Maximum-weight independent set is as "well" approximated as the unweighted one

Laboratoire d'Analyse et Modélisation de Systèmes pour

l'Aide à la Décision

CNRS UMR 7024

CAHIER DU LAMSADE

163

Juin 1999

Maximum-weight independent set is as “well”

approximated as the unweighted one

Résumé ii

Abstra t iii

1 Introdu tion 1

1.1 Theframeworkof the paper and the problems studied . . . 1

1.2 Ourmain ontributions . . . 2

2 Redu ing the weighted ase tothe unweighted one 3

2.1 Hereditaryindu ed-subgraph problems . . . 3

2.2 Arstimprovement forthe approximationofthe maximum-weight independent set 6

3 Towards

Ω(1/∆)

-approximation for IS and WIS 7

4 Further improvements 9

4.1 Aweighted version ofTurán's theorem . . . 9

4.2 Graph oloringand the approximationof WIS

k

. . . 10

4.3 Themain result . . . 13

5 Approximating maximum-weight lique 14

6 Maximum

l

- olorable indu ed subgraph and minimum hromati sum 15 6.1 Maximum

l

- olorable indu edsubgraph . . . 15 6.2 Minimum hromati sum . . . 16

Le stable maximum pondéré est approximable

((

aussi bien

))

que le stable

maximum non-pondéré

Résumé

Nous présentonstout d'abord une rédu tion d'expansion

O(log n)

(où

n

est l'ordre du graphe) préservant le rapport d'approximation entre les versions pondérée et non

pondé-rée d'une lassede problèmes de maximisation onsistant àtrouverun sous-grapheinduit

de poids maximum vériant une propriété héréditaire. Cette rédu tion nous permet

d'ef-fe tuer une première amélioration du meilleur rapport onnu pour le problème WIS du

stablemaximumpondéré.Ensuite,ens'appuyantsurlarédu tionproposée,nous on evons

un algorithme appro hé polynomial dont le rapport est égal au minimum de

O(log

k

n/n)

et

O(log µ/(k

2

µ(log log µ)

2

)

où

µ

désigneledegrémoyendugraphepourtoute onstante

k

. Danslesdeux as,l'améliorationdurapportestsigni ative:danslepremier as,lerapport

pour WIS sur lasse

O(log

2

n/n)

,le meilleur rapport onnupour IS(leproblème dustable non-pondéré) tandis que, dans le se ond as, nous obtenons le premier rapport de l'ordre

de

Ω(1/∆)

pour WIS (où

∆

est le degré maximum du graphe; le meilleur rapport onnu en fon tionde

∆

valait jusqu'i i

3/(∆ + 2)

). Par lasuite, àpartir duproblème de olora-tion, nous trouvons un algorithme appro hé polynomial pour WIS ayant pour rapport le

minimumentre

O(n

−

4

/5

)

et

O(log log ∆/∆)

.Ainsi, àmoinsque WISne soitapproximable à

O(n

−

4

/5

)

( e qui est peu probable, l'approximation de IS à rapport

n

ǫ−1

étant di ile

pourtout

ǫ > 0

),notrealgorithmeobtientlepremierrapport en

Ω(log log ∆/∆)

pourWIS (pourtout

∆

).Notonsquel'approximationduproblèmedustable pondéréounon pondéré àrapport

Ω(1/∆)

demeuraitpendantlongtempsunproblèmeouvert.Enn,nousproposons les premiers rapports non triviaux en

Ω(1/∆)

pour les problèmes de la lique maximum pondéréeetnonpondérée,leproblèmedusous-grapheinduit

l

- olorabledepoidsmaximum etleproblèmedelasomme hromatique.

Mots- lé: problèmes ombinatoires, omplexité,algorithmepolynomiald'approximation,

NP- omplétude,problèmehéréditaire,stable, lique,grapheinduit

l

- olorable,somme hro-matique.

unweighted one

Abstra t

We rst devise an approximation-preservingredu tion of expansion

O(log n)

(where

n

is the order of the input-graph) between weighted and unweighted versions of a lass of

problems alledweightedhereditaryindu ed-subgraphmaximizationproblems. Thisallows

us to perform arst improvement of the best approximation ratio for the weighted

inde-pendentsetproblem (WIS).Then, usingtheredu tiondeveloped,weproposeapolynomial

timeapproximationalgorithmforWISa hievingasratiotheminimumbetween

O(log

k

n/n)

and

O(log µ/(k

2

µ(log log µ)

2

)

where

µ

denotestheaveragegraph-degreeoftheinput-graph, for every onstant

k

. In any of the two ases, this is an important improvement sin e in the former one, the ratio for WIS outerperforms

O(log

2

n/n)

, the best-known ratio for (unweighted) independent set problem (IS), while in the latter ase, we obtain the

rst

Ω(1/∆)

ratio for WIS (where

∆

is the maximum graph-degree; the best-known ra-tio in terms of

∆

was

3/(∆ + 2))

. Next, basedupon graph- oloring, we devise a polyno-mialtimeapproximationalgorithmforWISa hievingratiotheminimumbetween

O(n

−

4

/5

)

and

O(log log ∆/∆)

. Here also, ex ept for the very unlikely ase where WIS an be ap-proximated within

O(n

−

4

/5

)

(approximation of IS within

n

ǫ−1

is hard for every

ǫ > 0

), our algorithm is the rst

Ω(log log ∆/∆)

-approximation algorithm for WIS (for every

∆

). Letusnote that approximationofbothindependentset versionswithin ratios

Ω(1/∆)

is a very well-knownopen problem. Finally, wepropose therst non-trivial

Ω(1/∆)

ratios for maximum-size and maximum-weight lique, for maximum-weight

l

- olorable indu ed sub-graphandfor hromati sum.

Keywords: ombinatorial problems, omputational omplexity,polynomial-time

approxi-mationalgorithm,NP- ompleteness,hereditaryproblem,independentset, lique,

l

- olorable indu edsubgraph, hromati sum.

1.1 The framework of the paper and the problems studied

Given a graph

G = (V, E)

and a set

V

′

_{⊆ V}

, a subgraph of

G

indu ed by

V

′

is a graph

G

′

= (V

′

, E

′

)

, where

E

′

_{= (V}

′

_{× V}

′

₎

_{∩ E}

. We onsider NP-hard graph-problems

Π

where the obje tive is to nd a maximum-order indu ed subgraph

G

′

satisfying a non-trivial hereditary

property

π

. For a graph

G

, anyone of its vertex-subsetsspe iesexa tlyone indu ed subgraph. Consequently, in what follows we onsider that afeasible solution for

Π

isthe vertex-set of

G

′

.

Let

G

bethe lassofallthegraphs. Agraph-property

π

isamapping from

G

to

{0, 1}

, i.e.,fora

G

∈ G

,

π(G) = 1

i

G

satises

π

and

π(G) = 0

, otherwise. Property

π

is hereditaryifwhenever it is satised by a graph it is also satised by every one of its indu ed subgraphs; it is

non-trivialifit istruefor innitely manygraphs andfalse for innitelymanyones ([4 ℄). Hereditary

indu ed-subgraph maximization problems have anatural generalization to graphswith positive

integral weights asso iated with their verti es (the weightsare assumedto be bounded by

2

n



the order of the input-graph  sothat every arithmeti al operation on them an be performed

in polynomialtime). Given agraph

G

,the obje tive ofaweighted indu edsubgraphproblemis to determine an indu ed subgraph

G

∗

of

G

su h that

G

∗

satises

π

and,moreover, the sum of the weightsofthe verti es of

G

∗

isthe largestpossibleamongthosesubgraphs. In whatfollows,

we denoteby

WΠ

theweighted version of

Π

.

Moreparti ularly, we onsiderin this paper the following indu ed-subgraph problems.

Maximum independent set. Given a graph

G = (V, E)

of order

n

, an independent set is a subset

V

′

_{⊆ V}

su h that no two verti es in

V

′

are linked by an edge in

E

, and the maximum independent setproblem (IS)isto nd amaximum-size independent set.

Maximum lique. Consider a graph

G = (V, E)

. A lique of

G

is a subset

V

′

_{⊆ V}

su h

that every pair of verti es of

V

′

are linked by an edge in

E

, and the maximum lique problem (KL) is to nd a maximum size set

V

′

indu ing a lique in

G

(a maximum-size lique).

Maximum

l

- olorable subgraph. Given a graph

G = (V, E)

and a positive onstant

l

, the problem of the maximum

l

- olorable indu ed subgraph (denoted by C

l

) is to nd a maximum-order indu ed subgraph

G

′

_{= G[V}

′

_]

of

G

(

V

′

_{⊆ V}

) su h that

G

′

is

l

- olorable (i.e., there existsa oloring for

G

′

of ardinality at most

l

).

Propertyisanindependentset ishereditary(thesubsetofanindependentsetisanindependent

set). Thesameholdsforpropertyisa lique (thevertex-subsetofa liqueindu esalsoa lique),

aswell asforpropertyis

ℓ

- olorable (ifthe verti es ofa graph

G

an be feasibly olored byat most

ℓ

olors, theneverysubgraph of

G

indu edby asubset ofits verti es anbe olored byat most

ℓ

olors).

Wealso onsiderthe following minimization problemand itsweighted version.

Minimum hromati sum. Given a graph

G = (V, E)

, an

l

- oloring is a partition of

V

into independent sets

C

1

, . . . , C

l

. The ost of an

l

- oloring is the quantity

P

l

i=1

i

|C

i

|

(in other words, the ost of oloring a vertex

v

∈ V

with olor

i

is

i

). The minimum hro-mati sum problem, denoted by CHS is to determine a minimum- ost oloring. For the

weighted versionof CHS,denoted byWCHS, every vertex

v

∈ V

isweighted byarational weight

w

v

, the ost of oloring

v

with olor

i

be omes

iw

v

, the value of an

l

- oloring be- omes

P

l

i=1

iw(C

i

)

,where

w(C

i

) =

P

v∈C

i

w

v

,andtheobje tivebe omesnowtodetermine a oloringofminimum value.

Given a problem

Π

dened on a graph

G = (V, E)

, andits weighted versionW

Π

, we denoteby

~

w

∈ IN

|V |

theve toroftheweights,by

w

v

theweightof

v

∈ V

andby

w

max

(G)

and

w

min

(G)

the largestandthesmallestvertex-weight,respe tively. Moreover, weadoptthe following notations:

w(V

′

)

: the total weight of

V

′

_{⊆ V}

, i.e.,the quantity

P

v∈V

′

w

v

;

β

w

(G)

: the valueof anoptimal solution for W

Π

;

∆(G)

: the maximum degree of

G

;

µ(G)

: the average degreeof

G

;

µ

w

(G)

: the quantity

P

v∈V

w(Γ(v))/w(V )

Γ(v)

: the neighborhoodof

v

∈ V

;

χ(G)

: the hromati number of

G

(the minimum number of olors with whi h one an feasibly olor the verti es of

G

);

¯

G

: the omplement of

G

dened by

¯

G = (V, ¯

E)

with

¯

E =

_{{ij ∈ V × V, i 6= j, ij /}

_{∈ E}}

(obviously,

G = G

¯¯

);

G[V

′

]

: the subgraph of

G

indu edby

V

′

_{⊆ V}

;

n

(k)

: the order ofthe graph

G[V

(k)

_]

,

V

(k)

_{⊆ V}

;

S

∗

(G[V

(k)

])

: an optimal(maximum-size)

Π

-solution in

G[V

(k)

_]

,

V

(k)

_{⊆ V}

.

Espe ially for IS and WIS, using standard notations, we will denote the size of a maximum

independent set by

α(G)

andthe valueof a maximum-weight independent set by

α

w

(G)

. Moreover, when no ambiguity an o ur, we will use

∆

,

µ

and

µ

w

instead of

∆(G)

,

µ(G)

and

µ

w

(G)

Given a square matrix

B = (m

ij

)

i,j=1,...n

, we denote by

Tr(B)

and

t

_B

the tra e and the

transposeof

B

, respe tively. Finally, givena ve tor

~u

, we denoteby

|~u|

its Eu lidean norm. 1.2 Our main ontributions

We are interested in devising polynomial time approximation algorithms (PTAAs) omputing

solutions, the values of whi h are as lose as possible to the value of an optimal one. The

quality of a PTAA is expressed, for every instan e

G

, by the ratio of the size (resp., weight) of the omputed solution to the optimal size (resp., weight). Given a PTAA A, this ratio will be

denotedby

ρ

A

(G)

. Foramaximization(resp.,minimization)problem

Π

,Aissaidtoguaranteean approximationratio

ρ(G)

ifforeveryinstan e

G

,

ρ

A

(G)

isboundedbelow(resp., above)by

ρ(G)

(

ρ(G)

denotes approximation ratios depending on some parameters of

G

). It is assumed that

ρ(G)

_{≤ ρ(G}

′

₎

(resp.,

ρ(G)

≥ ρ(G

′

₎

), foranysubgraph

G

′

of

G

. Finally, wedenote by

ρ

Π

(G)

the bestapproximationratio known for

Π

.

In whatfollows,wewill denoteby

β

′

w

(G)

the weight of anapproximated W

Π

-solution of

G

. On e more,thisweight will be denotedby

α

′

w

(G)

whendealing with WIS.

For IS, whi his the most famous amongthe problemsstudied here, the strongest

inapprox-imability result is the one in [11 ℄ arming that

0 < ǫ

≤ 1

, IS is not approximable within

n

ǫ−1

unlessNP=ZPP.Con erningpositiveapproximationresultsforIS,wegivein[5℄aPTAA

guar-anteeing asymptoti (

∆(G)

→ ∞

) approximation ratio

min

{k/µ(G), k

′

_{(log log ∆(G))/∆(G)}

_}

(where

k

′

of maximum degree is

3/(∆(G) + 2)

([8 ℄). A parti ularly interesting question frequently men-tioned in the relative literature about WIS-approximation is if it an be as well-approximated

asIS. Herewe prove thatthe answer is not sofarfrom being true.

The paper isorganized asfollows. Inse tion 2,we devisea redu tion ofexpansion

O(log n)

from weighted hereditary indu ed-subgraph problems to unweighted ones. This expansion an

beimprovedwhendealing with parti ularproblems;for example,itis

O(log log n)

whendealing with the pair IS-WIS (paragraph 2.2). Based upon this redu tion we draw rst improvements

for the ratioof WIS.In se tion3,we proposenewimproved approximationresults for IS. Next,

always basedupon the redu tion of se tion 2, we further improve approximations for WIS and

obtainanapproximationratioforWISwithvaluegreaterthantheminimumbetween

O(log

k

_n/n)

and

O(log µ(G)/(k

2

_{µ(G) log}

2

_{log µ(G)))}

. This is an important improvement sin e if the

WIS-ratio obtained is

O(log

k

_n/n)

, it outerperforms

O(log

2

_n/n)

, the best-known ratio for IS. On

the otherhand,ifthe ratio obtained is

O(log µ(G)/(k

2

_{µ(G) log}

2

_{log µ(G)))}

, thenwea hieve the

rst

Ω(1/∆(G))

ratio for WIS. Let us note that this is the rst time that non trivial results for WIS are produ ed by a redu tion to IS. In se tion 4, we propose another PTAA for WIS

whi h further improves results of se tion 3. Next, we generalize a result of [1℄ by linking the

approximation ofthe lassWIS

k

of WIS-instan es with weighted independen e number greater than

w(G)/k

to the approximation of a lass

G

l

of graph- oloring instan es in luding the

l

- olorable graphs. Combining this result with re ent works of [12 , 15℄ about

G

l

, we obtain the mainresultofthepaper,i.e.,anapproximationratioforWISofvaluegreater thantheminimum

between

O(n

−4/5

₎

and

O(log log ∆(G)/∆(G))

. Consequently, ex eptfor the very unlikely ase whereWIS anbeapproximatedwithin

O(n

−4/5

₎

(re allthatapproximationofISwithin

n

ǫ−1

is

hard forevery

ǫ > 0

, unlessNP=ZPP([11℄)),ouralgorithm isthe rst

Ω(log log ∆(G)/∆(G))

-approximationalgorithm forWIS.Inse tion5,wedeviseanewredu tionbetweenKLandWKL

and,baseduponitandusing ourresults onWIS,wededu ethe rst

Ω(1/∆(G))

approximation ratio for the maximum-size andmaximum-weight liqueproblems. We noteon e morethat the

results of thispaper work even for unbounded valuesof

∆(G)

. Finally, in se tion6 we improve re ent approximation results forWC

l

and WCHS.

2 Redu ing the weighted ase to the unweighted one

2.1 Hereditary indu ed-subgraph problems

Theorem 1. Consider a hereditary property

π

, an indu ed subgraph problem

Π

stated with respe t to

π

andthe weighted version

WΠ

of

Π

(wesuppose that weightsare positive). Forevery xed

M > 2

, every PTAAfor

Π

a hieving ratio

ρ

Π

(G)

an be transformed intoaPTAAfor W

Π

a hieving

ρ

WΠ

(G)

≥ Θ



_ρ

Π

(G)

log

_M

n



.

Proof. Set,for

i = 1, . . .

,

V

(i)

=



v

j

∈ V :

w

max

(G)

M

i

< w

j

≤

w

max

(G)

M

i−1



x = sup



ℓ : β

w



G



∪

1≤i≤ℓ

V

(i)



_<

β

w

(G)

2



G

x

= G



∪

1≤i≤x

V

(i)



G

x+1

= G



∪

1≤i≤x+1

V

(i)



G

d

= G



V

\ ∪

1≤i≤x

V

(i)



_.

Of ourse,

β

w

(G

d

)

≥ β

w

(G)/2

and

β

w

(G

x+1

)

≥ β

w

(G)/2

.

We rst remove verti es

v

k

su h that the graph

(

{v

k

}, ∅)

does not satisfy

π

. Then, the following lemmaholds.

Lemma 1. There exists a PTAA for

WΠ

a hieving approximation ratio

M

x

_/(2n)

.

Proof of lemma 1. Thealgorithm laimed onsistsof simplytaking

v

∗

_{∈ argmax}

v

i

∈V

{w

i

}

as

WΠ

-solution. Then,

β

′

w

(G) = w

max

(G)

and

β

w

(G)

≤ 2β

w

(G

d

)

≤ 2 |S

∗

(G

d

)

| w

max

(G

d

)

≤ 2 |S

∗

(G

d

)

|

w

max

(G)

M

x

.

Consequently,

β

_w

′

(G)

β

w

(G)

≥

M

x

2

_|S

∗

_(G

d

)

|

≥

M

x

2n

=

⇒ ρ

WΠ

(G)

≥

M

x

2n

(1) q.e.d.

Remark 1. For every

i

≥ 1

, the weight of any

Π

-solution

S

(i)

of

G[V

(i)

_]

lies in the interval

[

|S

(i)

_|w

max

(G)/M

i

,

|S

(i)

|w

max

(G)/M

i−1

]

: it is at least

|S

(i)

_|w

min

(G[V

(i)

])

≥ |S

(i)

|w

max

(G)/M

i

and atmost

|S

(i)

_|w

max

(G[V

(i)

])

≤ |S

(i)

|w

max

(G)/M

i−1

.

Letus nowprove the followinglemma whi his the entral partofthe proof ofthe theorem.

Lemma 2. Assume

x > 0

.

1. Let

β

w

(G)/2

≥ β

w

(G

x

)

≥ ((M − 2)/2M)β

w

(G)

and

p

∈ argmax

1≤i≤x

{β

w

(G[V

(i)

])

}

. If there existsa PTAA for

Π

guaranteeing approximation ratio

ρ

Π

(G[V

(p)

_])

in

G[V

(p)

_]

, then

one an solve

WΠ

in

G

, in polynomialtime, within ratio

((M

− 2)/2xM

2

_)ρ

Π

(G)

.

2. Let

β

w

(G

x

)

≤ ((M − 2)/2M)β

w

(G)

. If there exists a PTAA for

Π

guaranteeing ra-tio

ρ

Π

(G[V

(x+1)

_])

in

G[V

(x+1)

_]

, then one an solve

WΠ

in

G

, in polynomial time, within ratio

(1/M

2

_)ρ

Π

(G)

. Proof of item 1. Obviously,

β

w

(G

x

)

≤ xβ

w



G[V

(p)

]



remark 1

≤

x







S

∗



G

h

V

(p)

i





w

max

(G)

M

p−1

and,bythe hypothesis of theitem,

β

w

(G)

≤

2M

M

− 2

β

w

(G

x

)

≤

2M

M

− 2

x







S

∗



G

h

V

(p)

i





w

max

(G)

M

p−1

.

On the other hand, appli ation of a PTAA guaranteeing approximation ratio

ρ

Π

(G) < 1

for

Π

in

G[V

(p)

_]

onstru ts a solution

S

(p)

of

WΠ

of weight at least







S

(p)







w

min



G

h

V

(p)

i

≥







S

(p)







w

max

(G)

M

p

.

Note that

S

(p)

is

Π

-feasible for

G

. Moreover, starting from this solution, one an greedily augment itin orderto nallyprodu e amaximal

WΠ

-solutionfor

G

. Thisnalsolutionveries

β

′

Combination ofthe above expressions for

β

′

w

(G)

and

β

w

(G)

yields

β

′

w

(G)

β

w

(G)

≥



_M

_{− 2}

2xM

2













S

(p)











S

∗

G



V

(p)










≥

M

_{− 2}

2xM

2

ρ

Π



G

h

V

(p)

i

_≥

M

− 2

2xM

2

ρ

Π

(G)

and, onsequently,

ρ

WΠ

(G)

≥

M

_{− 2}

2xM

2

ρ

Π

(G).

(2) This on ludes theproof ofitem 1.

Proof of item 2. We now suppose that

β

w

(G

x

)

≤ ((M − 2)/2M)β

w

(G)

. Note thatsin e

x

is the largest

ℓ

for whi h

β

w

(G[

∪

1≤i≤ℓ

V

(i)

_{]) < β}

w

(G)/2

, set

V

(x+1)

isnon-empty. Let

S

opt

_(G

x+1

)

be an optimal W

Π

-solution in

G

x+1

(i.e.,

β

w

(G

x+1

) = w(S

opt

_(G

x+1

))

). Let

S(G

x

) = S

opt

(G

x+1

)

∩ V (G

x

)

(where

V (G

x

)

denotes the vertex-set of

G

x

) and

S(G[V

(x+1)

_{]) =}

S

opt

_(G

x+1

)

∩V

(x+1)

(inotherwords,

{S(G

x

), S(G[V

(x+1)

_])

_}

isapartitionof

S

opt

_(G

x+1

)

). Sin e

π

is hereditary, sets

S(G

x

)

and

S(G[V

(x+1)

_])

, being subsets of

S

opt

_(G

x+1

)

, also verify

π

(and, onsequently theyarefeasibleW

Π

-solutionsfor

G

x

and

G[V

(x+1)

_]

,respe tively). Wethen have:

w (S (G

x

))

≤ β

w

(G

x

)

w



S



G

h

V

(x+1)

i

_{≤ β}

w



G

h

V

(x+1)

i

β

w

(G

x+1

) = w (S (G

x

)) + w



S



G

h

V

(x+1)

i

_{≤ β}

w

(G

x

) + β

w



G

h

V

(x+1)

i

≤

M

− 2

2M

β

w

(G) + β

w



G

h

V

(x+1)

i

and also

β

w

(G

x+1

)

≥ β

w

(G)/2

. It follows from the above expressions that

β

w

(G[V

(x+1)

_])

_≥

β

w

(G)/M

and this together with Remark1 yield, aftersome easy algebra,

β

w

(G)

≤







S

∗



G

h

V

(x+1)

i





w

max

(G)

M

x−1

.

As previously, suppose that a PTAA provides, in polynomial time, a solution

S

(x+1)

for

Π

in

G[V

(x+1)

_]

, the ardinalityof whi h isat least

ρ

Π

(G[V

(x+1)

_])

_|S

∗

_(G[V

(x+1)

_])

_|

. Then,

β

′

w

(G)

≥

|S

(x+1)

|w

min

(G[V

(x+1)

])

≥ |S

(x+1)

|w

max

(G)/M

x+1

. Combination ofexpressions for

β

′

w

(G)

and

β

w

(G)

yields

β

_w

′

(G)

β

w

(G)

≥



₁

M

2













S

(x+1)











S

∗

G



V

(x+1)










≥

1

M

2

ρ

Π



G

h

V

(x+1)

i

≥

1

M

2

ρ

Π

(G).

Therefore,

ρ

WΠ

(G)

≥

1

M

2

ρ

Π

(G)

(3) and this on ludes the proofof item 2and of the lemma.

Remark 2 . For the asewhere

x = 0

, i.e.,

β

w

(G[V

(1)

_])

_{≥ β}

w

(G)/2

, arguments similarto the ones of the proof of item 2 in lemma 2 lead to

ρ

WΠ

(G) = β

′

w

(G)/β

w

(G)

≥ ρ

Π

(G)/2M

, better than the one ofexpression (3).

Considernowthe followingalgorithm wherewetakeuptheideasoflemmata1and2andwhere,

for a graph

G

′

, we denoteby

A(G

′

₎

the solution-set provided bythe exe ution of the

Π

-PTAA A on the unweighted versionof

G

′

.

fix a onstant

M

> 2

; partition V in sets

V

(i)

_{← {v}

k

: w

max

/M

i

< w

k

≤ w

max

/M

i−1

}

;

S

(0)

← v

∗

∈ {argmax

v

i

∈V

{w

i

}}

; OUTPUT

argmax

{w(S

(0)

_{), w(A(G[V}

(i)

_{])), i = 1, . . .}

_}

;

END. (*WA*)

Revisitexpressions (1),(2) and(3). It iseasy to seethat

ρ

WΠ

(G)

≥ ρ

WA

(G)

≥ max



_M

x

2n

, min



_M

− 2

2M

2

_x

ρ

Π

(G),

1

M

2

ρ

Π

(G)



.

(4) By expression (1) and by the fa t that the approximation ratio of any PTAA for W

Π

must be less than 1 (W

Π

being a maximization problem),

x = O(log

M

n)

. Taking this value for

x

into a ount in expression4, on ludesthe proofof the theoremwhi h, obviously, works alsoin the

asewhere weightsareexponential in

n

.

2.2 A rst improvement for the approximation of the maximum-weight

indepen-dent set

Itiswell-known ([16℄)that,

∀k ≥ 1

, thegeneral weighted independent setproblempolynomially redu es to PWIS

(k)

(the WIS-subproblem where the weights are bounded by

n

k

), by a simple

s aling and rounding pro ess. This redu tion preserves (within a fa tor of

(1

− ǫ)

) the ratios for WIS and PWIS, and works also for instan e-depending ratios. On the other hand, the

following approximationpreservingredu tion fromPWISto ISworking onlyfor onstant ratios

isestablished in [16 ℄.

Denition 1. Given a weighted graph

(G = (V, E), ~

w)

an unweighted graph

G

w

= (V

w

, E

w

)

an be onstru ted in the followingway:

V

w

=

{(u, i) : u ∈ V, i ∈ {1, . . . , w

u

}}

E

w

=

{(u, i)(v, j) : i ∈ {1, . . . , w

u

} , j ∈ {1, . . . , w

v

} , u 6= v, uv ∈ E} .

Inotherwords,everyvertex

u

of

V

isrepla edbyanindependentsetofsize

w

u

in

G

w

andevery edge of

E

orrespondsto a ompletebipartite graph in

G

w

.

One an easilyshowthatevery independent set

S

of

G

oftotal weight

w(S)

indu es,in

G

w

, the independent set

{(s, i) : s ∈ S, i ∈ {1, . . . , w

s

}}

of size

w(S)

, and onversely, for every indepen-dent set

S

w

of

G

w

, the set

S =

{u ∈ V : ∃i ∈ {1, . . . , w

u

}, (u, i) ∈ S

w

}

is an independent set of weight

w(S)

≥ |S

w

|

. Consequently,

α

w

(G) = α(G

w

)

and,byapplying a

ρ(G)

-approximation IS-algorithm to

G

w

, one an derive an approximated WIS-solution of

(G, ~

w)

guaranteeing ra-tio

ρ(G

w

)

.

By the above redu tion, a ratio

ρ(n, ∆)

, non-in reasing in

∆

, for IS transforms to a ra-tio

ρ(w(V ), w

max

(G)∆)

forWISand,ex eptfromthe aseof onstantapproximationratios, the redu tion above results in WIS-ratios depending on the weights. More pre isely, the following

result an beeasily proved.

Proposition 1. For every

ǫ > 0

andevery onstant

k > 0

, there exists a polynomialredu tion fromWIStoIS transformingevery IS-approximationratio

ρ(n, ∆, µ)

(where

ρ

isnonin reasing with respe t to its variables) into an approximation ratio

ρ(n

1+ǫ

_{w(V )/w}

max

(G), n

1+ǫ

∆, µ

w

)

≥

approximation ratiosfor WISusingthe redu tion ofproposition 1.

Revisitexpression 4and let

Π

beIS. Set, for every

k

,

M = 6

. Then:

•

if

x

≥ k log log n/ log M

, then

ρ

WIS

(G)

≥ log

k

_n/2n

;

•

if

x

≤ k log log n/ log M

, then

ρ

WIS

(G)

≥ 0.099ρ

IS

(G)/(k log log n)

and the following theorem holds.

Theorem 2. For every xed

ℓ

, every PTAA for IS a hieving ratio

ρ

IS

(G)

an be transformed into a PTAA for WISa hieving

ρ

WIS

(G)

≥ min

(

log

ℓ

n

2n

,

0.099ρ

IS

(G)

ℓ log log n

)

.

In terms of

n

, the best-known approximation ratio for IS is, to our knowledge,

Θ(log

2

_n/n)

a hievedbytheIS-PTAA of[3℄. Embedding itin

ρ

IS

(G)

-expressionof theorem2,weobtain the following on ludingtheorem.

Theorem 3.

ρ

WIS

(G)

≥ Θ

log

2

n

n log log n

!

.

Theabove resultimprovesbyafa tor

O(log log n)

thebest-knownapproximation ratiofun tion of

n

for WIS(

O(log

2

_{n/(n(log log n)}

2

₎₎

, dueto [8℄).

3 Towards

Ω(1/∆)

-approximation for IS and WIS

Inthisse tionwe showhowone anusethe lique-removalmethodof[3 ℄to obtain,for every

∆

,

Ω(1/∆)

-approximationsforISandWIS.Theauthorsof[3℄repeatedly allapro edure omputing either a

k

- lique ( liqueoforder

k

),oranindependentsetofexpe tivesize. Atmost

n/k

liques an so be dete ted, while at ea h lique-deletion the independen e number de reases no more

than 1. If the independen e number of the initial graph is large enough, a large independent

set is ne essary dete ted during one exe ution of the pro edure. In all, the following theorem

summarizes the thought pro ess of[3℄.

Theorem 4. ([3℄) There is a onstant

k

and an

O(nm)

algorithm (denoted by LARGEIS in what follows), omputing, for every graph

G

of order

n

with

α(G)

≥ n/k + m

and every

k

su h that

2

≤ k ≤ 2 log n

,an independent set of size

kO(m

1/(k−1)

₎

.

In[3℄

k

isnotsupposedtobe onstant;ratio

log

2

_n/n

forLARGEISisobtainedfor

k = O(log n)

. If weset

k = (log n/(ℓ(n) log log n))+1

,where

ℓ

isamappingsu hthat,

∀x > 0

,

0 < ℓ(x)

≤ log log x

and

m = n/k

2

, then an analysissimilarto the one for theorem4 leadsto the following.

Theorem 5. For every fun tion

ℓ

with

0 < ℓ(x)

≤ log log x

,

∀x > 0

, there exist onstants

C

and

K

su hthatalgorithmLARGEIS omputes,forevery graph

G

oforder

n > C

,anindependent set

S

su h that if

α(G)

≥ ℓ(n)n log log n/ log n

, then

|S| ≥ K log

ℓ(n)

_n

.

Let now

ℓ

,

C

and

K

be as in theorem 5, denote by GREEDY the natural greedy IS-algorithm andbyEXHAUST anexhaustive-sear halgorithm formaximumindependent set, and onsiderthe

IF

n

≤ C

THEN OUTPUT

S

← EXHAUST(G)

;

ELSE OUTPUT

S

← argmax{LARGEIS(G), GREEDY(G)}

; FI

END.(*STABLE*)

If

n

≤ C

, then EXHAUST omputes in onstant time a maximum independent set for

G

. So, suppose

n > C

and onsider ases

α(G)

≥ ℓ(n)n log log n/ log n

and

α(G) < ℓ(n)n log log n/ log n

.

•

If

α(G)

≥ ℓ(n)n log log n/ log n

, then, by theorem 5,STABLE guarantees

|S| ≥ K log

ℓ(n)

_n

;

sin e

α(G)

≤ n

,the approximation ratio obtained isat least

K log

ℓ(n)

_n/n

;

•

if

α(G) < ℓ(n)n log log n/ log n

, then by Turán's theorem ([17℄)

|S| ≥ n/(µ + 1)

, and the approximationratio obtained is

log n/(ℓ(n)(µ + 1) log log n)

.

Sin eGREEDYisoftimelinearin

|E|

,STABLEhasthesameworst- asetime- omplexityasLARGEIS. So,the following theorem on ludesthe above dis ussion.

Theorem 6. Consider

ℓ

su h that,

∀x > 0

,

0 < ℓ(x)

≤ log log x

. Then STABLE a hieves approximation ratio

ρ

STABLE

(G)

≥ min

(

K

log

ℓ(n)

_n

n

,

log n

ℓ(n)(µ + 1) log log n

)

.

If

log log n

≥ ℓ > 2

(inparti ularif

ℓ

is onstant),either

log

ℓ(n)

_n/n

,or

log n/(ℓ(n)(µ+1) log log n)

orresponds to a signi ant improvement with respe t to

O((log

2

_n)/n)

and

k/µ

, respe tively. In terms of graph-degree, it is, to our knowledge, the rst

Ω(1/µ)

approximation result for IS. Also,

log n

ℓ(n)(µ + 1) log log n

≥

log µ

(µ + 1) log

2

log µ

>

log log ∆

∆

.

Ratio

log log ∆/∆

forISwas,upto now, theonlyknown

Ω(1/∆)

ratioforIS(presented in[10℄). But the algorithm of [10℄ has the drawba k to be polynomial only if

∆

is bounded above. Moreover, the omplexity of algorithm STABLE is

O(nm)

, whereas the

O(k/µ)

ratio of [5℄ is obtainedbyaPTAAof omplexity

O(n

k

₎

. Of ourse,ourresultdoesnotguaranteearatioalways

bounded below by

log n/(ℓ(n)(µ + 1) log log n)

, but in the opposite ase, the ratio guaranteed is superior to the best-known

n

-depending ratio for IS (for instan e, onsider the ase

ℓ(n) =

log log n

). In fa t, if

K log

ℓ(n)

_n/n

_{≤ log n/(ℓ(n)(µ + 1) log log n)}

and

n

≥ C

0

, for a xed

C

0

, then

n/µ

≥ Kℓ(n)(log log n) log

ℓ(n)−1

_n

andin this aseGREEDYguarantees ratiobounded below

by

log

ℓ(n)−1

_n/n

. Inall,the following orollary an bededu ed.

Corollary 1. Given a graph

G

and

ℓ

su h that

2 < ℓ(n)

≤ log log n

, at least one of the two following onditions holds:

1. GREEDY guarantees ratio bounded below by

log

ℓ(n)−1

_n/n

(improving the best known

ρ(n)

-ratio if

ℓ > 3

);

2. STABLE guarantees ratio bounded below by

log n/(ℓ(n)(µ + 1) log log n)

(a hieving so ra-tio

Ω(1/µ)

).

onstantapproximationratios. Untilnow,studiesabout theapproximationofISwerelimited in

expressing ratio using one parameter (either the size or the degree). The results of this se tion

showthat onsidering both parameters, itispossible torea htighterapproximation ratios.

Combination of theorem6with theorem2 allows to obtainimportant approximationresults

also forWIS.

Theorem 7. For every onstant

k

,

ρ

WIS

(G)

≥ min

(

log

k

n

n

,

0.099 log n

k(k + 1)(µ + 1) log

2

log n

)

≥ min

(

log

k

n

n

,

0.099 log µ

k(k + 1)(µ + 1) log

2

log µ

)

.

In other words, theorem 7 guarantees for WIS the existen e of PTAAs a hieving either

n

-depending ratios mu h better than the ones known for IS (and omparable with the ones of

orollary1),or therst

Ω(1/µ)

ratiosforWIS(re allthatthe best-known

ρ(∆)

-ratioforWIS withoutrestri tions on

∆

 was, until now, the one of[8℄,bounded below by

3/(∆ + 2)

). 4 Further improvements

Inthisse tion, weproposeapolynomialapproximationresultfor maximum-weightindependent

set problem whi h is not dedu ed by redu tion from the unweighted ase. It improves all the

approximationresultsofse tion3,forbothweightedandunweighted ases,butthe orresponding

omplexity ishigher.

4.1 A weighted versionof Turán's theorem

Re allthat

µ

w

(G) =

P

v∈V

w(Γ(v))/w(V )

(notethat

µ

w

(G)

≤ ∆(G)

)and onsiderthefollowing algorithm for WIS.

BEGIN (*WGREEDY*)

S

_{← ∅}

; WHILE

V

6= ∅

DO

v

_{← argmin}

V

{w(Γ(v))/w(v)}

;

S

_{← S ∪ {v}}

;

V

← V \ ({v} ∪ Γ(v))

; update E; OD END.(*WGREEDY*)

Then,the followingeasytheoremholds(anunweighted versionofitseealso[9℄isthefamous

Turán's theorem).

Theorem 8. For every weighted-graph with maximum degree

∆

, algorithm WGREEDY omputes an independent set of weight atleast

w(V )/(µ

w

(G) + 1)

≥ w(V )/(∆ + 1)

.

Proof. Consider algorithm WGREEDY and suppose that its WHILE loop is exe uted

t

times. For

i

_{∈ {1, . . . t}}

letus denoteby

G

i

= (V

i

, E

i

)

thesurviving graph, by

v

i

the vertex sele tedduring the

i

th exe utionand by

Γ

i

(v)

the neighborhood of

v

in

G

i

. Then,

∀i ∈ {1, . . . t}

,

X

v∈({v

i

}∪Γ(v

i

))

w(Γ(v))

≥

X

v∈({v

i

}∪Γ

i

(v

i

))

w (Γ

i

(v))

≥

w (Γ

i

(v

i

))

w

v

i

(w

v

i

+ w (Γ

i

(v

i

)))

and, onsequently, byaddingside-by-sidetheexpressions above,for

i = 1 . . . t

, weget(notethat

V =

∪

i=1...t

{{v

i

} ∪ Γ

i

(v

i

)

}

),

w(V )µ

w

≥

t

X

i=1

w (Γ

i

(v

i

))

w

v

i

(w

v

i

+ w (Γ

i

(v

i

))) .

Onthe other hand,

w(V ) =

t

X

i=1

(w

v

i

+ w (Γ

i

(v

i

)))

and byCau hy-S hwarz inequalitywe get

w(V ) (µ

w

+ 1)

≥

t

X

i=1

(w

v

i

+ w (Γ

i

(v

i

)))

2

w

v

i

≥

(w(V ))

2

t

P

i=1

w

v

i

whi h on ludesthe proofsin e the weight ofthe greedy solution is

P

t

i=1

w

v

i

.

An easy orollary of theorem 8 is that whenever

α

w

(G)

≤ w(V )/k

, then WGREEDY a hieves in

G

ratio

k/(∆ + 1)

for WIS.Consequently, inordertodevise

O(k/∆)

-approximations forevery graph, one an fo us him/herself on the approximation of WIS in graphs with large weighted

independen enumber. For aninteger fun tion

k(n)

≤ n

, we denotebyWIS

k

the lassofgraphs with

α

w

(G) > w(V )/k

; for reasonsof simpli itywe assimilate this lasswith the orresponding WIS-subproblem. The work of [3 ℄ and its improvement by [1℄ show how IS

k

(the unweighted version of WIS

k

) an be approximated within

O(n

ǫ

k

−1

)

where

ǫ

k

depends only on

k

(note that using the redu tion of [16 ℄, one immediately gets a ratio

O(w(V )

ǫ

k

−1

₎

. Here, we perform a

further improvement bygeneralizingthe algorithm of[1 ℄ to the weighted ase.

4.2 Graph oloring and the approximation of WIS

k

Inthisse tion, weadapt themethodof[1℄to theweighted IS- aseandrelatetheapproximation

of WIS

k

to the oloring of a lassof graphs ( alled

G

k+1

in the sequel) ontaining the

(k + 1)

- olorable graphs. We rst re alltwo notions very loselyrelated, the Lovász

θ

-fun tionand the orthonormal representation of agraph.

Denition 2. ([14℄)Consider a graph

G = (V, E)

,

V =

{1, . . . n}

: the Lovász

θ

-fun tion

θ(G)

of

G

isthe maximum valueof

P

n

i,j=1

b

ij

where

B = (b

ij

)

i,j=1...n

rangesover allpositive semidenitesymmetri matri eswith tra e1andsu h that

b

ij

= 0

for every pair

(i, j)

,

i

6= j, ij ∈ E

;

the orthonormal representation of

G

is a system of

n

unit ve tors

( ~

u

i

)

i=1...n

in a

n

-dimensional Eu lidean spa e, su h that for every

(i, j)

su h that

ij /

∈ E

,

u

~

i

and

u

~

j

are orthogonal.

Proposition 2. ([14 ℄) Given a graph

G

, the following holds:

•

given an orthonormal representation

( ~

u

i

)

i=1...n

of

G

, and a unit ve tor

d

~

,

∃i ∈ {1, . . . n}

su h that

θ(G)

≤ 1/(~

d

· ~

u

i

)

2

;

For an integer fun tion

ℓ = ℓ(n)

≤ n

, set

G

ℓ

=

{G : θ( ¯

G)

≤ ℓ}

; by proposition 2, every

ℓ

- olorable graph belongs to

G

ℓ

. In [12℄ it is shown that every graph in

G

ℓ

with order

n

and maximum degree

∆

an be olored with

O(min

{∆

1−2/ℓ

_log

3/2

_{n, n}

1−3/(ℓ+1)

_log

1/2

_n

_})

olors in

randomized polynomial time ([12 ℄. The algorithm of[12℄ hasbeenderandomized laterby[15 ℄.

Theorem 9. Let

k = k(n)

≤ n

be an integer fun tion and

f

k

(x, y)

be a fun tion from

IN

_{× IN}

to

IN

, non-de reasing with respe t to both

x

and

y

. If there exists a

O(T (n))

algo-rithm A omputing, for every graph

G

∈ G

k+1

,

|V | ≤ n

, a

f

k

(n, ∆)

- oloring, then there exists a

O(max

{n

3

_{, T (n)}

_})

PTAA for WIS

k

guaranteeing ratio

1/(k(k + 1)f

k

(n, ∆))

.

Proof. Let

(G = (V, E), ~

w)

be an instan e ofWIS

k

of order

n

and

G

w

be asin denition 1 of se tion2. Sin e the weights aresupposedto be integral and

α

w

(G) > w(V )/k

, we have

α

w

(G) = α (G

w

)

≥

w(V )

k + 1

+

w(V )

k(k + 1)

+

1

k

.

In[7℄ itis shownthat:

• θ(G

w

)

is equalto the maximum valueof

P

n

i,j=1

√

w

i

w

j

b

ij

, where

b

ij

are asindenition 2;

•

one an ompute, in

O(n

3

₎

by the ellipsoid method, a positive semidenite symmetri

matrix

B = (b

ij

)

i,j=1...n

satisfying







n

P

i=1

n

P

j=1

√

_w

i

w

j

b

ij

≥ θ(G

w

)

−

_2k

1

> 0

b

ij

= 0

ij

∈ E

(5)

Given that

B

is symmetri and positive semidenite, there exist

n

ve tors

u

ˆ

i

, i

∈ {1, . . . n}

in

IR

n

(seen asEu lidean spa e) su h that,

∀(i, j) ∈ {1, . . . n}

2

,

u

ˆ

i

· ˆu

j

= b

ij

; in parti ular, we have

b

ii

≥ 0

. For our purpose we just need to ompute

n

ve tors

u

~

i

satisfying the following expression(6) for

ǫ = 1/(2knw(V ))

:



































~

u

i

· ~

u

j

= b

ij

i

6= j

|~

u

i

|

2

= b

ii

+ ǫ > 0

n

P

i=1

|~

u

i

|

2

= Tr(B) + nǫ = 1 + nǫ









n

P

i=1

√

_w

i

u

~

i









2

≥

P

n

i=1

n

P

j=1

√

_w

i

w

j

b

ij

(6)

Su h ve tors an be seen as non-zero approximations of

u

ˆ

i

,

i = 1 . . . n

. The system of

ve -tors

( ~

u

i

)

i=1...n

an be omputed ([6℄) by applying Cholesky's de omposition to the symmetri

positive denite matrix

B + ǫI

(where I is the identity matrix); one gets (in

O(n

3

₎

) an

n

-dimensional triangular matrix

U

su h that

B =

t

_{U U}

_{− ǫI}

. Let then

( ~

u

i

)

i=1...n

be the olumns of

U

; they learly satisfyexpression(6).

Set, for

i

∈ {1, . . . n}

,

~

d =

n

P

i=1

√

_w

i

u

~

i









n

P

i=1

√

_w

i

u

~

i









~

z

i

=

~

u

i

| ~

u

i

|

andnotethat

z

~

i

, onstitutesan orthonormalrepresentation of

G

¯

. Withoutlossofgenerality, we an assumethat

(~

d

· ~

z

1

)

2

_{≥ (~}

_d

_{· ~}

_z

j = max

(

i

∈ {1, . . . , n} :

i−1

X

l=1

w

l

≤

w(V )

k(k + 1)

)

K = G [

_{v

i

: i = 1, . . . , j

}] .

Then,

(~

d

· ~

z

j

)

2

_{≥ 1/(k + 1)}

whi himplies

K

∈ G

k+1

.

Proof of lemma 3. ByCau hy-S hwarzinequalityand expressions(5) and(6), we get:

(1 + nǫ)

n

X

i=1

w

i



~

d

_{· ~z}

i



2

=

n

X

i=1

|~

u

i

|

2

!

_n

X

i=1

w

i



~

d

_{· ~z}

i



2

≥

n

X

i=1

~

d

_·

√

w

i

u

~

i

!

2

=











n

X

i=1

√

w

i

u

~

i











2

≥

n

X

i=1

n

X

j=1

√

_w

i

w

j

b

ij

≥ θ (G

w

)

−

1

2k

≥ α (G

w

)

−

1

2k

≥

w(V )

k + 1

+

w(V )

k(k + 1)

+

1

2k

.

On the other hand, sin e for

i

∈ {1, . . . n}

,

d

~

· ~z

i

≤ 1

, (re all that

d

~

and

z

~

i

,

i = 1 . . . n

areunit ve tors),

(1 + nǫ)

n

X

i=1

w

i



~

d

· ~z

i



2

≤

n

X

i=1

w

i



~

d

· ~z

i



2

+

1

2k

.

Consequently,

n

X

i=1

w

i



~

d

_{· ~z}

i



2

≥

w(V )

k + 1

+

w(V )

k(k + 1)

.

(7) Re all that

(~

d

· ~

z

1

)

2

_{≥ (~}

_d

_{· ~}

_z

2

)

2

≥ . . . (~

d

· ~

z

n

)

2

. We have

(~

d

· ~

z

j

)

2

_{≥ 1/(k + 1) > 0}

sin e, in the opposite ase,

n

X

i=1

w

i



~

d

_{· ~z}

i



2

≤

j−1

X

i=1

w

i

+

n

X

i=j

w

i



~

d

_{· ~}

z

j



2

<

w(V )

k + 1

+

w(V )

k(k + 1)

whi h ontradi ts expression (7).

As noti ed in [1℄,

(~

d

· ~

z

j

)

2

_{≥ 1/(k + 1) > 0}

implies that the subgraph

K

of

G

indu ed by verti es

v

i

,

i = 1 . . . j

, satises

θ( ¯

K)

≤ k + 1

; this follows from the fa t that

z

~

i

,

i

∈ {1, . . . j}

, is an orthonormal representation of

¯

K

with value (see [14 ℄)less than

1/(~

d

· ~

z

j

)

2

_{≤ k + 1}

. Note

thatthisexpressionholdsfor the unweighted

θ

-fun tionof

G

; so,

K

∈ G

k+1

. This ompletes the proofof lemma3.

Lemma 3is originallyproved in [1℄ for the ase of(unweighted) IS. Asone an seefrom the

above proof,extension of itfor WISisnon-trivial.

Let us now ontinue the proof of theorem 9. By lemma 3, algorithm A ( laimed in the

statement of theorem 9) omputes a

f

k

(n, ∆)

- oloring of

K

. Then, the maximum-weight olor is an independent set of

K = (V

K

, E

K

)

(and of

G

) oftotal weight at least

w(V

K

)/f

k

(j, ∆(K))

. Sin e

w(V

K

)

≥ w(V )/(k(k + 1))

(see the denition of

j

),

j

≤ n

and

∆(K)

≤ ∆

, the maximum-weight olor is an independent set of

G

of weight at least

w(V )/(k(k + 1)f

k

(n, ∆))

. On the other hand

α

w

(G)

≤ w(V )

, and onsequently the following algorithm WIS_k guarantees ratio

(1) ompute the matrix

B

= (b

ij

)

i,j=1,...n

by the ellipsoid method; (2)

ǫ

← 1/(2knw(V))

;

(3) ompute the Cholesky's de omposition of

B

+ ǫI

; (4) ompute ve tors

~d

and

z

~

i

, i

∈ {1, . . . n}

;

(5) sort verti es in de reasing order with respe t to

(~d

· ~

z

i

)

2

; (6) ompute j and K; (7) IF

(~d

· ~

z

j

)

2

_{≥ 1/(k + 1)}

(8) THEN

(9) all A to ompute a oloring

(C

1

, . . . C

l

)

of K; (10)

S

← argmax{w(C

i

) : i = 1 . . . l

}

;

(11) omplete S to obtain a maximal independent set of G;

(12) ELSE

S

← ∅

; (13) FI

(14) OUTPUT S;

END. (*WIS_k*)

To on lude the proof, let us note that lines (1) and (3) are exe uted in

O(n

3

₎

, line (5)

in

O(n log n)

and line(11) in

O(n

2

₎

. Finally, the time- omplexityof line(9) isbounded above

by

T (n)

.

Of ourse, unless P=NP, in lusion of a graph in WIS

k

annot be polynomially de ided. However, algorithm WIS_k runson everygraph within the same omplexity; in fa t(by

instru -tion (7)), if

K

∈ G

k+1

, then algorithm A is alled to return a non-empty independent set

S

; otherwise,

S =

∅

. Consequently, if

S

6= ∅

(in parti ular if

G

∈ WIS

k

), algorithm WIS_k guar-antees the ratio laimed by theorem 9; in the opposite ase, the input-graph does not belong

to WIS

k

.

UsingforA,thederandomizedversionof[12 ℄presentedin[15℄,theorem9leadstothefollowing

approximation resultfor WIS

k

, for

2

≤ k(n) ≤ n

. Corollary 2.

ρ

WISk

≥ Θ

1

k(k + 1)∆

1−2/(k+1)

_log

3/2

_n

!

.

4.3 The main result

Considernowthe followingalgorithm, theworst-time omplexityofwhi histhesameastheone

of WIS_k.

BEGIN (*WIS*)

OUTPUT

S

← argmax{w(WIS

_

k(G)), w(WGREEDY(G))

}

; END. (*WIS*) By theorems8 and9,

∃c

′

su h that

ρ

WIS

≥ min

(

k

∆ + 1

,

c

′

k(k + 1)∆

1−2/(k+1)

_log

3/2

_n

)

.

By an easy but somewhat tedious algebra, one an prove that the righthand side of the above

inequalityis at leastaslarge as

min

(

k

∆ + 1

,

c

k

(k + 1)

(1+3k)/2

_log

3(k+1)/4

_n

)

Theorem 10. For any xed integer

k

≥ 2

and for

t = 3(k + 1)/4

ρ

WIS

(G)

≥ min



_k

(∆ + 1)

, O



log

−t

n





.

Furthermore,inthe asewhere

min

{k/(∆ + 1), O(log

−t

_n)

_{} = O(log}

−t

_n)

,

∆ + 1

≤ O(log

t

_n)

and

then algorithm WGREEDYalreadyguarantees awonderful (giventhe resultin [11℄)approximation

ratio.

Corollary 3. Consider agraph

G

and

k

≥ 2

. Then,there exists

t > 0

su hthat atleast oneof the two following onditions holds:

1. algorithm WGREEDY a hieves ratiobounded belowby

O(log

−t

_n)

;

2. algorithm WIS a hieves ratio bounded below by

k/(∆ + 1)

. Onthe other hand,letus revisit expression

min

(

k

∆ + 1

,

c

k

(k + 1)

(1+3k)/2

_log

3(k+1)/4

_n

)

and set

k = log n/(3 log log n)

; then,

∃n

0

≥ 1, ∀n ≥ n

0

,

(1/c

k

_{)(k + 1)}

(1+3k)/2

_log

3(k+1)/4

_n

_{≤ n}

4/5

and, sin e instan es with

n

≤ n

0

an be solved by exhaustive sear h in onstant time, the following theoremholds and on ludes the se tion.

Theorem 11.

ρ

WIS

(G)

≥ min



_{log n}

3(∆ + 1) log log n

, O



n

−4/5





.

5 Approximating maximum-weight lique

We des ribe in this se tion a new redu tion, working for both weighted and unweighted ases,

between independent setand lique. Letus notethatthe lassi al orresponden eindependent

setin

G

- liquein

¯

G

 preserves onstantand

ρ(n)

ratiosbutitdoesnotworkforratiosdepending on

∆

.

Let

G = (V, E)

be an instan e of KL and let

V =

{1, . . . , n}

. We onsider the

n

graphs

G

i

= G[

{i} ∪ Γ(i)]

,

i = 1, . . . , n

anddenote by

n

i

theirrespe tive orders; wealso onsiderthe

n

graphs

G

¯

i

. Then, the following proposition holds (re all that in every graph

G

, the size of a maximum liqueis never greater than

∆(G)

).

Proposition 3. For every

i

∈ {1, . . . , n}

, the following three fa ts hold: 1.

n

i

≤ ∆(G) + 1

and

∆( ¯

G

i

)

≤ ∆(G)

;

2. liques(resp., independent sets)of

G

i

(resp.,

G

¯

i

) are also liques(resp., independentsets) of

G

(resp.,

G

¯

);moreover,

∃i

∗

su hthatthemaximum lique(resp.,independentset)of

G

i

∗

(resp.,

G

¯

i

∗

) isexa tly the maximum lique (resp., independent set) of

G

(resp.,

G

¯

); 3. items1and2 holdalso forWKLandWISifwe onsiderweighted liquesandindependent

sets.

Let A be a PTAA for WIS a hieving ratio

ρ(n, ∆)

; we shall use itto produ e a WKL-solution for

G

. This anbe donebythe following algorithm.

OUTPUT

K

← argmax{w(A(¯G

i

)) : i = 1, . . . , n

}

; END. (*WKL*)

Sin e

K

is an independent set of

G

¯

, it is a lique (of the same weight) in

G

. Moreover, by items 1 and 2 of proposition 3, algorithm WKL a hieves, in polynomial time, approximation

ratio

ρ(∆ + 1, ∆)

for WKL.

Theredu tionjustdes ribedis,toourknowledge,therstonepreserving

ρ(∆)

ratiosbetween independent set and lique (inboth weighted and unweighted ases).

Given the ISresultof [3 ℄and the one oftheorem 3we on lude the following.

Theorem 12. There exist PTAAsfor weighted andunweighted versions of KL su h that

ρ

KL

(G)

≥ Θ

log

2

∆

∆

!

ρ

WKL

(G)

≥ Θ

log

2

∆

∆ log log ∆

!

.

Theresultsoftheorem12represent,toourknowledge,therst

ρ(∆)

ratiosforthe lique-problem. 6 Maximum

l

- olorable indu ed subgraph and minimum hromati sum

6.1 Maximum

l

- olorable indu ed subgraph

Let us note that we an assume

l < ∆

, be ause, ifnot, then

G

′

_{= G}

(see the denition of C

l

in se tion 1 and re all that there exists the polynomial-time oloring-algorithm of [13℄ always

guaranteeing a

∆

- oloring of

G

, unless

G

isa

(∆ + 1)

- lique). Consider now, for

l

∈ {1, . . . , ∆ − 1}

, the graph

l

_{G = (}

l

_V,

l

_E)

dened asfollows:

l

_V

_{= V}

× {1, . . . , l}

l

_{E =}



(v, i)(v

′

, j) :



(i = j)

∧ vv

′

∈ E



∨



(i

6= j) ∧ (v = v

′

)



For WC

l

the weight of vertex

(v, i)

equals

w

v

,

∀i = 1, . . . , l

. Clearly,

|

l

V

| = ln

∆(

l

G) = ∆(G) + l

_{− 1}

µ(

l

G) = µ + l

_{− 1}

(infa t, forall

(v

k

, i)

∈

l

_V

,

v

k

∈ V

,

i

≤ l

, the degreeof

(v

k

, i)

equalsthe degree of

v

k

plus

l

− 1

; the same holdsfor the averagedegree

µ(

l

_G)

).

The following holds:

•

let us onsider an independent set

S

⊂

l

_V

of

l

_G

; the family

S

i

=

{v ∈ V : (v, i) ∈ S}

,

i = 1, . . . , l

, isa olle tionofmutuallydisjoint independentsetsof

G

; sothe graph

G[

∪

i

S

i

]

is

l

- olorable;

•

onversely, for every

l

- olorable subgraph

G

′

_{= (V}

′

_{, E}

′

₎

of

G

and for every

l

- oloring

(S

1

, . . . , S

l

)

of

G

′

, the set

S =

{(v, i) : i ∈ {1, . . . , l}, v ∈ S

i

}

is anindependent set of

l

_G