Approximation of weighted hereditary induced subgraph maximization problems

Laboratoire d'Analyse et Modélisation de Systèmes pour

l'Aide à la Décision

CNRS UMR 7024

CAHIER DU LAMSADE

160

Janvier 1999

Approximation of weighted hereditary induced

subgraph maximization problems

Résumé ii

Abstra t ii

1 Preliminaries 1

2 The generi result 1 3 Important orollaries of Theorem 1 4 3.1 Maximum-weight independent setand maximum-weight lique . . . 4 3.2 Maximum-weight

ℓ

- olorable indu edsubgraph . . . 5 4 About ontinuous redu tions between weighted and unweighted versions of

re her he de sous-graphes induits pondérés

Résumé

Le but entral de et arti le est l'étude de l'approximation polynomiale de problèmes héréditaires de maximisation portant sur la re her he de sous-graphes induits pondérés. Aprèsavoirdonnéunrésultatgénérald'approximationpour ette lassedeproblèmes,nous montronsque elui- ia plusieurs orollairesimportants, parexemple, il améliored'un fa -teurd'

O(log log n)

lemeilleurrapportd'approximation onnupourlestablemaximum pon-déré.

Mots- lé: problèmes ombinatoires, omplexité, algorithmepolynomiald'approximation, NP- omplétude,stable,problèmehéréditaire.

Approximation of weighted hereditary indu ed-subgraph maximization problems

Abstra t

Themainpurposeofthispaperistostudypolynomialapproximationofweighted hered-itaryindu edsubgraphproblems. Aftergivingageneralapproximationresultforthis lass ofproblems, weshowthat theresult hasmany important orollariessin e, forinstan e, it allowsimprovementofthebest-knownapproximationratioforweightedindependentsetby afa torof

O(log log n)

.

Keywords: ombinatorial problems, omputational omplexity,polynomial-time approxi-mationalgorithm, NP- ompleteness,independentset, hereditaryproblem.

We onsider NP- omplete optimization graph-problems

Π

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

1

satisfying a non-trivial hereditary property

π

. Let

G

be the lass of all the graphs; agraph-property

π

is a mapping from

G

to

{0, 1}

, i.e., for a

G ∈ G

,

π(G) = 1

i

G

satises

π

and

π(G) = 0

, otherwise;

π

ishereditary ifwhenever itissatised by agraph itisalso satised byevery one ofits indu ed subgraphs;it isnon-trivial ifitis truefor innitelymany graphsand false for innitelymany ones([2℄). The property

π

: isa lique, or 

π

: isanindependent set aretypi alexamplesofsu hnon-trivial hereditaryproperties andthe maximum lique-problemor themaximum independent setarein their turntypi alexamplesof hereditaryindu edsubgraphproblems. Ageneralisation ofthe lassofproblemsjustintrodu ed isthe one where we onsiderthat positive weights areasso iatedwith the verti esof the input-graphs. Givenagraph

G

,the obje tiveofaweightedindu ed subgraphproblemisto determine an indu edsubgraph

G

∗

of

G

su h that

G

∗

satises

π

and,moreover, the sumof the weightsof the verti es of

G

∗

mustbethe largest possible.

Remark 1. By denitionof

π

(asa mapping fromthe lassof graphsto

{0, 1}

), itis obvious that

π

is ontext-free;inotherwords, ifagraph

G

satises

π

,then

G

satises

π

intoeverygraph whose

G

is anindu ed subgraph.

Given a polynomial-time approximation algorithm (PTAA) A that solves a maximization NP- omplete problem

Π

(we denote by

WΠ

the weighted version of

Π

), the quality of its approxi-mation behaviour is expressed by the ratio

ρ

of the value (size, or weight, or whatever) of the solution found by the algorithm to the optimal (obje tive) value of

Π

; the smallest su h ratio over all graphs is the approximation ratio

ρ

A

of the algorithm; the best-known ratio

ρ

Π

over allPTAA'ssolving

Π

istheapproximationratiofor

Π

. Instan e-dependingapproximationratios for NP- omplete graph-problems are ommonlyexpressed bymeans of(one or more) the three standardgraph-parameters, namelythe order

n

ofthe inputgraph, themaximum vertex-degree and the average vertex-degree of the input graph. In what follows, we will denote by

ρ

Π

(n)

(resp.,

ρ

WΠ

(n)

) the best-known

n

-depending approximation ratio for

Π

(resp., for

WΠ

). Note thatsin e we dealwith maximization problems ratio

ρ

isnon-in reasing in

n

.

Given a vertex-weighted graph

G = (V, E, ~

w)

of order

n

with weight-ve tor

w

~

, we denote by

w

i

the weight of avertex

v

i

∈ V

, and by

w

max

(G)

and

w

min

(G)

the largest and the smallest vertex-weights, respe tively. We denote by

β

w

(G)

and by

β

′

w

(G)

the weight of a best

WΠ

-solution and the weight of a maximal

2

Π

-solution of

G

, respe tively. For a

V

′

_{⊆ V}

, we denote by

G[V

′

_]

the subgraphof

G

indu ed by

V

′

and by

n

′

its order. Given a subgraph

G[V

(k)

_]

of

G

, we denote by

S

∗

_(G[V

(k)

_])

a maximum-size

Π

-solution of

G[V

(k)

_]

; also, given any

Π

-solution

S

′

of asubgraph

G

′

of

G

, wedenote by

w(S

′

₎

the weight of

S

′

. 2 The generi result

Thisse tionis mainly devotedto the proof ofthe following theorem.

Theorem 1. Consider a hereditary property

π

, an indu ed subgraph problem

Π

stated with respe t to

π

and the weighted version

WΠ

of

Π

(we suppose that weights are positive). Let

1

Inotherwords,afeasiblesolutionof

Π

isasubsetoftheverti esoftheinput-graph. 2

Throughoutthepaperweusethenotationmaximalsettodenoteasetwhi hismaximalforthein lusion withrespe ttoagivenproperty

π

;inotherwords,aset

S

ofitemsismaximalwithrespe tto

π

if(a)

S

fulls

π

and(b)ifweaddanyitemin

S

,thentheresultingsetdoesnotfull

π

;asetismaximumwithrespe tto

π

if itisthelargestbetweenthemaximalsets(withrespe tto

π

).

M > 2

be an integer andlet

V

(i)

_{= {v}

j

∈ V : w

max

(G)/M

i

< w

j

≤ w

max

(G)/M

i−1

}

,

i = 1, . . .

Finally, let

x = sup{ℓ : β

w

(G[∪

1≤i≤ℓ

V

(i)

_{]) < β}

w

(G)/2, β

w

(G[∪

1≤i≤ℓ+1

V

(i)

]) ≥ β

w

(G)/2}

. Then,

ρ

WΠ

(n) ≥ max



_M

x

2n

, min



_{M − 2}

2M

2

_x

ρ

Π

(n),

1

M

2

ρ

Π

(n)



.

Proof: For the sequel, we set

G

x

= G[∪

1≤i≤x

V

(i)

_]

,

G

x+1

= G[∪

1≤i≤x+1

V

(i)

_]

and

G

d

= G[V \

∪

1≤i≤x

V

(i)

]

(of ourse,

β

w

(G

d

) ≥ 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: The algorithm laimed onsists of simplytaking

v

∗

_{= argmax}

v

i

∈V

{w

i

}

as

WΠ

-solution andthenaddingverti esinsu hawaythatthesolutionnallyobtainedremainsfeasible

3 for

G

. Moreover, notethat, sin everti es

v

k

su hthatthe graph

({v

k

}, ∅)

doesnot satisfy

π

havebeen removed, the graph

({v

∗

_{}, ∅)}

is already a feasible solution. Consequently

β

′

w

(G) ≥ w

max

(G)

,

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,

ρ(n) =

β

′

w

(G)

β

w

(G)

≥

M

x

2|S

∗

_(G

d

)|

≥

M

x

2n

(1) q.e.d.

Remark 2. 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

.

For the rest of the proof, we distinguish two ases with respe t to

β

w

(G

x

)

, namely

β

w

(G)/2 ≥

β

w

(G

x

) ≥ ((M − 2)/2M )β

w

(G)

and

β

w

(G

x

) ≤ ((M − 2)/2M )β

w

(G)

, respe tively.

We rst onsider the ase

β

w

(G)/2 > β

w

(G

x

) ≥ ((M − 2)/2M )β

w

(G)

. Then the following lemmaholds.

Lemma 2. Let

β

w

(G)/2 ≥ β

w

(G

x

) > ((M − 2)/2M )β

w

(G)

,

p = argmax

1≤i≤x

{β

w

(G[V

(i)

])}

,

|V

(p)

_{| = n}

(p)

and

S

∗

_(G[V

(p)

_])

be themaximum-size

Π

-solutionin

G[V

(p)

_]

. IfthereexistsaPTAA for

Π

providing a maximal solution

S

(p)

su h that

|S

(p)

_{| ≥ ρ}

Π

(n

(p)

)|S

∗

(G[V

(p)

])|

, then one an solve

WΠ

in

G

, in polynomialtime, withinratio

((M − 2)/2xM

2

_)ρ

Π

(n)

. Proof: Obviously,

β

w

(G

x

) ≤ xβ

w

(G[V

(p)

_])

and

β

w

(G

x

) ≤ x|S

∗

_(G[V

(p)

_])|w

max

(G)/M

p−1

(by Remark2). So,

β

w

(G) ≤ (2M /(M − 2))x|S

∗

_(G[V

(p)

_])|(w

max

(G)/M

p−1

)

.

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

ρ

Π

(n) < 1

for

Π

in

G[V

(p)

_]

onstru ts a solution

S

(p)

of

WΠ

of weight at least

|S

(p)

_|w

min

(G[V

(p)

]) ≥

|S

(p)

|w

max

(G)/M

p

. Note that (byRemark 1)

S

(p)

is

Π

-feasible for

G

. Moreover, starting from this solution, one an greedily augment it in order to nally produ e a maximal

WΠ

-solution for

G

. This nalsolutionveries

β

′

w

(G) ≥ |S

(p)

|w

max

(G)/M

p

. Combination ofthe above expressions for

β

′

w

(G)

and

β

w

(G)

yields

ρ(n) =

β

′

w

(G)

β

w

(G)

≥



_{M − 2}

2xM

2



_|S

(p)

_|

|S

∗

_(G[V

(p)

_])|

!

≥

M − 2

2xM

2

ρ

Π

(n

(p)

_{) ≥}

M − 2

2xM

2

ρ

Π

(n)

(2) and this on ludes the proof of the ase

β

w

(G)/2 > β

w

(G

x

) ≥ ((M − 2)/2M )β

w

(G)

and of the lemma.

Lemma 3. Let

β

w

(G

x

) ≤ ((M − 2)/2M )β

w

(G)

and

S

∗

_(G[V

(x+1)

_])

be a maximum-size

Π

-solution of

G[V

(x+1)

_]

. If there exists a PTAA for

Π

providing a maximal

Π

-solution

S

(x+1)

of ardinality at least

ρ

Π

(n

(x+1)

₎

-times the ardinality of

S

∗

_(G[V

(x+1)

_])

, then one an solve

WΠ

in

G

, in polynomialtime, within ratio

(1/M

2

_)ρ

Π

(n)

.

Proof: Notethat sin 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)

is not 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[V

(x+1)

])) ≤ β

w

(G[V

(x+1)

])

β

w

(G

x+1

) = w(S(G

x

)) + w(S(G[V

(x+1)

])) ≤ β

w

(G

x

) + β

w

(G[V

(x+1)

])

≤

M − 2

2M

β

w

(G) + β

w

(G[V

(x+1)

_])

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 Remark2 yield, aftersome easy algebra,

β

w

(G) ≤ |S

∗

_(G[V

(x+1)

_])|w

max

(G)/M

x−1

.

Aspreviously,supposethataPTAAprovides,in polynomialtime,amaximalsolution

S

(x+1)

for

Π

in

G[V

(x+1)

_]

, the ardinalityofwhi hisatleast

ρ

Π

(n

(x+1)

_)|S

∗

_(G[V

(x+1)

_])|

. Then, one an greedily augment this solution in order to produ e a maximal

Π

-solution for

G

of total weight

β

′

w

(G) ≥ |S

(x+1)

|w

max

(G)/M

x+1

. Combination ofexpressions for

β

′

w

(G)

and

β

w

(G)

yields

ρ(n) =

β

′

w

(G)

β

w

(G)

≥



₁

M

2



_|S

(x+1)

_|

|S

∗

_(G[V

(x+1)

_])|

!

≥

1

M

2

ρ

Π

(n

(x+1)

_{) ≥}

1

M

2

ρ

Π

(n)

(3) and this on ludes the proofof the ase

β

w

(G

x

) ≤ ((M − 2)/2M )β

w

(G)

andof the lemma. Remark 3. For the ase where

x = 0

, i.e.,

β

w

(G[V

(1)

_{]) ≥ β}

w

(G)/2

, arguments similar to the ones of the proof of Lemma 3 lead to an approximation ratio

ρ

WΠ

(n) = β

′

w

(G)/β

w

(G) ≥

ρ

Π

(n)/2M

, better than the oneof expression(3).

Consider now the following algorithm where we take up the ideas of Lemmata 1, 2 and 3 and where, for a graph

G

′

, we denote by

A(G

′

₎

the solution-set provided by the exe ution of the

Π

-PTAA A onthe unweighted versionof agraph

G

′

. BEGIN (*WA*) 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)

← {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 is easy to see that, sin e

M

is xed, worst ases for the respe tive ratiosarerepresented byexpressions (1)and (2);so,

ρ

WΠ

(n) ≥ ρ

WA

(n) ≥ max



_M

x

2n

, min



_{M − 2}

2M

2

_x

ρ

Π

(n),

1

M

2

ρ

Π

(n)



(4)

areexponential in

n

.

The proof of Theorem 1 an be seen as a redu tion transforming every PTAA A a hieving ratio

ρ

A

(n)

for

Π

into a PTAA for W

Π

a hieving, at worst, ratio

e = Ω(ρ

A

(n)

/x)

. Quantity

1/x

is usually alled expansion of the redu tion (see [7℄). By expression (1) and by the fa t that the approximation ratio of any PTAA for W

Π

mustbe lessthan 1 (W

Π

being a maximization problem),

x = O(log

M

n)

(in fa t, as we will see in the next se tion, it an be mu h smaller depending on the form of

ρ

Π

(n)

). But even su h an expansion is a meaningful improvement with respe t to expansions indu ed by redu tions between weighted  no-weighted versions of problems whi h do not admit onstant-ratio PTAA's. For example, for the ase of IS, if we take into a ount thatnoPTAA an guaranteexed onstant ratio forit, redu tionsof [7℄have expansion

1/n

2

.

3 Important orollaries of Theorem 1

3.1 Maximum-weight independent set and maximum-weight lique

Consider now one of the most lassi al NP- omplete problems, the maximum independent set. Given agraph

G = (V, E)

, anindependent setisasubset

V

′

_{⊆ V}

su hthatwhenever

{v

i

, v

j

} ⊆

V

′

,

v

i

v

j

∈ E

/

, and the maximum independent set problem (IS) is to nd an independent set of maximum size. As we have already noted in se tion 1, property is an independent set is hereditary (the subset of an independent set is an independent set); moreover, it is non-trivial (notallsets

V

′

_{⊆ V}

areindependent). Averywell-knownandlargelystudiedgeneralizationofIS is its weighted version (denoted by WIS); here, we asso iate positive weights with the verti es of

G

andthe obje tive be omesto maximize the sum ofthe weights ofan independent set.

In terms of

n

, the best known approximation ratio for IS is, to our knowledge, a hieved by the IS-PTAA of Boppana and Halldórsson ([1 ℄):

ρ

IS

(n) = Θ(log

2

_n/n)

. Embedding it in expression(2), we get

ρ(n) ≥



_{M − 2}

2xM

2



Θ

log

2

_n

n

!

= Θ

log

2

_n

nx

!

.

(5)

For instan e, if

x ≤ log log n

, expression (5) indu es

ρ(n) ≥ Θ(log

2

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

, while, if

x ≥ log log n

, then expression (1) guarantees

ρ(n) ≥ Θ((log n)

log M

_/n)

; on the other hand, expression (3)always guarantees

ρ(n) ≥ Θ(log

2

_n/n)

. So, byexpression (4), the approximation ratioguaranteedbyalgorithm WA(supposing

M > 8

) alledwhenusedtosolveWISis,atworst,

ρ

WA

(n) ≥ Θ

log

2

n

n log log n

!

.

(6)

Expression(6)and the dis ussionaboveintrodu e the following orollary. Corollary 1.

ρ

WIS

(n) = Ω(log

2

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

.

Theaboveresultimprovesbyafa tor

O(log log n)

the best-knownapproximationratio(fun tion of

n

) for WIS(

Θ(log

2

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

2

₎₎

, dueto Halldórsson ([5℄)).

Finally,letusnotethatthesameapproximationresultholdsforanotherfamousNP- omplete problem, the maximum weighted lique(see [2℄for the statement ofthis problem).

3.2 Maximum-weight

ℓ

- olorable indu ed subgraph

Given a graph

G = (V, E)

and a positive onstant

ℓ

, the problem of the maximum

ℓ

- olorable indu ed subgraph (denotedbyC

ℓ

)istondamaximum-orderindu edsubgraph

G

′

_{= G[V}

′

_]

of

G

(

V

′

_{⊆ V}

)su h that

G

′

is

ℓ

- olorable (i.e., thereexistsa oloring for

G

′

of ardinality atmost

ℓ

). IntheweightedversionofC

ℓ

,denotedbyWC

ℓ

,positiveweightsareasso iatedwiththe verti es of

G

andtheobje tivebe omestondthemaximum-weight

ℓ

- olorableindu edsubgraph. On e more,propertyis

ℓ

- olorable ishereditary(ifthe verti es of agraph

G

an be feasibly olored byat most

ℓ

olors, then every subgraphof

G

indu edbyasubset of itsverti es an be olored by atmost

ℓ

olors)and non-trivial (given anumber

ℓ

every graph isnot

ℓ

- olorable).

The best-knownratiosfor C

ℓ

arethe onesof[4℄:

Θ((log n)

2

_/n)

(asfun tion of

n

) andof [3℄:

min{κ/µ, Θ((log log ∆)/∆)}

, for every xed

κ ∈ IN

+

, where

∆

,

µ

are the maximum and the average degrees of the graph, respe tively (as degree-fun tion). For WC

ℓ

, no approximation resultis, to our knowledge, mentioned until now.

Appli ation ofTheorem 1,usingthealgorithm of[4℄ asapproximationalgorithm for C

ℓ

(the algorithm A in algorithm WAof se tion2), leadsto the following orollary.

Corollary 2.

ρ

WCℓ

= Ω(log

2

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

.

4 About ontinuous redu tions between weighted and unweighted versions of independent set

In what follows, we use the term  ontinuous redu tion, introdu ed by Simon ([7℄), to denote the approximation-ratio-preserving redu tions. Informally, given two maximization problems

Π

and

Π

′

, a redu tion

Π ∝ Π

′

is alled ontinuous with an expansion of

e

ifea h polynomial-time

ρ

′

-approximation algorithm

A

′

for

Π

′

an be onverted into a polynomial-time

ρ

-approximation algorithm

A

for

Π

su h that

ρ ≥ eρ

′

. The orresponding notation is

Π

e

=⇒ Π

′

. It is very well-knownandlargelynoteduntilnowbymanyauthorsthatthe on eptof ontinuousredu tionsis strongly based upon the very restri tive hypothesisthat the approximation ratio

ρ

′

of

A

′

hasto beaxed onstant. Sin eISdoesnotadmit su halgorithms(in 1996ithasbeen provedthatit ishardtoapproximateISwithin

1/n

1−ǫ

,forany

ǫ > 0

,[6 ℄), ontinuousredu tionsproposeduntil nowbetweenISandWISfailtoprodu eWIS-ratiosasgoodastheonesknownforIS.Infa t,the bestknownratiosforISare

Θ(log

2

_n/n)

(intermsof

n

,[1℄)and

min{κ/µ, Θ((log log ∆)/∆)}

, for everyxed

κ ∈ IN

+

, where

∆

,

µ

arethemaximumandthe averagedegreesofthe graph, respe -tively([3 ℄). Ontheotherhand,thebest orrespondingratiosforWISare

Θ(log

2

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

(proved in this paper) and

3/(∆ + 2)

(see [5 ℄ for this last ratio), respe tively. We think that devising a ontinuous redu tion of onstant expansion between IS and WIS, or proving that su h a redu tiondoes not exist,is a very interesting open problem, the solution ofwhi h is far from beingobvious. In what follows, we give some easy ases of su h redu tions for restri tive WIS- lasses re ognizable in polynomial time. We denote by

α

w

(G)

and by

α

′

w

(G)

the weight of a maximum-weight independent set and the weight of an approximately maximum-weight independent set of

G

, respe tively.

Theorem 2. Consider a xed onstant

ℓ

andgraphs

G = (V, E)

of order

n

where atleast one of the following onditions isveried:

1. there exist atmost

ℓ

distin t weights, 2.

w

max

(G)/w

min

(G) ≤ ℓ

,

4.

w

max

(G) ≥ nw

′

, where by

w

′

we denote the se ond largest vertex weight.

For all these graph-families, there exist ontinuous redu tions of onstant expansion between IS andWIS.Consequently,in allthe above families of graphs,WIS an be approximated in polyno-mial timewithin

ρ

WIS

= ρ

IS

≥ max

(

Θ

log

2

_n

n

!

, min



_κ

µ

, Θ



_{log log ∆}

∆



)

for every xed

κ ∈ IN

+

.

Proof: We onsider the

ℓ

subgraphs,

G

(1)

_{, . . . , G}

(ℓ)

, of

G

, indu ed by the verti es of the same weight. Let

p = argmax

1≤i≤ℓ

{α

w

(G

(i)

)}

and

w

p

be the weight of the verti es of

G

(p)

; then,

α

w

(G) ≤ ℓα

w

(G

(p)

) = ℓw

p

|S

∗

(G

(p)

)|

. On the other hand, one an easily take

α

′

w

(G) =

max

1≤i≤ℓ

{w

i

|S

(i)

|}

, where

S

(i)

is the largest among the independent sets provided by the al-gorithms of [1℄ and of [3℄ for IS. It is easy to see that, in this ase,

ρ

WIS

= α

′

w

(G)/α

w

(G) ≥

w

p

|S

(p)

|/(ℓw

p

|S

∗

(G

(p)

)|) ≥ ρ

IS

/ℓ

and thatthis ratio veriesthe nal statement of the theorem, on ludingsothe proof ofitem 1.

For item 2, it su es to remove weights from the verti es of graph and to solve IS in

G

. Let

S(G)

be thelargestamong theindependent sets provided bythe algorithmsof[1 ℄and of[3℄ for ISand

S

∗

_(G)

bethe optimal one in

G

. Then, itiseasy to seethat

ρ

WIS

= α

′

w

(G)/α

w

(G) ≥

w

min

(G)|S(G)|/(w

max

(G)|S

∗

(G)|) ≥ ρ

IS

/ℓ

. On e more the nal statement of the theorem is veried and the proofof item 2is on luded.

The proof of item 3 isstandard and already used by many authors (see for example [7℄ for the proof of the result WIS(

k

)

1−ǫ

=⇒

IS, where WIS(

k

) denotes the version ofWIS where vertex-weights arebounded above by

n

k

,

k ≥ 1

; this proof unfortunately worksonly for xed- onstant ratios). Given aninstan e

G

w

ofWIS,one an onstru t aninstan e

G

of ISbyrepla ing every vertex

v

i

of weight

w

i

by an independent set

W

i

of size

w

i

; next, if two verti es

v

i

and

v

j

are linked by an edge in

G

w

, one draws a omplete bipartite graph between the two independent sets

W

i

and

W

j

ofsizes

w

i

and

w

j

,respe tively(inthe asewhereweightsarerationalswesimply repla e ea h weight by itself multiplied by the LCM of the weight-denominators). Now, note thatifanalgorithm deliversa solutionfor

G

(instan eof IS),one an dire tlyobtain asolution of

G

w

by simply repla ing the independent sets of the former by the orresponding verti es of the latter, this operation leading to a solutionfor

G

w

with the same valueasthe ardinality of the solution for

G

. Denoting by

n

w

and

∆

w

the order and the maximum vertex-degree of

G

w

, respe tively, and by

n

and

∆

the orresponding parameters of

G

, we have

n ≤ w

max

(G)n

w

,

∆ ≤ w

max

(G)∆

w

. It is easy to see that following the dis ussion just above and thanks to the fa tthatthe integralweightsarexed onstants,applyingthe algorithmsof[1℄and[3 ℄in

G

and retaining the best of the omputed solutions, a hieves for WISin

G

w

the same orders ofratios asfor ISin

G

, on ludingsothe proof ofitem 3.

For the proof of item 4, let us denote by

G

m

the subgraph of

G

indu ed by the verti es of weight

w

max

(G)

and by

S(G

m

)

the bestof the solutions provided whenrunning the algorithms of [1 ℄ and [3 ℄ on

G

m

. Obviously

α

′

w

(G) ≥ α

′

w

(G

m

) ≥ w

max

(G)|S(G

m

)|

. On the other hand,

α

w

(G) ≤ α

w

(G

m

)+(n−|S

∗

(G

m

)|)w

′

≤ w

max

(G)|S

∗

(G

m

)|+(n−|S

∗

(G

m

)|)w

max

(G)/n

. Divising, member-by-member, the above expressions for

α

′

w

(G)

and

α

w

(G)

, we get

ρ

WIS

=

α

′

_w

(G)

α

w

(G)

≥

|S(G

m

)|

|S

∗

_(G

_m

_{)| +}

n−|S

∗

_(G

m

)|

n

≥

|S(G

m

)|

|S

∗

_(G

m

)| + 1

.

ase,

α

′

w

(G)/α

w

(G) ≥ |S

∗

(G

m

)|/(|S

∗

(G

m

)| + 1) ≥ 1/2

(sin e

|S

∗

_(G

m

)| ≥ 1

). Consequently, on e more, the nalstatement of the theorem isveried and this on ludesthe proof of item 4 and ofthe theorem.

Referen es

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[2℄ P. Cres enzi and V. Kann. A ompendium of NP optimization problems. Available on www_address: http://www.nada.kth.se/~viggo/problemlist/ ompendium.html,1995. [3℄ M. Demangeand V.Th. Pas hos. Improved approximations for maximum independent set

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[4℄ M. M. Halldórsson. A still better performan e guarantee for approximate graph oloring. Inform. Pro ess. Lett.,45(1):1923, 1993.

[5℄ M. M. Halldórsson. Approximations via partitionning. JAIST Resear h Report IS-RR-95-0003F,Japan Advan ed Institute ofS ien e and Te hnology, Japan, 1995.

[6℄ J. Håstad. Clique is hard to approximate within

n

1−ǫ

. In Pro . FOCS'96, pages 627636, 1996.

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