Coloring k-colorable graphs using smaller palettes

Eran Halperin Ram Nathaniel Uri Zwick

Tel Aviv University

New coloring results

Coloring k-colorable graphs

of maximum degree using

-2/klog1/kcolors

(instead of -2/klog1/2colors [KMS])

New coloring results Coloring k-colorable graphs using

n(k) colors (instead of n(k)colors [KMS])

k (k) (k)

4 0.3684 0.4000

5 0.4686 0.5000

6 0.5443 0.5714

k (k) (k)

4 7/19 2/5

5 97/207 1/2

6 43/79 4/7

An extension of Alon-Kahale

AK: If a graph contains an independent set of size n/k+m, k integer, then an independent set of size m3/(k+1) can be found in polynomial time.

Extension: If a graph contains an independent set of size n then an independent set of size nf( can be found in polynomial time, where

][)(

)1()(

3)1)(1(

kkk

fkk

Graph coloring basics

If in any k-colorable graph on n vertices we can find, in polynomial time, one of

• Two vertices that have the same color under some valid k-coloring ;

• An independent set of size (n1-) ;

then we can color any k-colorable graph using O(n) colors.

Coloring 3-colorable graphs using O(n1/2) colors [Wigderson]

A graph with maximum degree can be easily colored using colors.

If <n1/2, color using colors.

Otherwise, let v be a vertex of degree hen, N(v) is 2-colorable and contains an independent set of size n1/2/2

Vector k-Coloring [KMS]

A vector k-coloring of a graph G=(V,E) is a sequence of unit vectors v1,v2,…,vn such that if (i,j) in E then <vi,vj>=-1/(k-1).

Finding large independent sets

Let G=(V,E) be a 3-colorable graph.

Let r be a random normally distributed vector in Rn. Let .

I’ is obtained from I by removing a vertex from each edge of I.

lnlnln 31

32c

}|{ crvViI i

Constructing the sets I and I’

riv

jv

Analysis

]2[]2)Pr[(

]Pr[]'[

)()(

)(]Pr[]'[

21

21

12111

21

1

2

2

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2

3

cmNcrvvm

crvcrvmmE

ecNe

cnNcrvnnEcc

ccc

Analysis (Cont.)

1cv

2cv

)(2 21 vvc

1cv

2cv

)(2 21 vvc

12cu

22cu

Analysis (Cont.)

221

21

)2(]22Pr[

]Pr[

cNcrucru

crvcrv

2

2

2

2

212

221

)2(

)2(c

c

cc

ecN

ecN

Analysis (Cont.)

))ln(

()(2

)(2

)2(2

)(]''[

Thus,

2)(

)2(

)(

,lnlnln With

3/12/

2111

2

2/3

2

21

21

2/

2111

2

31

32

2

3

2

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2

3

ne

n

cNn

cNn

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cN

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A simple observation

Either G[N(u,v)] is (k-2)-colorable,

or u and v get the same color under

any a k-coloring of G.

u

v)()(),( vNuNvuN

Suppose G=(V,E) is k-colorable.

A lemma of BlumLet G=(V,E) be a k-colorable graph with• minimum degree for every

Then, it is possible to construct, in polynomial time, a collection {Ti} of about n subsets of V such that at least one Ti satisfies:

• |Ti|=s)

• Ti has an independent subset of size

svNuN |)()(| Vvu ,

||))(( log1

11

ink TO

A lemma of Blum

v )(vN ))(( vNN

Graph coloring techniques

WigdersonKarger

Motwani Sudan

Blum

AlonKahale

Our Algorithm

Blum Karger

The new algorithm

Step 0:

If k=2, color the graph using 2 colors.

If k=3, color the graph using n3/14 colors using the algorithm of Blum and Karger.

The new algorithm

Step 1:

Repeatedly remove from the graph vertices of degree at most n(k)/(1-2/k). Let U be the set of vertices removed, and W=V-U.

Average degree of G[U] is at most n(k)/(1-2/k).

Minimum degree of G[W] at least n(k)/(1-2/k).

If |U|>n/2, use [KMS] to find an independent set of size n/D1-2/k= n1-k).

Step 1

UW

Average degree of G[U] is at most .

Minimum degree of G[W] at least .

Let n(k)/(1-2/k).

The new algorithm

Step 2:

For every u,v such that N(u,v)>n(1-(k)/(1-(k-2)),

apply the algorithm recursively on G[N(u,v)] and k-2.

If G[N(u,v)] is (k-2)-colorable, we get an independent set of size |N(u,v)|1-(k-2)>n1-(k).

Otherwise, we can infer* that u and v must be assigned the same color.

The new algorithm

Step 3: If we reach this step then |W|>n/2, the minimum degree of G[W] is at least n(k)/(1-2/k),

and for every u,v in W, N(u,v)>n(1-(k)/(1-(k-2)).

By Blum’s lemma, we can find a collection {Ti} of about n subsets of W such that at least one Ti

satisfies |Ti|=s) and Ti has an independent subset of size .

By the extension of the Alon-Kahale result,

we can find an IS of size

||))(( log1

11

ink TO

n k

k

k

k

k

)2(1

)(1

/21

)(23

The recurrence relation

)2(1)1(3

4

61)( 2

kk

kk

Karger]-[Blum )3(

0)2(

143

Hardness results

It is NP-hard to 4-color 3-colorable graphs [Khanna,Linial,Safra ‘93] [Guruswami,Khanna ‘00]

For any k, it is NP-hard to k-color

2-colorable hypergraphs

[Guruswami,Hastad,Sudan ‘00]