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A note on the approximation ratio of graph-coloring

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(1)
(2)

of graph- oloring

VangelisTh. Pas hos

LAMSADE,Université Paris-Dauphine

Pla e duMaré halDe Lattre deTassigny, 75775 Paris Cedex16,Fran e

pas hoslamsade.dauphine.fr

Mar h 17,2000

Abstra t

Thepurposeofthis noteistoprovethatminimumgraph oloring anbeapproximated

in polynomial time within approximation ratio the maximum between O(n=log `(n) 1

n)

and O(loglog=log) where n and  are the order and the maximum degree of the

input-graph. If the maximum is realized by the rst quantity, then it outer-performs the

best known approximationratiofun tion ofnfor oloring,while, ifit isrealizedbythe

se ondquantity,itouter-performsthebestknownapproximationratio,fun tionof.

Consider a graph G = (V;E) of order n. In minimum graph- oloring, denoted by C in what

follows,we try to olor V with asfew olorsaspossible sothatno twoadja ent verti esre eive

the same olor. Sin eadja ent verti es areforbidden to be olored bythe same olor, afeasible

solution of C is a partition of V into independent sets. So, the optimal solution of C, denoted

by (G), is a minimum- ardinality partition into independent sets. Given that C is NP-hard,

we are interested in polynomial time algorithms providing, for all instan es, feasible solutions

the values of whi h are as lose as possible to the value of the optimal ones. Su h algorithms

willbe alled polynomialtimeapproximation algorithms(PTAA).Theperforman eguaranteeof

aPTAA Afor an instan eI of aminimization (maximization)problem, alled approximation

ratio and denoted by 

A

(I), is the value of the solution omputed by A on I divided by the

optimal value ofI. Theapproximation ratio 

A

isthe largest (smallest)ratio, over all instan es

of agiven size, guaranteed by A.The approximation ratio for ,denoted by 



, is the smallest

(largest) 

A

, over all approximation algorithms A for . In the sequel we denote by IS the

maximum independent set problem. For a set U  V, we denote by G[U℄ the sub-graph of G

indu ed by U. Finally, we denote by (G) the stability number ( ardinality of a maximum

independent set) of Gandbythe maximum degree ofG. Itis well-knownthat

(G)(G)>n (1)

Intermsofn,the best-knownapproximationratio forCisO(nlog 2

logn=log3n)([4℄). Interms

of ,the best-known approximationratio for Cis=3 andensues from [6℄.

Thefollowingex avations hema,exe utedwithparametersGandINDEPENDENT_SETwhere

INDEPENDENT_SETisanyIS-algorithm,isoriginallyintrodu edin[5℄forapproximatelysolvingC.

BEGIN *EXCAVATION*

REPEAT

S INDEPENDENT_SET(G);

olor the verti es of S by the same not already used olor;

(3)

OUTPUT X the set of used olors;

END. *EXCAVATION*

Theex avation s hemaabove haveaninteresting property(provedin [5℄for the ase whereS is

amaximum independent setandgeneralized in [1℄forthe asewhere S isanyindependentset):

 EXCAVATION 6 lnn  INDEPENDENT_SET (2)

In[2℄aPTAA, alled CLIQUE_REMOVAL,isdevisedguaranteeingapproximationratioO(log 2

n=n)

for IS. In[3 ℄,we renethe analysis of[2℄ andprove that

for every fun tion ` su h that, 8x>0 , 0<`(x)loglogx there exist onstants 

and K su h that algorithm CLIQUE_REMOVAL omputes, for every graph G of

or-der n>, an independent set S su h that if (G)>`(n)nloglogn=logn, then

jSj >Klog `(n)

n.

In what follows, we onsider `(n) onstant and denote it by `; we denote by EXHAUST, an

exhaustive-sear h algorithm for C and by _COLOR the - oloring algorithm of [6℄. Without

lossofgeneralitywe supposethatverti esare olored by1;2;::: Moreover,let K andbeasin

the quoted propositionabove.

BEGIN *COLOR*

IF n6 THEN OUTPUT EXHAUST(G) FI

S CLIQUE_REMOVAL(G); i 1; ^ X ; ; WHILEjSj>Klog ` n DO olor S by olor i; ^ X ^ X[fig ; i i+1; G G[VnS℄; IF G=; THEN OUTPUT ^ X FI S CLIQUE_REMOVAL(G); OD ~ X _COLOR(G); OUTPUT X ^ X[ ~ X; END. *COLOR*

The WHILE-loop of the algorithm above is an appli ation of EXCAVATION(G,CLIQUE_REMOVAL).

Observe alsothat, for every iteration iof the WHILE-loop,ifwe denote by G

i

the graph input

of iteration i(G

1

=G)and byn

i

itsorder, then

 CLIQUE_REMOVAL (G i )> Klog ` n i n i (3) Denote now by ^

G the subgraph of G indu ed by the union of the independent sets S olored

duringtheexe utionsofthe WHILE-loop,andbyn^itsorder. Then,bythepropertyofex avation

s hema (expression(2)) and byexpression (3):

 EXCAVATION  ^ G  6 lnn^ Klog ` ^ n ^ n = logn^ logeKlog ` ^ n ^ n > ^ n klog ` 1 ^ n (4)

(4)

LetG be the subgraphof Ginputof algorithm _COLOR (i.e., G=G[V nV(G)℄),n~ be itsorder

and ~

be its maximum degree. Observe that, following the quoted proposition above,

 ~ G  6 `~nloglogn~ logn~ (1) =)  ~ G  > logn~ `loglogn~ (5) Appli ation of _COLOR in ~

Gwill ompute aset ~

X of olors verifying (usingexpression (5))

 _COLOR  ~ G  = ~ X   ~ G  6 ~  logn~ `loglogn~ = ~ `loglogn~ log~n 6 ~ `loglog ~  log ~  (6)

Usingexpressions (4)and(6), thefollowingholds forthesetX of olors omputedbyalgorithm

COLOR: jXj = ^ X + ~ X 6 EXCAVATION  ^ G    ^ G  + _ COLOR  ~ G    ~ G  6 max n  EXCAVATION  ^ G  ; _ COLOR  ~ G o   ^ G  +  ~ G   (7) Obviously, both ( ^ G)and ( ~

G)aresmaller than (G);hen e, by expression(7)one gets

 COLOR = jXj (G) 6max ( 2^n klog ` 1 ^ n ; 2 ~ `loglog ~  log ~  ) 6max  2n klog ` 1 n ; 2`loglog log  :

Inall, thefollowing theorem hasbeen proved bythe abovedis ussion.

Theorem 1.  C 6max  O  n log ` 1 n  ;O  loglog log  :

Let `>5. Ifthe maximum in theorem 1 is realized by the term n=(log ` 1

n), then, theorem 1

improvestheratio of[4℄)bya fa tor(log 2

lognlog ` 4

n). Ontheotherhand,supposethatthe

maximum is realized by loglog=log. Then, theorem 1 largely improves the ratio of [6℄)

and onstitutesthe rst o()ratio for C.

Referen es

[1℄ L.AlfandariandV.T.Pas hos. Master-slavestrategy andpolynomialapproximation.

Com-put. Opti. Appl. Toappear.

[2℄ B. B. Boppana and M. M. Halldórsson. Approximating maximum independent sets by

ex- luding subgraphs. BIT,32(2):180196,1992.

[3℄ M.DemangeandV.T.Pas hos. Maximum-weight independentsetisaswell-approximated

as the unweighted one. Cahier duLAMSADE 163, LAMSADE, Université Paris-Dauphine,

1999.

[4℄ M. M. Halldórsson. A still better performan e guarantee for approximate graph oloring.

Inform. Pro ess. Lett.,45(1):1923, 1993.

[5℄ D. S. Johnson. Approximation algorithms for ombinatorial problems. J. Comput. System

S i., 9:256278,1974.

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