 vertices in A have≤ δ

4m neighbors in B

≤ m/4 vertices in A have≤ δ

4m neighbors in C

Hence ≥ m/2 vertices in A haveboth ≥ δ

4m neighbors in B and

≥ δ4m neighbors in C .

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 49: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of the Triangle Removal Lemma (contd.)

Since “most degrees into a largeset are large”

≤ m/4 vertices in A have≤ δ

4m neighbors in B

≤ m/4 vertices in A have≤ δ

4m neighbors in C

Hence ≥ m/2 vertices in A haveboth ≥ δ

4m neighbors in B and

≥ δ4m neighbors in C .

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 50: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of the Triangle Removal Lemma (contd.)

Consider a such vertex from A.

How many edges go between Sand T?

S ≥ δ4m and T ≥ δ

4m

d(B, C ) ≥ δ2 and (B, C ) is

δ10 -regularhence e(B, C ) ≥

( δ2 − δ

10)|S ||T | ≥ δ3

64m2

Total # triangles≥ δ3

64m2 · m2 = δ3

128k3 n3.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 51: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of the Triangle Removal Lemma (contd.)

Consider a such vertex from A.

How many edges go between Sand T?

S ≥ δ4m and T ≥ δ

4m

d(B, C ) ≥ δ2 and (B, C ) is

δ10 -regularhence e(B, C ) ≥

( δ2 − δ

10)|S ||T | ≥ δ3

64m2

Total # triangles≥ δ3

64m2 · m2 = δ3

128k3 n3.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 52: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Proof of the Triangle Removal Lemma (contd.)

Consider a such vertex from A.

How many edges go between Sand T?

S ≥ δ4m and T ≥ δ

4m

d(B, C ) ≥ δ2 and (B, C ) is

δ10 -regularhence e(B, C ) ≥

( δ2 − δ

10)|S ||T | ≥ δ3

64m2

Total # triangles≥ δ3

64m2 · m2 = δ3

128k3 n3.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 53: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Is the Triangle Removal Lemma important? YES!

The Triangle Removal Lemma

For all 0 < δ < 1, there exists ǫ = ǫ(δ), such that for every n-vertex graph G ,at least one of the following is true:

1. G can be made triangle-free by removing < δn2 edges.

2. G has ≥ ǫn3 triangles.

The graph property “triangle-free” is “testable”.

Yet the complexity has dependence of towers of δ.

e.g., 128k3

δ3 , k is tower of 2’s of height depending on O(1/δ).

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 54: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Is the Triangle Removal Lemma important? YES!

The Triangle Removal Lemma

For all 0 < δ < 1, there exists ǫ = ǫ(δ), such that for every n-vertex graph G ,at least one of the following is true:

1. G can be made triangle-free by removing < δn2 edges.

2. G has ≥ ǫn3 triangles.

The graph property “triangle-free” is “testable”.

Yet the complexity has dependence of towers of δ.

e.g., 128k3

δ3 , k is tower of 2’s of height depending on O(1/δ).

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 55: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Conclusion and remarks

A LOT OF applications of the Regularity Lemma in the fieldof property testing.

Counting the number of forbidden subgraphs, testingmonotone graph properties, dealing with partition-typeproblems, etc.

Excellent surveys for the Regularity Lemma: [5, 6]; and nicelecture notes: [1] (by Luca Trevisan); also Luca Trevisan’sBlog: “in theory” (http://lucatrevisan.wordpress.com).

Question

Is it possible to apply the Regularity Lemma to design

fixed-parameter algorithms for graph problems?

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 56: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

Thank you!

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 57: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

References

[1] Additive Combinatorics and Computer Science. Minicourse: August23–24 at Princeton University (immediately after RANDOM +APPROX’07). Lecturers: Boaz Barak, Luca Trevisan, and Avi Wigderson

[2] N. Alon, R. A. Duke, H. Lefmann, V. Rodl, and R. Yuster: Thealgorithmic aspects of the Regularity Lemma. J. Algorithms 16 (1994)80–109.

[3] N. Alon, E. Fischer, M. Krivelevich, and M. Szegedy: Efficient testing oflarge graphs. Combinatorica 20 (2000) 451–476.

[4] W. T. Gowers: Lower bounds of tower type for Szemeredi’s uniformitylemma. Geom. Funct. Anal. 7 (1997) 322–337.

[5] J. Komlos and M. Simonovits: Szemeredi’s regularity lemma and itsapplications in graph theory. Bolyai Society Mathematical Studies 2,Combinatorics, Paul Erdos is Eighty (Volume 2) (D. Miklos, V. T. Sos, T.Szonyi eds.), Keszthely (Hungary) (1993), Budapest (1996), pp. 295–352.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma

Page 58: Regularity Lemma

IntroductionRegular pairs and their properties

Szemeredi’s Regularity LemmaA simple application

Conclusion and remarks

[6] J. Komlos, A. Shokoufandeh, M. Simonovits, and E. Szemeredi: Theregularity lemma and its applications in graph theory. Theoretical Aspectsof Computer Science, Lecture Notes Comput. Sci., Vol. 2292, pp.84–112, 2002.

[7] E. Szemeredi: Regular partitions of graphs. In Proc. Colloque. Inter.CNRS (J. C. Bermond, J. C. Fournier, M. Las Vergnas, and Sotteaueds.), 1978, pp. 399–401.

Computationa Theory Lab, CSIE, CCU, Taiwan Introduction to the Regularity Lemma


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