Polynomial approximation and graph-coloring

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Laboratoire d'Analyse et Modélisation de Systèmes pour

l'Aide à la Décision

CNRS UMR 7024

CAHIER DU LAMSADE

187

Janvier 2002

Polynomial approximation and graph coloring

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Résumé i

Abstra t i

1 Introdu tion 1

I Graph- oloring in standard approximation 3 2 How an one always olor a graph with

∆(G)

olors? 4

2.1 Anon- onstru tive theorem andits ulterior onstru tive proof . . . 4

2.1.1 Coloring 3- onne tedgraphs . . . 4

2.1.2 Coloring onne ted graphs ontaining verti esof degreeless than

∆(G)

. 5 2.1.3 Theoverallalgorithm . . . 5

2.2 A

∆(G)/3

approximationratio for graph- oloring . . . 6

3 Towards ratios of

o(∆(G))

6 3.1 Asemidenite relaxationfor graph- oloring . . . 7

3.2 Therounding phase . . . 8

4 The greedy oloring-algorithm 9 4.1 Anex avations hemafor graph- oloring . . . 9

4.2 Instantiating INDEPENDENT_SETbythe greedy algorithm . . . 10

5 Improving the ratio for graph- oloring 11 5.1 Coloring

k

- olorable graphs . . . 11

5.2 Expanding k_COLOR to runfor all graphs . . . 11

5.3 Thewholeimprovement . . . 12

6 A better approximation ratiofor oloring 13 6.1 Improving GREEDY_Cin

k

6.2 Modifying algorithmk_COLORINGto runon all graphs . . . 14

6.3 Thewholeimprovement . . . 14

7 A still better approximation ratio for graph- oloring 15 7.1 Findinglarge independent sets ... . . 15

7.1.1 ...Byremoving liques. . . 16

7.1.2 ...In

k

7.2 Theoverall oloring-algorithm . . . 18

8 A further improvement of the performan e guarantee for the approximation of graph- oloring 19 9 Inapproximability results 21 9.1 Negative results via graph-theoreti gap te hniques . . . 21

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ber of unused olors 23 10A few words about the worst-value solution for graph- oloring 23 11A dierential-approximation algorithm based upon ane-equivalen e 24 12Frommat hing to 3 - independent sets 25 13Coloring graphs via set- overing 25 14Coloring, set- overing and semi-lo al optimization 26 15Unifying the abovealgorithms 28

15.1 Re overing algorithmD_COLOR4 . . . 29

16Using

∆

_COLOR to olor sparse graphs 30 17Negative results 30 18Final remarks 31 Referen es 33 A Proof of the results of se tion 2 36 A.1 Proofof lemma2 . . . 36

A.2 Proofof lemma3 . . . 36

A.3 Proofof theorem2 . . . 36

A.3.1

∆(G[A]) < 3

. . . 36

A.3.2

|A| < ∆(G) + 1

. . . 36

A.3.3

|A| = ∆(G) + 1

. . . 36

A.3.4

G[A]

is 3- onne ted . . . 36

A.3.5

G[A]

is bi onne ted . . . 36

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Résumé

Considérons un graphe

G = (V, E)

d'ordre

n

. Le problème de la oloration minimum onsisteàattribuerd'unnombreminimumde ouleursauxsommetsde

V

defaçon à eque deuxsommetsadja entsnesoientpas oloriésave lamême ouleur.Ceproblèmeestparmi lespremiers démontrés intrinsèquement di iles et, par onséquent, il est très improbable qu'ilpuisse êtrerésolu optimalementparunalgorithme polynomial. Dans et arti le,nous faisons un tour d'horizon des prin ipaux algorithmes d'approximation ( eux pour lesquels des rapports d'approximation théoriques ont été étudiés) pour le problème de oloration etdis utonsleurs rapports d'approximation etleur omplexité. Enn,nous proposons une améliorationdurapportd'approximationpour eproblème.

Mots- lé: graphe, oloration, omplexité, NP- omplet,algorithmed'approximation.

Polynomial approximation and graph- oloring

Abstra t

Consider a graph

G = (V, E)

oforder

n

. Inminimumgraph- oloringproblemwetryto olor

V

withasfew olorsaspossiblesothatnotwoadja entverti esre eivethesame olor. Thisproblemis amongtherstonesprovedtobeintra tableandhen e,itisveryunlikely that anoptimal polynomial-time algorithm ould everbedevisedfor it. Inthis paper,we surveythemain polynomial timeapproximationalgorithms(theones forwhi htheoreti al approximabilityboundshavebeenstudied)fortheminimumgraph- oloringandwedis uss their approximation performan e and their omplexity. Finally, we further improve the approximationratioforgraph- oloring.

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Consider a graph

G = (V, E)

of order

n

. In minimum graph- oloring problem (C), we wish to olor

V

with as few olors as possible so that no two adja ent verti es re eive the same olor. Thisproblem was shownto be NP-hard in Karp'soriginal paper ([40℄),and remains NP- omplete even restri ted to graphsof onstant (independent on

n

) hromati number at least 3 (formoreinformationsaboutthe omplexityofnumerousrestri tions,orgeneralizationsofC,the interestedreaderisalsorefereedto [35 ℄). The hromati number ofagraph,denotedby

χ(G)

,is the smallestnumber of olorsthat an feasibly olor itsverti es. Agraph

G

is alled

k

- olorable if its verti es an be legally olored by

k

olors, in other words if its hromati number is at most

k

; itwill be alled

k

- hromati if

k

is its hromati number.

Sin e adja ent verti es are forbidden to be olored with the same olor, a feasible solution of C an be seen as a partition of

V

into vertex-sets su h that, for ea h one of these sets, no two of its verti es aremutually adja ent. Su h sets areusually alled independent sets. So, the optimal solutionof Cisa minimum- ardinality partition intoindependentsets.

Another ombinatorialquantitydenedondis retestru turesthatwillbeusefulinthesequel is the one of set- overing. Given a family

S

of sets drawn from a ground set

C

(satisfying

∪

Si∈S

S

i

= C

),a set- overing isa family

S

′

_{⊆ S}

su h that

∪

Si∈S

′

S

i

= C

.

Both quantities, independent set andset- overing, give rise to two well-known NP-hard op-timization problems, namely the one of nding the maximum- ardinality independent set of a graph, denoted by

α(G)

, and the one of nding the minimum- ardinality set- overing of a set system. Inwhat follows wewill denote byIS the formerandbySCthe latter.

Given the omputationalhardness of C, if onewishes to devise fastalgorithms, then he/she hastodeveloppolynomialtimemethodsproviding,forallinstan es,feasiblesolutions,thevalues of whi hare as lose as possible to the valueof the optimal ones. Thisiswhat in the literature is ommonly alled polynomial time approximation algorithms (PTAA).

The goodness of a PTAA A is measured by the so- alled approximation ratio. Two types of ratios have been used until now: the one that, in some way, ompares the value

A(I)

of the approximate solution for

I

, provided by A, with the value

OPT(I)

of the optimal one, and the one omparing

A(I)

not only with

OPT(I)

but also with

WORST(I)

, the value of the worst feasible solution of

I

. Ea h one of these measures draws its proper working-framework into whi h every NP-hard problem an be analyzed and implies its proper approximation results. This is, as we shall see in the sequel, parti ularly true for C (in general, given an NP-hard problem, approximation results an be very dierent whenusing the one working-framework or the other). For reasonsofsimpli ity, wewill speakabout

ρ

-framework to denotetheformerand about

δ

-framework to denotethe latter. Informally:

•

in the

ρ

-framework, the omparison between

A(I)

and

OPT(I)

is usually performed by answering the following question: what is the ratio between

A(I)

and

OPT(I)

?; this question indu es the so- alled approximationratio;

•

in the

δ

-framework, the omparison between

A(I)

,

OPT(I)

and

WORST(I)

is performed by answering the question: what is the least

ǫ

for whi h A is to an extent of

(1 − ǫ)

like the worst algorithm, and to an extent of

ǫ

like the ideal one?; this question indu es the dierential-approximation ratio.

LetA be aPTAA for an NP-hardproblem

Π

, let

I

be the set ofinstan es of

Π

, let

A(I)

be the value of the solution provided by A on an instan e

I ∈ I

,

OPT(I)

be the value of the optimal solution for

I

and

WORST(I)

be the value of the worst solution of

I

. Moreover, letfor axed onstant

M

,

I

M

be the subset of

I

dened as

I

M

= {I ∈ I : OPT(I) > M }

, and let

σ(I)

be

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•

Approximation ratio

ρ

:

the approximation ratio

ρ

A

(I)

of the algorithm A on an instan e

I ∈ I

is dened as

ρ

A

(I) =

A(I)

OPT(I)

the approximation ratio

ρ

A

of A for

Π

isdened as

ρ

A

=

sup {r : ρ

A

(I) > r, ∀I ∈ I} Π

a maximizationproblem

inf {r : ρ

A

(I) < r, ∀I ∈ I} Π

a minimizationproblem the asymptoti approximationratio

ρ

∞

A

of A for

Π

isdened as

ρ

∞

_A

=

sup {r : ∃M, ρ

A

(I) > r, ∀I ∈ I

M

} Π

a maximization problem

inf {r : ∃M, ρ

A

(I) 6 r, ∀I ∈ I

M

} Π

a minimization problem the approximation ratio

ρ

Π

for

Π

isdened as

ρ

Π

=

max {ρ

A

:

Aa PTAAfor

Π} Π

a maximization problem

min {ρ

A

:

Aa PTAAfor

Π} Π

a minimizationproblem

•

Dierential-approximation ratio

δ

:

the dierential-approximation ratio

δ

A

(I)

of the algorithm A on an instan e

I ∈ I

is dened as

δ

A

(I) =

WORST(I) − A(I)

WORST(I) − OPT(I)

the dierential-approximation ratio

δ

A

of A for

Π

isdened as

δ

A

= sup {r : δ

A

(I) > r, I ∈ I}

the asymptoti dierential-approximation ratio

δ

∞

A

of A for

Π

is dened as

δ

∞

_A

= lim

k→∞

sup

I

σ(I)>k

WORST(I) − A(I)

WORST(I) − OPT(I)

the dierential approximation ratio

δ

Π

for

Π

isdened as

δ

Π

= max {ρ

A

:

Aa PTAA for

Π}

Let us note that the best expe ted ase for the behavior of an approximation algorithm is that it ould guarantee an approximation ratio tending to 1. More formally, the ideal would be that we had a PTAA A re eiving as inputs an instan e

I

of an NP-hard problem and a xed onstant

ǫ

and guaranteeing approximation ratio (resp., dierential-approximation ratio)

1 + ǫ

or

1 − ǫ

depending on if the problem at hand is a minimization or a maximization one (resp.,

1 − ǫ

, for the

δ

-framework), for every

ǫ > 0

. In this ase, we have, in fa t, a se-quen e

(A

ǫ

)

ǫ>0

of approximation algorithms having the desired approximation properties just des ribed. Su hasequen eis alled apolynomial timeapproximation s hema (PTAS)or,in the

δ

-framework, dierential PTAS (DPTAS). Moreover, we an further lassifysu h s hemata fol-lowingthetime- omplexityofalgorithm

A

ǫ

. WesospeakaboutPTAS(DPTAS)ifits omplexity is

O(f (|I|)

Θ(1/ǫ)

₎

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is

O(g(1/ǫ)O(|I|

k

₎₎

, where

|I|

isthe size of

I

,

f

and

g

polynomials not depending on

|I|

and

k

a xed onstant (not depending neither on

|I|

, nor on

ǫ

). A PTAS, or FPTAS (resp., DPTAS, or FDPTAS) is alled asymptoti if, for every instan e

I

, it guarantees approximation ratio

1 + ǫ + c/OPT(I)

or

1 − ǫ − c/OPT(I)

,depending onif the problem athand is aminimization or amaximizationone (resp.,

1 − ǫ − c/(WORST(I) − OPT(I))

, forthe

δ

-framework),for some xed onstant

c

and for every

ǫ > 0

.

The approximation of C by e ient algorithms is a entral problem in omplexity theory. In

ρ

-framework, people knewquite early that a polynomial time approximation s hema annot exist for it, sin e alower bound 2 was proved for the ratio of every PTAA supposed solving C; this result, due to Gareyand Johnson, has been published in 1976. But even if the resear hers onje turedthatapproximationratiosgreater than2wereequally unlikely,su haresultwasnot formallyprodu edupto the early90'swhentheimpossibilityofapproximatingCbya onstant ratio approximation algorithm is proved byLund and Yannakakis. This result, motivates from then on, many resear hers for sear hing either for approximation algorithms with improved approximation ratios, or to strengthen the existing inapproximability results. On the other hand, in

δ

-framework, C is better-approximable than in the

ρ

-one, sin e onstant dierential-approximationratio PTAAsexist sin e1994.

This paper, even if it surveys both positive and negative approximation results 1

about C in both the approximation frameworks, is rather oriented towards the positive ones. For this reason,positiveresultsarepresented and ommentedindetailandtheunderlyingalgorithmsare spe iedinakindofpseudo-PASCAL,whiletheinapproximabilityonesaresimplymentioned. Beforepresentingpositiveresults,wegivetheintuition(orkey-idea)behindthem. Theobje tive of this survey is double: to present already known results and to introdu e some new ones. In order to be as short and easy to read as possible, only new results are proved, while already existingresults arestatedwithoutproofs. Also,inse tion3a oloringmethodisdis ussedusing a randomized algorithm that has been derandomized later by rather ompli ated te hniques. Sin e both random algorithm itself and its derandomization are long, and on the other hand, paper does not dealwith probabilisti methods,only the underlying idea is dis ussed,while we omitto spe ify the overallalgorithm.

Let us onsider asimple undire ted onne ted graph

G = (V, E)

of order

n

. Sometimes,we will denote by

|G|

the order of

G

and by

V (G)

its vertex-set. Given a set

V

′

of verti es of a graph

G

,wewilldenoteby

G[V

′

_{] = (V}

′

_{, E(G[V}

′

_]))

the partialsubgraphof

G

indu edby

V

′

. For everyvertex

v ∈ V

,wedenoteby

Γ

G

(v)

thesetofneighborsof

v

(neighborhoodof

v

)andby

d

◦

_(v)

the quantity

|Γ

G

(v)|

, this quantity is lassi ally alled the degree of

v

;

∆(G) = max

v∈V

{d

◦

_(v)}

is the maximum degree of

G

. The notion of neighborhood an be extended to apply to a set

V

′

⊆ V

;wedenoteby

Γ

G

(V

′

₎

thesetofneighborsoftheverti esof

V

′

,i.e.,

Γ

G

(V

′

_{) = ∪}

v∈V

′

Γ

_G

(v)

. Whenevernoambiguity ano urweusenotation

Γ(v)

insteadof

Γ

G

(v)

. Givenverti es

v

and

u

, we denoteby

d(v, u)

their distan e,i.e., thelength (number ofedges) oftheshortest elementary path linking

v

and

u

(sin e

G

is undire ted

d(v, u) = d(u, v)

). By

K

t

, we denote a omplete graphon

t

verti es. Alllogarithmsofthepaperaretothebase2unlessotherwisenoted. Finally, the following well-knownexpression ([10 ℄)links

χ(G)

and

α(G)

and willbe usedlater:

α(G)χ(G) > |G|

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1

For anNP-hard minimization (resp.,maximization) problem

Π

, an approximationresultis alledpositive if, givenan approximationalgorithmA, it provides an upper(resp., lower) bound onthe approximationratio guaranteedby Awhenusedtosolve

Π

;itis allednegative orinapproximability resultwhenitprovidesalower (resp.,upper)boundfortheratioofanalgorithmorofawhole lassofapproximationalgorithmsfor

Π

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Graph- oloring in standard approximation 2 How an one always olor a graph with

∆(G)

olors?

We rst re all some standard graph-theoreti on epts used in what follows in this se tion (for more details one an be referred to [10 , 15℄). Consider a onne ted graph

G = (V, E)

. A subset

A ⊆ V

is alled arti ulation set if

G[V \ A]

is not onne ted; an arti ulation set of size 1 will be alled arti ulation point. Agraph

G

is

k

- onne ted i

|G| > k + 1

andi itdoesnot ontain arti ulation set of size less than

k

. As a onsequen e,

G

is bi onne ted i

|G| > 3

and it does not admitarti ulation points(i.e., for everypair

(x, y) ∈ V × V

, there exist twovertex-disjoint elementary paths between

x

and

y

). A blo in

G

is a set

A ⊆ V

su h that

G[A]

is onne ted, without arti ulation points, and maximal (i.e., for any

v ∈ V \ A

,

G[A ∪ {v}]

is either not onne ted, orit has at least an arti ulationpoint). Thenif

|A| > 2

,

G[A]

is bi onne ted, while, if

|A| = 2

, it is an isthmus, i.e., an edge whose removal in reases the number of the onne ted omponentsof thegraph. Obviously, the setofblo sof

G

overs

V

. A blo

B

is alled extremal ifit ontains avertex

z

su hthatforanyotherblo

B

′

,either

B

and

B

′

arevertex-disjoint, or

z

istheir only ommonvertex.

2.1 A non- onstru tive theorem and its ulterior onstru tive proof

Theorem 1 below (known also as Brook's theorem and non- onstru tively proved in [17 ℄) was onstru tively proved by Lovász in [42 ℄ in a rather ondensed way. We give in this se tion a personal interpretationof this result. The proofs aregiven in the appendix.

Theorem 1. ([17 ℄)If

G

is onne ted with

∆(G) > 3

and ifit ontains no subgraph

K

∆(G)+1

, then it is

∆(G)

- olorable.

Consider a graph

G

, and order its verti es, say

x

1 , x

2 , . . . , x

n

. One an olor them one-by-one in the following way: olor

x

1

with1; then, olor

x

2

with1, if

x

1 x

2 ∈ E

/

, with2 otherwise and ontinue oloringea hvertexwiththesmallest olorit anbeassignedatthatstage. Denotethis algorithm by LEGAL_C and suppose it is alled withinputs

G

and

x

1 , x

2 , . . . , x

n

. The following easy lemmaholds for LEGAL_C.

Lemma 1. LEGAL_C olors the verti es of any graph

G

with at most

∆(G) + 1

olors, in polynomial time.

Proof. Sin e

∆(G)

isthe maximumgraph-degree,

∆(G) + 1

olorsaresu ient tolegally olor anyvertexof

G

andall its neighbors.

The key-ideaofthe algorithmof[42℄ isto onstru tgood orderingsofthe verti es of

G

, so thatLEGAL_C olorsthem withnomore

∆(G)

olors.

2.1.1 Coloring 3- onne ted graphs

Consideragraph

G = (V, E)

su hthat: itis onne ted,

∆(G) > 3

,itdoesnot ontaina

K

∆(G)+1

anditadmitsatleastapairofverti es,the removalofwhi hdoesnotdis onne t

G

. Notethata 3- onne tedgraph verifyingthe onditions of theorem1 an besu h agraph. Also onsider the followingalgorithm3C_COLOR.We onsiderin whatfollowstwotypesofexe utionfor3C_COLOR. The rst one is parameterized by a graph

G

, when it is exe uted starting from line (1). The se ond one is parameterized by a graph

G

and two verti es

v

and

u

, when we suppose that lines(1)and(2)areomittedanditsexe utionstartsfromline(3). Thisisthe aseofexe ution of 3C_COLORin the bodyofalgorithm

∆

_COLORin paragraph 2.1.3.

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(1) let

{1, . . . , ∆(G)}

be the set of olors to be used; (2) hoose verti es v and u su h that

d(v, u) = 2

and

G

′

_{← G[V \ {v, u}]}

remains onne ted; (3) let w be a vertex adja ent to both v and u; (4) arrange the verti es of

G

′

in a sequen e

(w = x

1 , x

2 , . . . , x

n−2

)

su h that ea h vertex

x

i

, i > 2

is adja ent to a vertex

x

j

, j 6 i

;

*this an be done breadth-first-sear h (BFS,[1℄) starting from w* (5) olor v and u with the olor 1;

(6) OUTPUT

{1} ∪ LEGAL

_

C(G

′

_{, x}

n−2

, x

n−3

, . . . , x

1 )

; END. *3C_COLOR*

Lemma 2. Let

G

be a 3- onne ted graph not ontaininga

K

∆(G)+1

. Then 3C_COLOR omputes a legal

∆(G)

- oloring for

G

in polynomial time.

2.1.2 Coloring onne ted graphs ontaining verti es of degree less than

∆(G)

We now give another algorithm assigning at most

∆(G)

olors to the verti es of a onne ted graph withat least one vertexof degree

< ∆(G)

.

BEGIN *1V_COLOR*

(1) let

{1, . . . , ∆(G)}

be the set of olors to be used; (2) hoose a vertex v su h that

d

◦

_{(v) < ∆(G)}

;

(3) arrange the verti es of

G

in a sequen e

(v = x

1 , x

2 , . . . , x

n

)

su h that ea h vertex

x

i

, i > 2

is adja ent to a vertex

x

j

, j 6 i

;

*this an be done by BFS starting from v* (4) OUTPUT LEGAL_C(

G, x

n

, x

n−1

, . . . , x

1

);

END. *1V_COLOR*

Lemma 3. Let

G

be a bi onne ted graph, not ontaining a

K

∆(G)+1

, with at least one vertex of degree

< ∆(G)

. Then 3C_COLOR omputes a legal

∆(G)

- oloring for

G

inpolynomial time. 2.1.3 The overall algorithm

Given a oloring,wewill all olor-permutation theex hangeofa olor

i

to

j

andof

j

to

i

onall verti es olored withthese olors (when

j

has not been used,then olor-permutation between

i

and

j

be omesinusing

j

instead of

i

). Wearewell-preparednowto spe ify analgorithmlegally oloringthe verti esof anygraph

G

byusingat most

∆(G)

olors.

BEGIN *

∆

_COLOR*

(1) ompute all the blo s of G; (2) FORevery blo A of G DO (3) IF

∆(G[A]) < 3

THEN

(4) order arbitrarily the verti es of A, say

x

1 , x

2 , x

3

; (5) LEGAL_C(

G[A], x

1 , x

2 , x

3

);

(6) GOTO (2); (7) FI

(8) CASE |A| DO

(9)

|A| < ∆(G) + 1

: olor A with olors 1,...,

∆(G)

and GOTO (2); (10)

|A| = ∆(G) + 1

: 1V_COLOR(G[A℄) and GOTO (2);

(10)

(13) IFG[A℄ is bi onne ted THEN

(14) hoose arbitrarily a vertex x with

d

◦

_{(x) > 3}

; (15) IF

G[A \ {x}]

is bi onne ted THEN

(16) let

y

∈ A

be su h that d(x,y) = 2; (17) 3C_COLOR(G[A℄,x,y) and GOTO (2); (18) FI

(19) IF

G[A \ {x}]

is not bi onne ted THEN (20) let

B

1

and

B

2

two extremal blo s of G; (21) let

z

1 ∈ B

1

and

z

2 ∈ B

2

adja ent to x; (22) 3C_COLOR(G[A℄,

z

1 , z

2

);

(23) FI (24) FI (25) OD

(26) OD

(27) order the blo s in su h a way that ea h blo has at most a vertex in ommon with the blo s pre eding it in the ordering;

(29) OUTPUT the union of the olors used by permuting them if ne essary; END. *

∆

_COLOR*

Theorem 2. Let

G

bea onne tedgraphwith

∆(G) > 3

andnot ontainingsubgraph

K

∆(G)+1

. Then, algorithm

∆

_COLORlegally olorsthe verti esof

G

with atleast

∆(G)

olors,inpolynomial time.

2.2 A

∆(G)/3

approximation ratio for graph- oloring

Suppose now that 1_COLOR is an algorithmre eiving an input-graph

G

and de iding if

G

is an independent setand, if yes, oloring its verti es with one olor. Obviously, this an be done in polynomial time. Also,let2_COLOR be anotheralgorithmde iding if

G

isbipartite. This an be done by starting with two olors and by oloring a vertex with one of them and its neighbors withtheotherone. Ifattheend alltheverti esof

G

arelegally olored,then itisbipartite and, moreover, a2- oloring is dis overed. Finally, onsiderthe following PTAA for C.

BEGIN *APPROX_COLOR* 1_COLOR(G); 2_COLOR(G);

∆

_COLOR(G);

OUTPUT the smallest of the legal olorings produ ed; END. *APPROX_COLOR*

Theorem 3.

ρ

APPROX

_

COLOR

6 ∆(G)/3

. 3 Towards ratios of

o(∆(G))

Awell-knownmethodfor solvingnumerousNP-hardproblemsapproximately usesthe following s hema, denoted bySCHEMA in the sequel:

[Phase1℄ optimallysolve the linear-programmingrelaxationof theproblem (this an bedone in polynomial time([5 ℄));

[Phase2℄ round this solution to a feasible (but approximate) solution for the original (integer programming)problem.

(11)

the so- alled semidenite programming relaxations ([3 , 30 ℄). As opposed to the formers, the latters oer rounding te hniques leading to feasible integer solutions thatare guaranteed to be withinaspe iedfra tion tothe optimalones. Semideniteprogrammingin omputing approx-imatesolutionsfor ombinatorialproblems isoriginallyusedbyGoemans andWilliamson([29 ℄) formaximum utandmaximum2-satisabilityproblems. Basedupontheworkof[29℄,Kargeret al.([39 ℄)devisearandomizedapproximationalgorithmfor

C

a hievingapproximationratio

o(∆)

. Theirmethod an be shortlydes ribed as follows.

3.1 A semidenite relaxationfor graph- oloring

Consider a

k

- olorable graph. Instead of assigning integers (or olors) to the verti es of the graph, a unitve tor

~v

i

∈ R

n

isassigned to anyvertex

v

i

∈ V

,

i = 1, . . . , n

. In order to apture thefa tthatadja ent verti esareassignedwithdierent olors,theve tors assignedto adja ent verti es have to be dierent in some spe i (but natural) way. This requirement leads to the ve tor

k

- oloring.

Given a graph

G = (V, E)

of order

n

and a real

k > 1

, a ve tor

k

- oloring of

G

is an assignmentofunitve tors

~v

i

∈ R

n

toanyvertex

v

i

∈ V

,su hthatforanytwoadja entverti es

v

i

and

v

j

the dot produ t of

~v

i

and

~v

j

satises

hv

i

, v

j

i 6 −1/(k − 1)

(in other words, the angle between the ve tors orresponding toadja ent verti esmustbe su iently large).

The so-dened ve tor

k

- oloring is seen in [39 ℄ as a kind of relaxation for C that plays the role that a hypotheti al fra tional

k

- oloring would play if one used a onventional linear-programmingrelaxation for the problem. Thisis justied bythe following lemma.

Lemma 4. ([39℄)

•

Every

k

- olorable graph has a ve tor

k

- oloring.

•

Moreover, for all positiveintegers

k

and

n

with

k 6 n + 1

,there exist

k

unit ve torsof

R

n

su h that the dot produ tof any distin t pair is

−1/(k − 1)

.

Remarkthatthe

k

ve tors ofthe se onditemoflemma4fulll thespe i ation oftheve tor

k

- oloring(theemphasized propositionjustabove). Furthermore,one an immediatelybije tively mapthese ve tors to

k

distin t olorsin order to produ e a

k

- oloring of agraph

G

oforder

n

.

Following [Phase1℄ of SCHEMA, one has now to determine a ve tor

k

- oloring of

G

. This an be done using the following auxiliary problem. Given a graph

G = (V, E)

of order

n

, a matrix

k

- oloring of

G

is an

n × n

symmetri positive semidenite matrix

M

with

m

ii

= 1

and

m

ij

6 −1/(k − 1)

for

v

i

v

j

∈ E

. The key-point of the relationship between matrix and ve tor olorings ofa graph isgiven by the following lemma.

Lemma 5. ([39℄)

•

A graph has a ve tor

k

- oloring ifand only if it has a matrix

k

- oloring.

•

If a graph has a matrix

k

- oloring, then a ve tor

(k + ǫ)

- oloring an be onstru ted from this matrix oloring in time polynomial in

n

andin

log(ǫ

−1

₎

.

•

Conversely, if a graph

G

has a ve tor

k

- oloring, then a matrix

(k + ǫ)

- oloring of

G

an be onstru ted in time polynomial in

n

and in

log(ǫ

−1

₎

.

We have supposedat the beginning of the urrent paragraph that

G

is

k

- olorable. Therefore, by the se ond item of lemma 4, there exists a ve tor

k

- oloring for

G

and, by the rst item

(12)

of lemma 5, there exists a matrix

k

- oloring of

G

. This matrix oloring an be onstru ted by solving the following semideniteoptimization problem:

SDP =











min σ

M = {m

ij

}

positive semidenite

m

ij

6 σ

for

v

i

v

j

∈ E

m

ij

= m

ji

m

ii

= 1

Remarkthat,bydenitionofthematrix oloringgiven justabovelemma5,the solutionofSDP spe ies the entries of a matrix

k

- oloring

M

. Sin e

G

has a ve tor (and, by the rst item of lemma 5, a matrix) oloring, there exists a solution to SDP with

σ

∗

_{= −1/(k − 1)}

. If one uses a linear-programmingmethod, she/he is ableto determine a feasible solution ofSDP with

σ 6 −1(k + ǫ − 1)

, for some arefully hosen

ǫ ∈ R

. This solution, by the denition of the matrix oloring, spe ies a matrix

(k + ǫ)

- oloring of

G

. This, by the se ond item of lemma 5, an produ e a ve tor

(k + 2ǫ)

- oloring of

G

. In order to simplify presentation, the error

ǫ

an be ignored sin e it an be made very small as to be irrelevant to the analysis in [39 ℄. So, the following propositionholds.

Proposition 1. ([39 ℄)Givena

k

- olorable graph

G

ave tor

k

- oloringof

G

anbe determined in polynomial time.

3.2 The rounding phase

On e[Phase1℄a omplished, onehastopro ess[Phase2℄thatimpliestheroundingoftherelaxed solutiontoafeasibleintegerone. Theoriginalroundingte hniqueproposed, alledsemi- oloring in [39℄, is randomized and produ es an assignment of olors with relatively few identi ally olored adja ent verti es. This semi- oloring is then transformed into a legal graph- oloring. Moreformally, a

k

-semi- oloring ofa graph

G

isan assignment of

k

olorsto atleast halfof its verti es su h that no two adja ent verti es re eive the same olor. Dealing withsemi- olorings, the following holds.

Lemma 6. ([39 ℄) If an algorithm SEMICOLOR an

k

p

-semi- olor any subgraph of order

p

of a graph

G

in randomized polynomial time (where

k

p

in reases with

p

), then SEMICOLOR an polynomially olor the wholegraph

G

with

O(k

n

log n)

olors.

In other words, a semi- oloring of

G

an be transformed into a legal oloring by losing only a logarithmi fa tor with respe t to the olors used for the semi- oloring. The randomized algorithm transforming ve tor olorings into semi- olorings is not given here. The interested reader an be referred in [39 ℄. Inany ase, the following an be shown.

Proposition 2 . ([39℄) For every integer fun tion

k = k(n)

, a ve tor

k

- olorable graph

G

an be semi- olored with atmost

O(∆(G)

1−(2/k)

_{plog ∆(G))}

olors in randomized polynomial time. Combination of lemma 6 withthe rst item of lemma 4 and with propositions 1 and 2 implies the following.

Proposition 3. ([39 ℄) Any

k

- olorable graph

G

of order

n

an be olored in randomized polynomial time with atmost

O(∆(G)

1−(2/k)

_{plog ∆(G) log n)}

olors.

Two last points remain to be settled: (i) an the randomized method proposed be transformed into a deterministi one, and (ii) how one an make this method to run for any graph (re all thatuntil now the graph issupposed

k

- olorable

2 )? 2

(13)

nomial derandomization ofthe algorithms in [39 ℄ a hieving the same approximation guarantees thanthe original(random)ones. Ontheotherhand,forpoint(ii),the followingpro edure,that will be usedlater also, an be used. Sin e the omplexity of the overall

k

- oloring algorithmis polynomial in

k

, one an run it for all

k ∈ {2, . . . n}

in this order (multiplying so its worst- ase omplexityby

n

), and stop it for the rst

k

for whi h a feasible solution is produ ed. Asit is pointed out in [51 ℄, this thought pro ess an be implemented by divide-and- onquer in su h a waythatthe whole omplexityofthederived algorithmwillbemultiplied notby

k

but by

log k

. So, in the light of the above settlements, proposition 3 holds for any graph and the algorithm laimed runsin deterministi polynomial time.

Finally,runbothalgorithm

∆

_COLORandthe s hemadealtin the urrent se tion (parameter-ized by the algorithms of [39 ℄, derandomized bythe method of [44 ℄, and extended so as it runs for any graph bythe remark settling point (ii)above),and take nallythe bestof the solutions produ ed. Then, the following on ludingtheorem holds.

Theorem 4. C an be approximately solved in polynomial time and in any graph

G

within approximation ratio

ρ

C

6 min

(

∆(G)

χ(G)

, O

∆(G)

1−

2 k

plog ∆(G) log n

χ(G)

!)

.

4 The greedy oloring-algorithm

4.1 An ex avation s hema for graph- oloring

When one triesto approximately solve C, the rst algorithm omingin mind isthe greedy one des ribedbythefollowingex avations hema exe utedwithparameters

G

andanIS-algorithm INDEPENDENT_SET.

BEGIN *EXCAVATION* REPEAT

S

← INDEPENDENT

_

SET(G)

;

olor the verti es of S with the same not already used olor; remove S from G;

UNTIL G be omes empty;

OUTPUT X the set of used olors; END. *EXCAVATION*

AlgorithmEXCAVATIONhastwointerestingproperties,expressedbyitems1and2ofproposition4 below. Item1isproved in[31 ℄,but hasbeenimpli itlyusedin [11 ,37 ,51℄andexpli itlyin[31 ℄. Item 2 is proved in [36 ℄ for the ase where

S

is a maximumindependent set and is generalized in [2 ℄ forthe ase where

S

is anyindependent set. Here itismainly usedin se tion 8.

Proposition 4.

1. Anyiterativeappli ation ofalgorithm INDEPENDENT_SET that omputes an independent set of size

ι

k

(G) = Θ(n

1/t

₎

,

t > 1

,in a

k

- olorable graph

G

of order

n

,produ es a oloring

X

verifying

|X| 6 2n/ι

k

(G)

.

2.

ρ

EXCAVATION

6 ln n/ρ

INDEPENDENT

_

SET

(14)

An interesting polynomial instantiation of the s hema des ribed above, where the independent setisfoundviaagreedyalgorithm,hasbeen studiedbyJohnsonin [37 ℄. Wegiveinwhatfollows an outlinefor both the greedy IS-algorithm andthe greedy C-algorithmthat ensues.

BEGIN *GREEDY_IS*

S

← ∅

; REPEAT

v

← argmin

v

i∈V

{d

◦

(v

i

)}

;

S

← S ∪ {v}

;

V

← V \ ({v} ∪ Γ(v))

;

G

← G[V]

; UNTIL

V

= ∅

; OUTPUT S; END. *GREEDY_IS* BEGIN *GREEDY_C* REPEAT

S

← GREEDY

_

IS(G)

;

olor the verti es of S with the same non already used olor;

V

← V \ S

;

G

← G[V]

; UNTIL

V

= ∅

;

OUTPUT the set

X

J

of used olors; END. *GREEDY_C*

The omplexity of algorithm GREEDY_IS is

O(|E|)

and will be alled at most

n

times within the REPEATloop of algorithmGREEDY_C,ea h su h all stri tly de reasing

V

. Hen e, the overall worst- ase omplexityof GREEDY_Cis

O(n|E|)

.

Let us denote by

V

(ℓ)

the vertex-set of the urrent, surviving, graph at the beginning of the

ℓ

thiterationofGREEDY_C.Thekey-pointforthestudyofitsapproximationperforman eisthe following. Ifagraphis

k

- olorable,then,atea hiteration

ℓ

of GREEDY_C,

d

◦

_(v

j

) 6 |V | − ⌈|V |/k⌉

, where

v

j

is the minimum- urrent-degree vertex hosen by GREEDY_IS. Therefore, the verti es oloredduring

ℓ

thiteration(i.e., the onstru tedmaximalindependent setduringthisiteration) will be ofsize atleast

log

k

|V |

. Thisleads tothe following lemma,provedbyWigderson in [51 ℄. Lemma 7. ([51℄) Algorithm GREEDY_C olors any

k

- olorable graph

G

with

|X

J

| 6 3n/ log

k

n

olors.

Bylemma7,theapproximationratioofalgorithmGREEDY_Cina

χ(G)

- hromati graphbe omes:

ρ

GREEDY

_

C

6 |X

J

|

χ(G)

6

1 χ(G)

3n log χ(G)

log n

6 n

log n

(2)

leading so to the following on ludingtheorem. Theorem 5. ([37℄)

ρ

GREEDY

_

C

6 n/log n

.

(15)

TheimprovementoftheapproximationratioofChasremainedopenfor9yearsuntil1982when Wigdersonhasshownin[51 ℄howtoobtainbetterperforman eguarantees. Hismethod,outlined in what follows, isbaseduponthe following observations:

1. the neighborhood of any vertexin a

k

- olorable graph is

(k − 1)

- olorable; 2. 2- oloring ispolynomial (see se tion 2).

These observations (the se ond being a termination ondition) lead, as we will see, to a ni e re ursive approximationstrategy for C.

5.1 Coloring

k

- olorable graphs

We rst present the following pro edure, alled withparameters

k

and

G

and

i

(where

i

means that

G

will be olored with olors

i, i + 1, . . .

), that olors a

k

- olorable graph with relatively few olors. Set,for

k = 2, 3, . . .

,

f

k

(n) = n

{1−(1/(k−1))}

. BEGIN *k_COLOR*

(1) CASEk DO

(2) k = 2: OUTPUT 2_COLOR(G);

*verti es of G will be olored with olors i and i+1* (3)

k > log|G|

: olor V(G) with a distin t olor per vertex;

*verti es of G are assigned olors i, i+1,...,i+|G|-1* (4)

k 6 log|G|

: WHILE

∆(G) > ⌈f

k

(|G|)⌉

DO

(5) let v be su h that

d

◦

_{(v) = ∆(G)}

;

(6)

H

← G[Γ

G

(v)]

;

(7)

{i, i + 1, . . . , i + j − 1} ← k

_

COLOR(k − 1, H, i)

; (8) olor v with i+j;

(9)

i

← i + j

; (10)

G

← G[V \ (Γ

G

(v) ∪ {v})]

; (11) OD (12)

{i, i + 1, . . . , } ← ∆

_

COLOR(G)

; (13) OUTPUT

X

k

_

C

the set of used olors; (14) OD

END. *k_COLOR*

In the initial graph, algorithm is alled as k_COLOR(k,G,1). Line (12) of k_COLOR will be exe uted on graphs with

∆(G) < ⌈f

k

(|G|)⌉

, using so less than

⌈f

k

(|G|)⌉

unused olors. The algorithm alled in line (2) is the one de iding if a graph is bipartite, and if yes, omputing a 2- oloring of itsverti es.

Lemma 8. ([51 ℄) k_COLOR assigns, in

O(k(n + |E|))

, the verti es of any

k

- olorable graph with at most

2k⌈f

k

(n)⌉ = 2k⌈n

(1−(1/(k−1)))

_⌉

olors. 5.2 Expanding k_COLOR to run for all graphs

Aswehave already mentioned at the end of paragraph 3.2, one an expand algorithmk_COLOR (destinated to olor

k

- olorable graphs) to run for any graph. Re all that she/he only has to run itfor all

k ∈ {2, . . . n}

in this order, andstop itfor the rst

k

for whi h pro edure k_COLOR produ esa feasiblesolution. This an beimplemented as follows.

(16)

(1) FOR

p

← 1

TO

⌈logn⌉

DO (2)

X

← k

_

COLOR(2

p

_{, G, 1)}

;

(3) IF X is feasible THEN

p

0 ← p

and GOTO (5) FI; (4) OD

(5) applybinary sear h in

{2

p0−1

_{+ 1, . . . 2}

p0

_}

to find the least

k

← k

0

for whi h k_COLOR produ es a legal oloring;

(6) OUTPUT

X

E

← k

_

COLOR(k

0 , G, 1)

; END. *Ek_COLOR*

Sin e, givena graph

G

:

(i) theexe ution ofline(2)ofalgorithmEk_COLORprodu esalegal oloringforall

k > χ(G)

, (ii)

G

is always

χ(G)

- olorable,and

(iii)

k

0

isthe smallest

k

for whi h the exe ution ofline (5) willprodu e a feasible oloring, the following lemmaholds.

Lemma 9. ([51℄)

k

0

6 χ(G)

.

Combining lemmas 8 and 9, we get the following on luding theorem for the approximation performan e of algorithmEk_COLOR.

Theorem 6. ([51 ℄) Ek_COLOR olors, in

O((n + |E|)χ(G) log χ(G))

, the verti es of

G

with at most

2χ(G)⌈n

(1−(1/(χ(G)−1)))

_⌉

olors. 5.3 The whole improvement Bytheorem6,

ρ

Ek

_

COLOR

6 2⌈n

(1−(1/(χ(G)−1)))

_⌉

andfun tion

f (x) = 2⌈n

(1−(1/(x−1)))

_⌉

isin reasing in

x

. On the other hand, by lemma 7and expression(2),

ρ

GREEDY

_

C

6 3n log χ(G)/(χ(G) log n)

and fun tion

g(x) = 3n log x/(x log n)

is de reasing in

x

. Let us ombine the two algorithms to produ e the following nalalgorithm.

BEGIN *W_COLOR*

X

E

← Ek

_

COLOR(G)

;

X

J

← GREEDY

_

C(G)

;

OUTPUT

X

W

= argmin{|X

E

|, |X

J

|}

; END. *W_COLOR*

Of oursealgorithmW_COLOR omposedbypolynomial omponent-algorithmsisalsopolynomial. A little algebra shows that the interse tion point of the urves

f (x)

and

g(x)

is in the neighborhood of

x = ⌈n log log n/2 log n⌉

. Algorithm Ek_COLOR is superior to GREEDY_C for

χ(G) 6 log n/ log log n

while, for

χ(G) > log n/ log log n

GREEDY_C is more performant than Ek_COLOR.For

χ(G) = log n/ log log n

, both

ρ

Ek

_

COLOR

and

ρ

GREEDY

_

C

areof

O(n log

2 _{log n/log}

2 _n)

and the following theorem on ludesthe se tion. Theorem 7. ([51℄)

ρ

W

_

COLOR

6 3n log

2 _{log n/ log}

2 _n

(17)

Seven years later, Bergerand Rompel, using thought pro esses quite lose to the onesadopted in [51 ℄ (i.e., oloring rst

k

- olorable graphs, extending the result to work for every graph, the hromati number of whi h belongs to a ertain interval of values, and next omposing the algorithm produ ed with another well-working in graphs with hromati -number values out of the interval onsidered), perform, in [11 ℄, a further notable improvement of the approximation ratio ofC.

The key-observation in [11 ℄ is a kind of renement of the respe tive observation of [37 ℄. Re all that, as we have seen in se tion 4, Johnson observed that if

G

is

k

- olorable, then there exists an independent set of size at least

⌈|V |/k⌉

and, onsequently, any node

v

in this set has

d

◦

_{(v) 6 |V | − ⌈|V |/k⌉}

; this observation enabled him, using algorithm GREEDY_IS, to nd an independent set

S

of

O(log

k

|V |)

small-degree verti es to whi h gave the same olor. The renement lying at the heart of the method proposed in [11 ℄ is that anysubset

S

′

of

S

veries

|Γ(S

′

)| 6 |V | − ⌈|V |/k⌉

. Thisallows them to somewhatmodify algorithmGREEDY_IS to hoose at ea hstepa set(insteadof one)of

O(log

k

|V |)

small-degreeverti esthatareindependent and havesmall neighborhood. Theyareso abletolegally olor

O[(log

k

|V |)

2 ]

verti eswiththe same olor.

LetusnowoutlinehowBergerandRompelprodu e feasible oloringsfor

k

- olorable graphs. We onsider, without lossof generality, thatthe olors aredrawnfromthe set

{1, 2, . . .}

. 6.1 Improving GREEDY_C in

k

- olorable graphs

Considerthefollowingalgorithmk_COLORINGparameterizedbyaninteger

k

,axed

α > 0

anda graph

G

. Therst onditionin line(10)re eivesTRUEif the set

S

isanindependent set, FALSE otherwise.

BEGIN *k_COLORING* (1) IF

k

> min{n

1/2

_{, n}

α

_}

THEN OUTPUT GREEDY_C(G) FI (2)

m

← ⌊αlog

k

n⌋

; (3)

c

← 1

; (4)

UNCOLORED

← V

; (5) WHILE

|UNCOLORED| > km

DO (6)

U

← UNCOLORED

; (7) WHILE

|U| > km

DO (8)

H

← G[U]

;

(9) FOR

S

⊆ H

su h that |S| = m DO

(10) IF S AND

|Γ

H

(S)| 6 |U| − (|U|/k)

THEN GOTO (13) FI (11) OD

(12) OUTPUT G not k- olorable; (13) olor S with olor ;

(14)

U

← U \ [S ∪ Γ

H

(S)]

;

(15)

UNCOLORED

← UNCOLORED \ S

; (16) OD

(17)

c

← c + 1

; (18) OD

(19) olorthe UNCOLORED-verti es with olors , ..., +|UNCOLORED|-1; *assigns an unused olor per UNCOLORED-vertex*

(18)

partition the verti es of U into

ℓ = ⌊|U|/km⌋

sets

B

i

,

i

= 1, . . . , ℓ

, su h that

B

j

,

j

= 1, . . . , ℓ − 1

, ontains km elements and

B

ℓ

ontains the remaining ones (

km 6

|B

ℓ

| < 2km

); FOR

j

← 1

T0

ℓ

DO FOR

S

⊆ B

j

su h that

|S| = m

DO GOTO line (10);

Bythe pigeonholeprin iple,atleast oneofthesets

B

i

will ontainaset

S

ofsize

m

andthisset an be found by exhaustively sear hingthe

C

kα log

k

n

α log

k

n

= O(n)

subsets ofea h

B

i

. Consequently, implementation of line(9) of algorithmk_COLORING an be done in polynomial time.

Theorem 8. ([11 ℄) For any

α > 0

, algorithm k_COLORING olors, in

O[n

3+3α

_{/(k log}

k

n)]

, the verti es of any

k

- olorable graph with

2n/(α log

2 k

n) + O(n/ log

3 k

n)

olors. 6.2 Modifying algorithm k_COLORING torun on all graphs

The following algorithm (parameterized by

α

and

G

) modies, in the spirit of [51 ℄, algorithm k_COLORINGto workon anygraph. Algorithms1_COLOR and2_COLOR alled byk_COLORINGare as in se tion2.

BEGIN *Ek_COLORING*

(1) IF 1_COLOR(G) feasible THEN OUTPUT 1_COLOR(G) FI (2) IF 2_COLOR(G) feasible THEN OUTPUT 2_COLOR(G) FI (3) FOR

p

← 1

TO

⌈logn⌉

DO

(4)

X

← k

_

COLORING(2

p

_{, α, G)}

;

(5) IF X is feasible THEN

p

0 ← p

and GOTO (7) FI; (6) OD

(7) applybinary sear h in

{2

p

0−1

_{+ 1, . . . 2}

p

0 }

to find the least

k

← k

0

for whi h k_COLORING produ es a legal oloring;

(8) OUTPUT

X

E

← k

_

COLORING(k

0 , α, G)

; END. *Ek_COLORING*

The omplexity of algorithm Ek_COLORING is

O(n

4+3α

_{/ log n)}

(it alls at most

n − 2

times k_COLORING). Moreover, with the same arguments as the ones for lemma 9,

k

0

6 χ(G)

. So, the oloringprodu ed satises

|X

E

| 6

2n

α log

2 _χ(G)

n

+ O

n

log

3 _χ(G)

n

!

(3) 6.3 The whole improvement

Re all that

ρ

Ek

_

COLOR

6 2⌈n

(1−(1/(χ(G)−1)))

_⌉

and this bound is in reasing in

χ(G)

. On the other hand, by theorem 8 and expression (3),

ρ

Ek

_

COLORING

6 2n/(αχ(G) log

2 χ(G)

n) +

o(n/ log

2 _χ(G)

n)

and this last bound is de reasing in

χ(G)

. For

χ(G) = O(⌈log n/ log log n⌉)

, both bounds are at most

O(n log

3 _{log n/ log}

3 _n)

. The following algorithm, BR_COLOR, ombines algorithms Ek_COLOR and Ek_COLORING into an algorithmi s hema for C. Its running time is of

O(max{n

(4+(21/η))

_{/ log n, (n + |E|)χ(G) log χ(G)})}

([11 ℄). BEGIN *BR_COLOR* fix a large

η > 0

; OUTPUT

X

E

← Ek

_

COLORING(7/η, G)

; OUTPUT

X

′

E

← Ek

_

COLOR(G)

; OUTPUT

argmin{|X

E

|, |X

′

E

|}

; END. *BR_COLOR*

(19)

Theorem 9. ([11℄)

ρ

BR

_

COLOR

6 ηn log

3 _{log n/ log}

3 _n

.

Let us observe that algorithmBR_COLOR has a fairly serious drawba k sin e its exe ution time requirements(beingpolynomialin

n

for xed

η

)isexponential in

η

. Thismeans thatthebetter the approximation ratioa hieved,the higher its exe ution time.

7 A still better approximation ratio for graph- oloring

In1993,afurtherimprovementof oloring'sapproximationratiohasbeenpublishedby Halldórs-son in [31 ℄. Thespirit ofthis work isquitesimilar to the previousones(ex ept the one of[42 ℄), i.e., one olors a graph by ex avating independent sets, but the used IS-algorithms are quite dierent from the greedy one used until then. It is well-known to people working on the de-sign of approximation algorithms that very frequently the e ien y of an algorithm strongly depends on the value of the optimal solution of an instan e. Some algorithms work well for smalloptimal values,whilesome otheronesworkbetteron instan es withlargeoptimal values. The key idea of [31 ℄ is to ombine into an ex avation s hema two IS-algorithms, one of them (OPT-CLIQUE_REMOVAL) behaving e iently in graphs with small hromati numbers, while the otherone(COLOR_IS) behavingwellingraphswithlarge hromati numbers. Simultaneous run-ning, atea hiteration ofthe ex avations hema,of both algorithms and oloring largestamong the independent sets omputed with an unused olor leads to an improved ratio for any value of

χ(G)

.

7.1 Finding large independent sets ...

Werstbrieydes ribethe twoIS-algorithms,OPT-CLIQUE_REMOVALandCOLOR_IS,usedin[31 ℄ for ex avating independent sets.

The ideaofusing theformerisbased upon thefa tthatindependent sets in graphswithout liques or with small liques are larger than independent sets in general graphs. Furthermore satisfa torily large independent sets (a hieving improved approximation ratios) an be polyno-mially found there (see for example [49 , 50 ℄, where the author deals with triangle-free graphs). Hen e,givenagraph

G

,one anredu eitbyremoving liquesofa ertainsize

ℓ

. Inthesurviving graph (that is

ℓ

- lique-free), in parti ular if its size remains large (in some sense), she/he an apply some e ient algorithm omputing a large independent set. Ex avation of su h large independent setsispossibleas longasthe initial andthe onse utive(surviving) graphs ontain few disjoint liques. Of ourse, if large independent sets are ex avated, by item 1 of proposi-tion4,thegraph anbe oloredwithrelativelyfew olors. Ontheotherhand,iftheinitialgraph ontains manydisjoint liques, then

α(G)

must be small and, byexpression (1),

χ(G)

must be large. Inboth ases the approximationratio for C an so be improved.

AlgorithmCOLOR_ISoriginallyoperatesin

k

- olorable graphs. Informally,itre ursivelynds an independent set

S

of a ertain size and takes the union of

S

with the resultof its re ursive running on the graph

G[S ∩ Γ(S)]

. This is done as long as the order of the surviving graph ex eedsaxedthreshold

t

. Assoonas the orderofthe survivinggraph be omessmallerthan

t

, OPT-CLIQUE_REMOVAL is alled. The nal independent set is the union of the independent sets re ursively omputed by COLOR_IS together with the one omputed by OPT-CLIQUE_REMOVAL. RunningCOLOR_ISforanyvalueof

k

andretainingthebestresult,one anobtainanindependent set at least as good as the result of the exe ution

COLOR

_

IS(χ, G)

, and then using item 1 of proposition4, one an hope to ompute a small oloringfor

G

.

(20)

We rst present an intermediate algorithm, originally devised in [16 ℄, and used, dire tly or undire tly, by both OPT-CLIQUE_REMOVAL and, COLOR_IS. It is based upon the following result due to Ramsey.

Theorem 10. For any pair

(s, t)

of integers, there is an integer

n

for whi h every graph of order

n

ontains either a lique

K

s

,or an independent set

S

of size

t

.

Ifwe denoteby

R(s, t)

theminimalvalueof

n

forwhi h theabovetheoremholds, then the best knownupperboundfor

R(s, t)

,dueto ErdösandSzekeres([25℄),is

R(s, t) 6 C

s−t+2

t−1

. Letusset

C

_t−1

s−t+2

= r(s, t)

for all positive integers

s

and

t

(by onvention, if one of

s

and

t

isnegative or zero, then

C

s−t+2

t−1

= 1

),and let

t

s

(n) = min{t : r(s, t) > n}

.

Algorithm RAMSEY (parameterized by a graph

G

and an integer

s

) developed in [16 ℄ and stronglyinspiredbytheproofoftheupperboundfor

R(s, t)

([25℄)nds,intime

O(|E|)

, eithera lique

K

of order

s

, or anindependent set

S

of size

t

. Moreover,

r(s, t) > n

and

s.t > c(log n)

2

, for some onstant

c

.

BEGIN *RAMSEY*

S

← ∅

;

K

← ∅

; WHILE

|V(G)| > 1

DO hoose

v

∈ V(G)

;

t

← t

s

(n)

; IF

|Γ(v)| > r(s − 1, t)

THEN

K

← K ∪ {v}

;

G

← G[Γ(v)]

;

s

← s − 1

; ELSE

S

← S ∪ {v}

;

G

← G[V \ ({v} ∪ Γ(v))]

; FI OD OUTPUT

K

← K ∪ V(G)

; OUTPUT

S

← S ∪ V(G)

; END. *RAMSEY*

It is next ombined witha greedy me hanism of lique removaland the following IS-algorithm (parameterized byagraph

G

andan integer

s

) isderivedin [16 ℄.

BEGIN *CLIQUE_REMOVAL*

(S, K) ←

RAMSEY

(G, s)

; WHILE

|K| > s

DO

G

← G[V \ K]

;

(S, K) ←

RAMSEY

(G, s)

; OD OUTPUT S; END. *CLIQUE_REMOVAL*

Inorderthatthe nal oloring-algorithmisthe bestpossible,oneneedsto ndthe valueof

s

for whi htheapproximationratioofalgorithmCLIQUE_REMOVAListhebestpossible(inotherwords, the independentset omputed isthe largestpossible). Then,one anfollowthethought pro ess originally proposed in [51 ℄, i.e., one an use binary sear h to dene a good

s

. This leads to the following nal version of CLIQUE_REMOVAL, alled OPT-CLIQUE_REMOVAL and parameterized

(21)

by a graph

G

, of omplexity

O(|E|n log n)

, where we denote by

ρ

i

the approximation ratio of CLIQUE_REMOVAL(G,i). BEGIN *OPT-CLIQUE_REMOVAL* guess an s;

S

← CLIQUE

_

REMOVAL(G, s)

; IF

ρ

s

> (n/s

2 )

(1/s)

THEN RETURN S FI

REPEATrun CLIQUE_REMOVAL(G,p) with

p

= 2

ℓ

,

ℓ = ⌈logk⌉, . . .

; UNTIL the first

ℓ ← ℓ

0

for whi h

ρ

p

> (n/p

2 ₎

(1/p)

; REPEAT apply binary sear h in

{2

ℓ0−1

_{+ 1, . . . 2}

ℓ0

_}

; UNTIL the least

p

← p

0

for whi h

ρ

p

O

> (n/p

2

0 )

(1/p0)

; OUTPUT

S

← CLIQUE

_

REMOVAL(G, p

0 )

;

END. *OPT-CLIQUE_REMOVAL*

Lemma 10. ([31 ℄) Running algorithm OPT-CLIQUE_REMOVAL on a graph

G

of order

n

with

α(G) > tn

,

t > 1/ log n

, returns an independent setof size atleast

e

−1

_n

t

_/t

. 7.1.2 ...In

k

- olorable graphs

Letus now onsiderthe se ond IS-algorithm(parameterized byaninteger

k

and byagraph

G

) presented in [31 ℄ and running in

k

- olorable graphs. As for algorithmk_COLORING in se tion 6, the rst onditionin line (3)re eivesTRUEif the set

S

is anindependent set, FALSE otherwise. BEGIN *k-COLOR_IS*

(1) IF

|V| = 1

THEN OUTPUT V FI; (2) FOR

S

⊆ V

su h that

|S| = log

k

n

DO (3) IF S THEN

(4) IF

|G[V \ (S ∪ Γ(S))]| > nlogn/(2kloglogn)

(5) THENOUTPUT

S

∪

k-COLOR_IS

(k, G[V \ (S ∪ Γ(S))])

; (6) ELSE

I

←

OPT-CLIQUE_REMOVAL

(G[V \ (S ∪ Γ(S))]) ∪ S

;

(7) IF

|I| > (logn)

3 _/(6loglogn)

THEN OUTPUT

S

∪ I

FI; (8) FI

(9) FI (10) OD

(11) OUTPUT G not k- olorable; END. *k-COLOR_IS*

Notethatremark1holdsalsoforalgorithmk-COLOR_IS.Consequently, the exe utionoftheFOR loop ofline (2) isperformedin polynomial time.

We an modify algorithm k-COLOR_IS to work for all graphs (i.e., for any

k

) produ ing so the following IS-algorithm (parameterized by

G

) of omplexity

O(n|E|χ(G))

. The IF- ondition in this algorithmre eives TRUEif

S

k

isindependent.

BEGIN *COLOR_IS*

FOR

k

← 1

TO n DO

S

k

←

k-COLOR_IS

(k, G)

; IF

S

k

THEN store

S

k

FI

OUTPUT the best among the

S

k

's stored; OD

END. *COLOR_IS*

Lemma 11. ([31 ℄) The appli ation of algorithm COLOR_IS in

G

produ es an independent set

(22)

We areready nowto outlinethe overallC-algorithmof [31 ℄,the worst- ase omplexityofwhi h is

O(n

2 _|E|)

. BEGIN *H_COLOR* REPEAT

SCR

←

OPT-CLIQUE_REMOVAL

(G)

;

SCS

← COLOR

_

IS(G)

;

S

← argmax{|SCR|, |SCS|}

;

olor the verti es of S with the same unused olor;

V

← V \ S

;

G

← G[V \ S]

; UNTIL

V

= ∅

;

OUTPUT the set of the olors used; END. *H_COLOR*

Suppose rst

χ(G) 6 log n/(2 log log n)

. Then, by lemma 10, algorithm OPT-CLIQUE_REMOVAL guarantees an independent setof size

|SCR| > e

−1

n

1 χ(G)

= e

−1

χ(G)n

χ(G)

1 _.

Byitem1ofproposition4,anex avations hemausingOPT-CLIQUE_REMOVAL(algorithmH_COLOR isinstantiation ofsu ha s hema) omputesa oloring ofsize

|X| = O

n

e

−1

_χ(G)n

χ(G)

1 !

= O

n

1−

1 χ(G)

χ(G)

!

(4) On the other hand, if

χ(G) > log n/(2 log log n)

, then by lemma 11, algorithm COLOR_IS guar-anteesan independent set ofsize

|SCS| >

log

2 _n

log χ(G)

1 2 max

n

log

2χ(G) log log n

_{log n}

, 1

o .

By item1of proposition4,algorithmH_COLOR (using COLOR_IS) willprodu e a oloringwith

|X| = O





n log χ(G) max

n

log

2χ(G) log log n

_{log n}

, 1

o

log

2 n





(5)

The orrespondingratiosaretheright-handsidesofexpressions(4)and(5)dividedby

χ(G)

,the formerbeing de reasing and the se ond in reasing in

χ(G)

. For

χ(G) = log n/(2 log log n)

, the values of the two ratiosare both of

O(n log

2 _{log n/ log}

3 _n)

and the following theorem on ludes the se tion.

Theorem 11. ([31 ℄)

ρ

H

_

COLOR

= O(n log

2 _{log n/ log}

3 _n)

(23)

tion of graph- oloring

RevisitforawhilealgorithmOPT-CLIQUE_REMOVALofse tion7. In[23 ℄,thefollowingproposition isproved.

Proposition 5. ([23 ℄) Forevery fun tion

ℓ

su hthat,

∀x > 0

,

0 < ℓ(x) ≤ log log x

, thereexist onstants

κ

and

K

su hthat algorithmOPT-CLIQUE_REMOVAL omputes,forevery graph

G

of or-der

n > κ

,anindependentset

S

su hthatif

α(G) > ℓ(n)n log log n/ log n

,then

|S| > K log

ℓ(n)

_n

. In what follows, we denote by EXHAUST, an exhaustive-sear h algorithm for C. Without loss of generalitywe supposethatverti esare olored with

1, 2, . . .

Moreover, let

K

and

κ

be as in the quoted propositionabove.

BEGIN *COLOR*

(1) IF

n 6

κ

THEN OUTPUT EXHAUST(G) FI (2)

S

←

OPT-CLIQUE_REMOVAL(G); (3)

i

← 1

; (4)

^

X

← ∅

; (5)

V(^

G) ← ∅

; (6) WHILE

|S| > Klog

ℓ

_n

DO

(7) olor S with olor i; (8)

V(^

G) ← V(^

G) ∪ S

; (9)

^

G

← G[V(^

G)]

; (10)

^

X

← ^

X

∪ {i}

; (11)

i

← i + 1

; (12)

G

← G[V \ S]

; (13) IF

G

= ∅

THEN OUTPUT

^

X

FI (14)

S

←

OPT-CLIQUE_REMOVAL(G); (15) OD (16)

~

X

← ∆

_

COLOR(G)

; (17) OUTPUT

X

← ^

X

∪ ~

X

; END. *COLOR* Theorem 12.

ρ

C

6 max

2n

k log

ℓ(n)−1

n

,

2∆(G)ℓ(n) log log n

log n

.

Proof. Obviously,ifline(1)ofalgorithmCOLORisexe uted, thenitreturnsaminimum oloring for

G

in polynomial time. The WHILE-loop of the algorithm above (lines (6) to (15)) is an appli ation of EXCAVATION(G,OPT-CLIQUE_REMOVAL). Observe also that, for everyiteration

i

of the WHILE-loop,if we denoteby

G

i

the graph

G[V \ V ( ˆ

G)]

(

G

1 = G

),then

ρ

OPT-CLIQUE_REMOVAL

(G

i

) >

K log

ℓ(|Gi|)

|G

i

|

|G

i

|

(6) Thenbyitem2 ofproposition 4and by expression(6):

ρ

EXCAVATION

ˆ

G

6 ln

G

ˆ

K log

ℓ

(|

G

ˆ

|)

_|

_G

ˆ

_|

|

_G

ˆ

_|

=

log

G

ˆ

log eK log

ℓ

(|

G

ˆ

|)

_|

_G

ˆ

_|

|

_G

ˆ

_|

6 G

ˆ

k log

ℓ

(|

G

ˆ

|)

−1

ˆ

G

(7)

(24)

Set

˜

G = G[V \ V ( ˆ

G)]

(in other words,

˜

G

is the subgraph of

G

input of algorithm

∆

_COLOR in line(16)). Then,byproposition5andexpression(1),appli ationof

∆

_COLORin

G

˜

will ompute a set

X

˜

of olorsverifying

ρ

∆

_

COLOR

˜

_G

=

˜

X

χ

˜

G

6 ∆

˜

G

log

_|

G

˜

_|

ℓ

(|

G

˜

|)

log log

|

G

˜

|

=

∆

˜

G

ℓ

˜

G

log log

˜

G

log

˜

G

(8) Thefollowing holds for the set

X

of olors omputed byalgorithmCOLOR:

|X| =

X

ˆ

+

X

˜

6 ρ

EXCAVATION

ˆ

G

χ

ˆ

G

+ ρ

∆

_

COLOR

˜

_G

χ

˜

G

6 max

n

ρ

EXCAVATION

ˆ

G

, ρ

∆

_

COLOR

˜

_G

o

χ

ˆ

G

+ χ

˜

G

(9) Obviously, both

χ( ˆ

G)

and

χ( ˜

G)

are smaller than

χ(G)

. So, using expressions (7), (8 and (9), one gets

ρ

COLOR

=

|X|

χ(G)

6 max







2 G

ˆ

k log

ℓ

(|

G

ˆ

|)

−1

G

ˆ

,

2∆

˜

G

ℓ

G

ˆ

log log

G

˜

log

G

˜







6 max

2n

k log

ℓ(n)−1

n

,

2∆(G)ℓ(n) log log n

log n

(10) This ompletesthe proof ofthe theorem.

As we have already seen, in terms of

n

the best known approximation ratio for C is

O(n log

2 _{log n/ log}

3 _n)

(se tion 7) and the very tight analysisof [31 ℄ does not allow improve-ment of this ratio even in parti ular lasses of graphs. Let

ℓ > 5

and suppose rst that the maximum in expression (10) is realized by the term

O(n/ log

ℓ−1

_n)

. Then, for graphs with

∆(G) = O(n/ log

ℓ−2

n)

, Cisapproximablewithinratio

O(n/ log

ℓ−1

_n)

andtheorem12improves the ratio of[31 ℄ byafa tor

Ω(log

2 _{log n log}

ℓ−4

_n)

.

Revisit nowthe ratioin theorem4and remark thatthe rst fun tion isde reasing in

χ(G)

, whilethe se ond one isin reasing for

χ(G) 6 2 log ∆(G)

and de reasing for

χ(G) > 2 log ∆(G)

.

•

If

χ(G) 6 2 log ∆(G)

, the ratio expression in theorem 4 attains its minimumvalue when the two terms are equal, in other words when

(∆(G))

2/χ(G)

_{= Θ(plog ∆(G) log n)}

, i.e., when

χ(G) = Θ(log ∆(G)/ log log n)

. Inthis ase, the minimumandthereforethe valueof the ratio guaranteed bythe expression in theorem4 be ome