Coloring k-colorable graphs using smaller palettes Eran Halperin Ram Nathaniel Uri Zwick Tel Aviv...
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Transcript of Coloring k-colorable graphs using smaller palettes Eran Halperin Ram Nathaniel Uri Zwick Tel Aviv...
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
2
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
2
2
2
3
ne
n
cNn
cNn
cnNmnE
cee
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cN
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c
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cc
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cc
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]