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;
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)
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
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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
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