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Coloring sparse random k -colorable graphs in polynomial expected time Julia Böttcher, TU München, Germany MFCS, Gdansk, 2005

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Page 1: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

Coloring sparse random k-colorablegraphs in polynomial expected time

Julia Böttcher, TU München, Germany

MFCS, Gdansk, 2005

Page 2: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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Definitions

V (G) = {1, . . . , n} = vertext set of G.

χ(G) = chromatic number of G.

MFCS, Gdansk, 2005

Page 3: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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Definitions

V (G) = {1, . . . , n} = vertext set of G.

χ(G) = chromatic number of G.

Algorithm A runs in polynomial expected running time:∑

|G|=n

tA(G) · P[ G ] is polynomial.

(tA(G): running time of A on input G)

MFCS, Gdansk, 2005

Page 4: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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Approximating the chromatic number

Coloring graphs:

THEOREM HALLDÓRSON 1993

χ(G) can be approximated within n(log log n)2

(log n)3 .

MFCS, Gdansk, 2005

Page 5: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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Approximating the chromatic number

Coloring graphs:

THEOREM HALLDÓRSON 1993

χ(G) can be approximated within n(log log n)2

(log n)3 .

THEOREM ENGEBRETSEN,HOLMERIN 2003

Approximating χ(G) within n1−O(1/√

log log n) is hard.

MFCS, Gdansk, 2005

Page 6: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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Approximating the chromatic number

Coloring graphs:

THEOREM HALLDÓRSON 1993

χ(G) can be approximated within n(log log n)2

(log n)3 .

THEOREM ENGEBRETSEN,HOLMERIN 2003

Approximating χ(G) within n1−O(1/√

log log n) is hard.

Coloring 3-chromatic graphs:

THEOREM BLUM,KARGER 1997

G can efficiently be colored with n3/14+o(1) colors.

THEOREM KHANNA, LINIAL, SAFRA 1993

4-coloring G is hard.MFCS, Gdansk, 2005

Page 7: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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k-coloring random k-colorable graphs

Coloring uniformly distributed k-colorable graphs:

� Algorithms that work w.h.p. KUCERA 1977; TURNER 1988

� In polynomial expected time. DYER AND FRIEZE 1989

MFCS, Gdansk, 2005

Page 8: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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k-coloring random k-colorable graphs

Coloring uniformly distributed k-colorable graphs:

� Algorithms that work w.h.p. KUCERA 1977; TURNER 1988

� In polynomial expected time. DYER AND FRIEZE 1989

u

v

MFCS, Gdansk, 2005

Page 9: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

--------------

k-coloring random k-colorable graphs

Coloring uniformly distributed k-colorable graphs:

� Algorithms that work w.h.p. KUCERA 1977; TURNER 1988

� In polynomial expected time. DYER AND FRIEZE 1989

u

vN(v) ∩ N(u)

MFCS, Gdansk, 2005

Page 10: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

--------------

k-coloring random k-colorable graphs

Coloring uniformly distributed k-colorable graphs:

� Algorithms that work w.h.p. KUCERA 1977; TURNER 1988

� In polynomial expected time. DYER AND FRIEZE 1989

w u

vN(v) ∩ N(u)

MFCS, Gdansk, 2005

Page 11: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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The random k-colorable graph Gn,p,k

Construct Gn,p,k as follows:

� Partition V into k color classesV1, . . . , Vk of equal size.

� For i 6= j insert edges between Viand Vj with probabilityp = d/n.

MFCS, Gdansk, 2005

Page 12: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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The random k-colorable graph Gn,p,k

Construct Gn,p,k as follows:

� Partition V into k color classesV1, . . . , Vk of equal size.

� For i 6= j insert edges between Viand Vj with probabilityp = d/n.

V1

V2

V3

MFCS, Gdansk, 2005

Page 13: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

--------------

The random k-colorable graph Gn,p,k

Construct Gn,p,k as follows:

� Partition V into k color classesV1, . . . , Vk of equal size.

� For i 6= j insert edges between Viand Vj with probabilityp = d/n.

V1

V2

V3

MFCS, Gdansk, 2005

Page 14: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

--------------

k-coloring random k-colorable graphs

Coloring uniformly distributed k-colorable graphs:

� Algorithms that work w.h.p. KUCERA 1977; TURNER 1988

� In polynomial expected time. DYER AND FRIEZE 1989

MFCS, Gdansk, 2005

Page 15: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

--------------

k-coloring random k-colorable graphs

Coloring uniformly distributed k-colorable graphs:

� Algorithms that work w.h.p. KUCERA 1977; TURNER 1988

� In polynomial expected time. DYER AND FRIEZE 1989

Coloring Gn,p,k:

� W.h.p.� for p ≥ nǫ/n, BLUM AND SPENCER 1995

� for p ≥ c/n. ALON AND KAHALE 1997

� In polynomial expected time� for p ≥ nǫ/n, SUBRAMANIAN 2000

� for p ≥ c · ln n/n, COJA-OGHLAN 2004

� for p ≥ c/n.

MFCS, Gdansk, 2005

Page 16: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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An Algorithm for 3-Coloring Gn,p,3

Algorithm 1 : COLORGn,p,3(G)Input : a graph Gn,p,3 = G = (V,E)

Output : a valid coloring for Gn,p,3

begin1 for 0 ≤ y ≤ n do2 foreach 3-coloring of each Y ⊆ V of size y do

/∗∗ The initial phase ∗∗/3 Construct a coloring that fails on < ǫn vertices ;

/∗∗ The iterative recoloring procedure ∗∗/4 Refine the initial coloring ;

/∗∗ The uncoloring procedure ∗∗/5 Repeatedly uncolor vertices having few neighbors of

some other color ;/∗∗ The extension step ∗∗/

6 Color the components of uncolored vertices ;7 if we have a valid coloring of G then return ;

end

MFCS, Gdansk, 2005

Page 17: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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An Algorithm for 3-Coloring Gn,p,3

Algorithm 1 : COLORGn,p,3(G)Input : a graph Gn,p,3 = G = (V,E)

Output : a valid coloring for Gn,p,3

begin1 for 0 ≤ y ≤ n do2 foreach 3-coloring of each Y ⊆ V of size y do

/∗∗ The initial phase ∗∗/3 Construct a coloring that fails on < ǫn vertices ;

/∗∗ The iterative recoloring procedure ∗∗/4 Refine the initial coloring ;

/∗∗ The uncoloring procedure ∗∗/5 Repeatedly uncolor vertices having few neighbors of

some other color ;/∗∗ The extension step ∗∗/

6 Color the components of the uncolored vertices ;7 if we have a valid coloring of G then return ;

end

MFCS, Gdansk, 2005

Page 18: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

--------------

An Algorithm for 3-Coloring Gn,p,3

Algorithm 1 : COLORGn,p,3(G)Input : a graph Gn,p,3 = G = (V,E)

Output : a valid coloring for Gn,p,3

begin1 for 0 ≤ y ≤ n do2 foreach 3-coloring of each Y ⊆ V of size y do

/∗∗ The initial phase ∗∗/3 Construct a coloring that fails on < ǫn vertices ;

/∗∗ The iterative recoloring procedure ∗∗/4 Refine the initial coloring ;

/∗∗ The uncoloring procedure ∗∗/5 Repeatedly uncolor vertices having few neighbors of

some other color ;/∗∗ The extension step ∗∗/

6 Color the components of the uncolored vertices ;7 if we have a valid coloring of G then return ;

end

MFCS, Gdansk, 2005

Page 19: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

--------------

An Algorithm for 3-Coloring Gn,p,3

Algorithm 1 : COLORGn,p,3(G)Input : a graph Gn,p,3 = G = (V,E)

Output : a valid coloring for Gn,p,3

begin1 for 0 ≤ y ≤ n do2 foreach 3-coloring of each Y ⊆ V of size y do

/∗∗ The initial phase ∗∗/3 Construct a coloring that fails on < ǫn vertices ;

/∗∗ The iterative recoloring procedure ∗∗/4 Refine the initial coloring ;

/∗∗ The uncoloring procedure ∗∗/5 Repeatedly uncolor vertices having few neighbors of

some other color ;/∗∗ The extension step ∗∗/

6 Color the components of the uncolored vertices ;7 if we have a valid coloring of G then return ;

end

MFCS, Gdansk, 2005

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--------------

An Algorithm for 3-Coloring Gn,p,3

Algorithm 1 : COLORGn,p,3(G)Input : a graph Gn,p,3 = G = (V,E)

Output : a valid coloring for Gn,p,3

begin1 for 0 ≤ y ≤ n do2 foreach 3-coloring of each Y ⊆ V of size y do

/∗∗ The initial phase ∗∗/3 Construct a coloring that fails on < ǫn vertices ;

/∗∗ The iterative recoloring procedure ∗∗/4 Refine the initial coloring ;

/∗∗ The uncoloring procedure ∗∗/5 Repeatedly uncolor vertices having few neighbors of

some other color ;/∗∗ The extension step ∗∗/

6 Color the components of the uncolored vertices ;7 if we have a valid coloring of G then return ;

end

MFCS, Gdansk, 2005

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An SDP for MAX- 3-CUT

Problem: Find a 3-cut⋃

Vi = [n] s.t.∑

i6=j e(Vi, Vj) is maximal.

Idea: Use variables taking one of 3 different valuesIdea: (3 unit vectors s1, s2, s3 with 〈si |sj〉 = −1/2).

MFCS, Gdansk, 2005

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An SDP for MAX- 3-CUT

Problem: Find a 3-cut⋃

Vi = [n] s.t.∑

i6=j e(Vi, Vj) is maximal.

Idea: Use variables taking one of 3 different valuesIdea: (3 unit vectors s1, s2, s3 with 〈si |sj〉 = −1/2).

Max-3-cut:max

ij∈E(G)

2

3(1 − 〈vi |vj〉) ,

s.t.vi ∈ {s1, s2, s3} ,∀i ∈ V

MFCS, Gdansk, 2005

Page 23: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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An SDP for MAX- 3-CUT

Problem: Find a 3-cut⋃

Vi = [n] s.t.∑

i6=j e(Vi, Vj) is maximal.

Idea: Use variables taking one of 3 different valuesIdea: (3 unit vectors s1, s2, s3 with 〈si |sj〉 = −1/2).

Max-3-cut:max

ij∈E(G)

2

3(1 − 〈vi |vj〉) ,

s.t.vi ∈ {s1, s2, s3} ,∀i ∈ V

Relaxation:max

ij∈E(G)

2

3(1 − 〈xi |xj〉) ,

s.t. xi ∈ Rn, ‖xi‖ = 1 ∀i ∈ V

〈xi |xj〉 ≥ −1

2∀i, j ∈ V

(SDP3)FRIEZE,JERRUM 1997

MFCS, Gdansk, 2005

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SDP3 and 3-colorable graphs

� For a 3-colorable graph Max-3-cut = |E|

MFCS, Gdansk, 2005

Page 25: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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SDP3 and 3-colorable graphs

� For a 3-colorable graph Max-3-cut = |E|

� Realization of a maximum 3-cut in SDP3: map the colorclasses to the vectors s1, s2, s3.

V1

V2

V3

120◦

s1

s2

s3

An optimal solution for SDP3 on 3-colorable graphs with color classes V1,V2,V3

MFCS, Gdansk, 2005

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SDP3 and 3-colorable graphs (ctd.)

Positions of the vectors of an optimal solution to SDP3 are “faraway” from this ideal picture in general:

u

vw

u′

v′

w′

xu

xv

xwxu′

xv′

xw′

An optimal solution for SDP3 on a 3-colorable graphs in 3 dimensions

MFCS, Gdansk, 2005

Page 27: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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An Algorithm for 3-Coloring Gn,p,3

Algorithm 1 : COLORGn,p,3(G)Input : a graph Gn,p,3 = G = (V,E)

Output : a valid coloring for Gn,p,3

begin1 for 0 ≤ y ≤ n do2 foreach 3-coloring of each Y ⊆ V of size y do

/∗∗ The initial phase ∗∗/3 Construct a coloring that fails on < ǫn vertices ;

/∗∗ The iterative recoloring procedure ∗∗/4 Refine the initial coloring ;

/∗∗ The uncoloring procedure ∗∗/5 Repeatedly uncolor vertices having few neighbors of

some other color ;/∗∗ The extension step ∗∗/

6 Color the components of uncolored vertices ;7 if we have a valid coloring of G then return ;

end

MFCS, Gdansk, 2005

Page 28: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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Finding an initial coloring of Gn,p,3

� X = (xv)v∈V : optimal solution to SDP3(G)

� µ-neighborhood of v : Nµ(v) := {v′ ∈ V | 〈xv |xv′〉 > 1 − µ}

MFCS, Gdansk, 2005

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Finding an initial coloring of Gn,p,3

� X = (xv)v∈V : optimal solution to SDP3(G)

� µ-neighborhood of v : Nµ(v) := {v′ ∈ V | 〈xv |xv′〉 > 1 − µ}

LEMMA

For all ǫ < 1/2 there is a µ < 1/2 s.t. t.f. holds with prob-ability greater than 1 − exp(−4n/3): For each i ∈ {1, 2, 3}there is a vertex vi ∈ Vi with

� |Nµ(vi) ∩ Vi| ≥ (1 − ǫ)n/3 and

� |Nµ(vi) ∩ Vj | < ǫn/3 for all j 6= i

MFCS, Gdansk, 2005

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Finding an initial coloring of Gn,p,3

Goal: A coloring of SDP3 that fails on < ǫn vertices.

THEOREM COJA-OGHLAN,MOORE,SANWALANI

With probability at least 1 − exp(−2n) t.f. holds

SDP3(Gn,p) ≤2

3

(

n

2

)

p + O(

n3p(1 − p)

)

.

MFCS, Gdansk, 2005

Page 31: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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Finding an initial coloring of Gn,p,3

Goal: A coloring of SDP3 that fails on < ǫn vertices.

THEOREM COJA-OGHLAN,MOORE,SANWALANI

With probability at least 1 − exp(−2n) t.f. holds

SDP3(Gn,p) ≤2

3

(

n

2

)

p + O(

n3p(1 − p)

)

.

Idea: Construct G∗ ∈ Gn,p from G ∈ Gn,p,3 by insertingadditional edges with probability p within each color class.

LEMMA

With probability at least 1 − exp(−3n/2) t.f. holds

SDP3(G∗) − SDP3(G) ≤ O(n

√pn) .

MFCS, Gdansk, 2005

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Concluding Remarks

� k-coloring is hard / difficult to approximate.

� k-colorable graphs can be k-colored in polynomialexpected time.

� On sparse graphs semidefinite programming helps.

MFCS, Gdansk, 2005

Page 33: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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Concluding Remarks

� k-coloring is hard / difficult to approximate.

� k-colorable graphs can be k-colored in polynomialexpected time.

� On sparse graphs semidefinite programming helps.

There are only 3 colors, 10 digits,

and 7 notes; it’s what we do with

them that’s important.

RUTH ROSS

MFCS, Gdansk, 2005

Page 34: Coloring sparse random -colorable graphs in polynomial ...personal.lse.ac.uk/boettche/docs/Boettcher_ColoringSparseRandomG… · Coloring sparse random k-colorable graphs in polynomial

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Concluding Remarks

� k-coloring is hard / difficult to approximate.

� k-colorable graphs can be k-colored in polynomialexpected time.

� On sparse graphs semidefinite programming helps.

There are only 3 colors, 10 digits,

and 7 notes; it’s what we do with

them that’s important.

RUTH ROSS

MFCS, Gdansk, 2005