Towards a general formal framework for polynomial approximation

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Laboratoire d'Analyse et Modélisation de Systèmes pour

l'Aide à la Décision

CNRS UMR 7024

CAHIER DU LAMSADE

177

Mars 2001

Towards a general formal framework for polynomial

approximation

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Résumé iii

Abstra t iv

I Introdu tion 1

1 A few words about polynomial approximation 1

1.1 Standardapproximation . . . 1

1.2 Dierential approximation . . . 2

2 A list of NPO problems 2 2.1 Hereditaryindu ed-subgraph maximization problems . . . 2

2.2 Someother NPOproblems. . . 3

3 Notations 4 4 The s ope of the paper and its main ontributions 5 II Roughing out a new approximation framework 7 5 Approximation hains: a generalization of the approximation algorithms 7 6 Approximation level 7 7 Convergen e and hardness threshold 8 8 Fun tionalapproximation-preserving redu tions 10 III A hieving approximation results in the new framework 11 9 Indu ed hereditary subgraph maximization problems 12 10Maximum independent set 14 10.1 A rst improvement of the approximation of the maximum-weight independent setvia theorem1 . . . 14

10.2 Approximation hains for maximum independent set with ratios fun tions of graph-degree . . . 16

10.2.1 An approximation hainwith ratio fun tion of(G). . . 16

10.2.2 An approximation hainwith improved ratio fun tion of(G) . . . 17

10.3 Towards (1=)-approximations . . . 17

10.4 Further improvements . . . 19

10.4.1 A weighted version ofTurán's theorem . . . 19

10.4.2 Graph oloringand the approximationof max_WIS k . . . 20

10.4.3 Themain result. . . 24

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12.1 `isa xed onstant . . . 26

12.2 `dependson graph-parameters . . . 26

13Minimum hromati sum 27

14Minimum oloring 29

14.1 An algorithmi hainfor minimum oloring with improved

standard-approxima-tionratio . . . 29

14.2 Afurther improvement ofthe standard-approximation ratio forgraph- oloring . . 31

15FP-redu tions between standard and dierential approximation 32

15.1 Su ient onditions for transferringresults betweenstandard anddierential

ap-proximation . . . 32

15.2 Binpa king . . . 33

15.3 Minimum vertex- overing . . . 33

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polynomiale

Résumé

Nousproposonsuneextensionduformalismedel'approximationpolynomialepermettant

d'envisagerdenouvelles lassesderésultats.Nous montronsd'une part ommentles

résul-tatsexistantss'intègrentdans e adreetd'autrepartquelstypesderésultatsouquestions,

jusqu'alors di iles àexprimer, y trouventune pla e naturelle. Nousexploitons e

forma-lisme pour établir des résultats d'approximation pour diérents problèmes NP-di iles.

Nous nous intéressons d'abord à la lasse des problèmes de sous-graphe induit de poids

maximum vériant une propriété héréditaire. Elle omprend notamment les problèmes de

stable, lique ou en ore sous-graphe induit `- olorable de valeur maximum. Nous

propo-sons d'abord une rédu tion en approximation qui transforme un rapport pour un

pro-blème non pondéré de la lasse ( as où tous les poids sont égaux) en un rapport de la

forme O(=logn) pour sa version pondérée générale.Cette appro he nous permet

d'amé-liorer le rapport d'approximation duproblème de stable de poids maximum (max_WIS):

le rapport nalement établi est le minimum entre une expression du type O(log k n=n) et O(log(G)=(k 2 (G)(loglog(G)) 2

),où(G) désigneledegrémoyendugrapheinstan e

et où k est une onstante quel onque. Cha un des deux termes orrespond àune avan ée

signi ativeparrapportàl'existant.Eneet,lepremiertermeestà ompareràO(log 2

n=n),

lemeilleurrapportfon tiondenétablipourle asparti uliernonpondéré(max_IS)

(pro-blèmedestable maximum).Le se ondtermequantàlui orrespondaupremierrapportdu

type(1=(G))pourmax_WIS((G)estledegrémaximumdugrapheinstan e).Dansun

se ondtemps, ennousappuyantsurdesrésultatspourla olorationdesgraphes,nous

pro-posons unalgorithmepolynomialpourmax_WISgarantissant ommerapportleminimum

entreO(n 4=5

)etO(loglog(G) =(G)).Cettefoisen ore,hormisle asoùmax_WISpeut

être appro hé ave le rapport O(n 4=5

) (l'approximation de max_IS garantissantle

rap-portn 1

estNP-di ilepourtout>0)notrealgorithmedonnea èsaupremierrésultat

d'approximationimpliquant un rapport (loglog(G) =(G)) pour max_WIS ( e i pour

tout(G)).Notonsquel'approximationde ha unedesversions(pondéréeetnonpondérée)

duproblèmedestablemaximumave unrapport(1=(G))estunproblèmeouvert onnu.

Nous nous intéressons alors au problème de lique maximum pour lequel au un résultat

d'approximationimpliquantun rapport non-trivialfon tion de (G)n'est jusqu'àprésent

établi. Nous proposons des algorithmes polynomiaux garantissant respe tivement les

rap-ports O(log 2

(G)=(G))et O(log 2

(G)=(G)loglog(G)) pourlesproblèmesde lique

maximumetde liquedepoidsmaximum.Nousobtenonslemêmetypederésultatspourle

problèmedesous-grapheinduit `- olorablemaximumetsa versionpondérée.Sortantde la

lassedesproblèmesdesous-grapheinduitmaximumvériantunepropriétéhéréditaire,nous

nousintéressonsalorsàdeuxproblèmesdeminimisation:leproblèmedesomme hromatique

minimumetleproblèmede olorationminimum.Pourlepremier,nousétablissonsune

rédu -tionentresaversionpondéréeetmax_WISquipréserve(àune onstantemultipli ativeprès)

lerapport d'approximation. Ellepermet le transfert desrésultats obtenus pour max_WIS

auproblèmedesomme hromatiquepondérée.Enn,pourla olorationminimum,nous

pro-posonsdesalgorithmesd'approximationaméliorantlesrésultatsexistants.Pluspré isément,

nous obtenons un rapport de la forme minfO(n

);O(logn=(G)loglogn)g. Dans le as

oùleminimum orrespondautermeO(n

), etteexpressiondominelesmeilleursrésultats

possibles fon tionde n(pour tout >0,l'approximationgarantissantpourtoute instan e

de la oloration le rapport n

est NP-di ile).Par ontre, si le rapport orrespond à la

quantitéO(logn=(G)loglogn),ilaméliorelesmeilleursrésultatsa tuelsfon tionde(G)

et orrespondaupremierrapport delaforme(1=(G))pourla olorationminimum.

Mots- lé: algorithme d'approximation,rapportd'approximation,problèmeNP- omplet,

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Abstra t

In a rst time we draw a rough shape of a general formal framework for polynomial

approximationtheorywhi hen ompassestheexistingonebyallowingtheexpressionofnew

typesof results. We show how thisframework in orporates allthe existing approximation

resultsand, moreover, how new typesof results an beexpressed within it. Next, we use

theframework introdu edto obtainapproximationresultsfor anumberof NP-hard

prob-lems. Inthisse ondpartofthepaper,werstdealwitha lassofproblems alledweighted

hereditary indu ed-subgraph maximizationproblems, notable representativesof whi h are

maximumindependent set, maximum lique, maximum `- olorableindu edsubgraph, et .

We devise apolynomial approximation-preserving redu tiontransforming any

approxima-tion ratio for any unweightedproblem of the lass, into approximationratio O(=logn)

for its weighted version. This allows us to perform subsequent improvements of the

ap-proximationratiofortheweightedindependentsetproblem(max_WIS),inordertonally

obtainasratiotheminimumbetweenO(log k n=n)andO(log(G)=(k 2 (G)(loglog(G)) 2 ),

where (G) denotes theaveragegraph-degree of theinput-graph, forevery onstantk. In

any of the two ases, this is an important improvementsin e in theformer one, the ratio

formax_WISouter-performsO(log 2

n=n), thebest-knownratioforthe(unweighted)

inde-pendentsetproblem(max_IS),whileinthelatter ase,weobtaintherst(1=(G))ratio

formax_WIS(where(G)isthemaximumgraph-degree). Next,basedupongraph- oloring,

wedeviseapolynomialtimeapproximationalgorithmformax_WISa hievingratiothe

min-imumbetweenO(n

4=5

)andO(loglog(G) =(G)). Here also,ex eptfortheveryunlikely

ase where max_WIS an be approximated within O(n 4=5

) (approximation of max_IS

within n 1

is hard for every > 0), our algorithm is the rst (loglog(G)

=(G))-approximation algorithm for max_WIS (for every (G)). Let us note that

approxima-tion of bothindependent set versions within ratios (1=(G)) is a very well-known open

problem. Then we deal with maximum lique problem for whi h no non-trivial

approxi-mationratios,fun tions of(G)are presentlyknown. Weproposeherealgorithms

a hiev-ingratios O(log 2

(G)=(G))and O(log

2

(G)=(G)loglog(G)) for maximum-sizeand

maximum-weight lique, respe tively. We do the same for the unweighted and weighted

versions of maximum`- olorableindu ed subgraph. We then leavethe lass of hereditary

indu ed-subgraph maximization problems and deal with two minimization problems, the

minimum hromati sumandtheminimum oloring. For theformer,weshowtheexisten e

ofapolynomialredu tionbetweenitsweightedversionandformax_WISpreserving(upto

multipli ative onstant)theapproximationratiosforbothproblems. Thisredu tionallows

transferof the results obtainedfor max_WIS to minimum-weight hromati sum. Let us

note that this resultalso is entirelynewand non-triviallyobtained. Finally, for minimum

oloring, weprodu e approximationratios improving all the known ones. Morepre isely,

weobtainaratioofmin fO(n

);O(logn=(G)loglogn)g. Inthe asewhere theminimum

is realized by the quantity O(n

), it outer-performs all the ratios, fun tions of n, while

if the minimum is realized by the quantity O(logn=(G)loglogn), it outer-performs the

best knownratio,fun tion of(G)and onstitutestherst(1=(G)) ratiofor minimum

oloring.

Keywords: approximationalgorithm, approximationratio, NP- omplete problem,

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Introdu tion

1 A few words about polynomial approximation

AnNPoptimization(NPO)problemis ommonlydened(see,forexample,[8 ℄)asafour-tuple

(I;S;v I

;opt ) su h that:

I isthe setof instan es of andit anbe re ognizedin polynomialtime;

given I 2 I, S(I) denotes the set of feasible solutions of I; for every S 2 S(I), jSj is

polynomial in jIj; given anyI and any S polynomial in jIj, one an de ide in polynomial

time ifS2S(I);

given I 2 I and S 2 S(I), v

I

(S) denotes the value of S; v I

is polynomially omputable

and is ommonly alled obje tive fun tion;

opt2fmax;ming.

The lass of NP optimization problems is ommonly denoted by NPO. The most interesting

sub- lass of NPO is the lass of the NP-hard optimization problems, known to be unsolvable

in polynomial time unlessP=NPand people widelythinksthat thisfa tis very unlikely. This

probable intra tability of NP-hard problems motivates both resear hers and pra titioners in

trying to approximately solve su h problems, i.e., in trying to nd in polynomial time, not the

best solutionbut onesolution whi h, insome sense, isnear the optimal one.

Given an instan e I of a ombinatorial optimization problem , !(I), (I) and (I) will

denotethe values ofthe worst solution ofI (inthe sense ofthe obje tive fun tion),the

approx-imated one (provided by a polynomial time approximation algorithm (PTAA) A supposed to

feasibly solve problem ), and the optimal one, respe tively. There exist mainly two thought

pro essesdealing with polynomialapproximation.

1.1 Standardapproximation

Here, the quality ofan approximationalgorithm A is expressed bythe ratio

A (I)=min (I) (I) ; (I) (I)

and the quantity A

=supfr : A

(I)>r;I 2Ig isthe onstant approximation ratio of A for .

Theratio indu ed by the standardapproximation will be denotedby.

Starting from the basi notion of approximation ratio, one an dene the one of onstant

asymptoti approximation ratio expressingthe performan e ofa PTAA whenworking on

limit-instan es. Thisratiois dened ([23℄)as

supfr:9M; A (I)>r;I 2I M g where I M =fI 2I:(I)>Mg ([23 ℄).

A parti ularly interesting ase of PTAA (representing an ideal approximation behavior),

is the one of the polynomial time approximation s hema (PTAS). A PTAS is a sequen e A

ofPTAAsguaranteeing,forevery>0,approximationratio A

=1 with omplexityO(n

k ),

wherek isa onstantnotdependingonnbuteventuallydependingon1=. Afurtherrenement

of PTAS is the one of fully PTAS (FPTAS), i.e., of a PTAS of omplexity O((1=) k 0 O(n k )), where k and k 0

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Here, the quality ofan approximationalgorithm A is expressed bythe ratio Æ A (I)= j!(I) (I)j j!(I) (I)j : The quantity Æ A =inffr : Æ A

(I) >r;I 2Ig is the onstant dierential-approximation ratio of A

for . The ratio adopted bythe dierential approximation will be denoted by Æ. The onstant

asymptoti dierential-approximation ratio isdened as([19 ℄)

lim k!1 inf I jS(I)j>k !(I) (I) !(I) (I) :

ThedierentialPTAS andFPTASaredenedanalogouslyto thestandardones. When A

,orÆ A

areequal to 0,we onsider instan e-depending ratios, i.e., ratiosexpressed in terms ofinstan e

parameters.

In what follows, when we indierently are referred to either the former or the latter

ap-proximation, or when approximationratios in both theories oin ide, we will use instead of

or Æ. Moreover, S-APX (resp., D-APX) will denote the lass of problems for whi h the best

knownstandard (resp.,dierential) PTAAsa hieve xed onstantapproximationratios.

Analo-gously, S-(resp.,D-)PTASorFPTASwilldenotethe lassesofproblemsadmitting standard

(resp., dierential) PTASs or FPTASs, et . We will also denote by

(I) (resp., Æ

(I)), the

beststandard-approximation(resp., dierential approximation)ratio knownfor . Whenratios

depend onparameters of I, it willbeassumed that(I)6(I 0

) foranysub-instan eI 0

ofI.

2 A list of NPO problems

In this paper we speak, in more or less details, about a number of ombinatorial problems.

Although all these problems arevery well-known in omplexity theory, in order that the paper

isself- ontained,we dene them in whatfollows.

2.1 Hereditary indu ed-subgraph maximization problems

Given a graph G = (V;E) and a set V

0

V, the subgraph of G indu ed by V

0 is a graph G 0 = (V 0 ;E 0 ), where E 0 = (V 0 V 0

)\E. Let G be the lass of all the graphs. A

graph-property is a mapping from G to f0;1g, i.e., for a G 2 G, (G) = 1 i G satises and

(G) = 0, otherwise. Property is hereditary if whenever it is satised by a graph it is also

satised by every one of its indu ed subgraphs; it is non-trivial if it is true for innitely many

graphs and false for innitelymany ones ([8 ℄). We onsider NP-hard graph-problems where

theobje tiveistondamaximum-orderindu edsubgraphG 0

satisfyinganon-trivialhereditary

property . These problemsare alled hereditary indu ed-subgraph maximization problems. For

a graph G, anyone of its vertex-subsets spe ies exa tlyone indu ed subgraph. Consequently,

in what follows we onsider that a feasible solution for is the vertex-set of G 0

. Hereditary

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

integral weights asso iated with their verti es (the weights are assumed to be bounded by 2 n

,

where n is the order of the input-graph, so that every arithmeti al operation on them an be

performedinpolynomialtime). Given agraph G, theobje tiveof aweighted indu edsubgraph

problem is to determine an indu ed subgraph G

of G su h thatG

satises and, moreover,

the sumof the weights of theverti esof G

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Given a graph G = (V;E), an independent set is a subset V 0

V su h that whenever

fv i ;v j g V 0 , v i v j =

2 E, and the maximum independent set problem is to nd an

inde-pendent set ofmaximum size. Inthe weighted versionof max_IS,denoted bymax_WIS,

positive weights areasso iated with the verti es of the input graph and the obje tive

be- omes to determine an independent set for whi h the sum ofthe weights of its verti es is

the largestpossible.

Maximum lique (max_KL).

Consider a graph G = (V;E). A lique of G is a subset V 0

V su h that every pair of

verti es of V 0

are linked by an edge in E, and the maximum lique problem (max_KL)

is to nd a maximum size set V 0

indu ing a lique in G(a maximum-size lique). In the

weighted versionofmax_KL,denotedbymax_WKL,positiveweightsareasso iatedwith

the verti es of the inputgraph and the obje tive be omesto determine a liquefor whi h

the sumof the weights ofits verti es isthe largest possible.

Maximum `- olorable indu ed subgraph (max_C`).

GivenagraphG=(V;E)anda onstant`<(G)(themaximumgraph-degree),max_C`

onsistsin ndinga maximum-order subgraph G 0

of Gsu h thatG 0

is `- olorable.

Propertyisan independent set ishereditary(a subsetofanindependentsetisanindependent

set). Thesameholdsforpropertyis a lique (avertex-subset ofa liqueindu esalsoa lique),

aswell asforpropertyis`- olorable (ifthe verti es ofa graph G an be feasibly olored byat

most ` olors, theneverysubgraph ofG indu edby asubset ofits verti es anbe olored byat

most ` olors).

2.2 Some other NPO problems

Minimum vertex- overing (min_VC).

Given a graph G =(V;E), a vertex over is a subset V 0

V su h that, 8uv 2E, either

u2V

0

, or v2V 0

, and the minimum vertex- overing problem is to determine a

minimum-size vertex over.

Minimum set- overing (min_SC).

Given a olle tionS ofsubsetsofanite setC, aset over isasub- olle tion S 0 S su h that [ S i 2S 0S i

= C, and the minimum set- overing problem onsists in nding a over of

minimum size. We denotebymin_3-SCthe restri tion of min_SC on3-element sets.

Minimum oloring (min_C).

Considera graphG=(V;E) ofordern. Wewishto olor V withasfew olorsaspossible

sothatnotwo adja ent verti es re eive the same olor. The hromati number ofa graph

is the smallest number of olors whi h an feasibly olor its verti es. A graph G is alled

k- olorable ifitsverti es an be legally olored byk olors,in otherwords ifits hromati

number isat most k; itwill be alled k- hromati ifk isits hromati number.

Minimum hromati sum (min_CHS).

GivenagraphG=(V;E),anl- oloringisapartitionofV intoindependentsetsC 1

;:::;C l

.

The ostof an l- oloring isthe quantity P

l i=1

ijC i

j (in other words, the ost of oloringa

vertexv2V with oloriisi). Theminimum hromati sumproblem,denotedbyCHSisto

determinea minimum- ost oloring. Forthe weighted versionof CHS,denotedbyWCHS,

every vertexv2V is weighted bya rationalweight w v

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be omes iw v

, the value of an l- oloring be omes l i=1 iw(C i ), where w(C i ) = v2C i w v ,

and the obje tive be omes nowto determine a oloringof minimum value.

Bin pa king (BP).

We are given a nite set L = fx

1 ;:::;x

n

g of n rational numbers and an unbounded

number of bins, ea h bin having apa ity1. We wishto arrange all these numbers in the

leastpossible binsin su ha waythatthe sumofthe numbers inea h bin doesnot violate

its apa ity.

Minimum and maximum traveling salesman problem (min_TSP and max_TSP).

Given a omplete graph on n verti es, denoted by K

n

, with positive distan es on its

edges, min_TSP (resp., max_TSP) onsists in minimizing (resp., maximizing) the ost

ofaHamiltonian y le 1

,the ostofsu ha y lebeingthesumofthedistan esofitsedges.

Aninteresting sub- aseofTSPistheonein whi hedge-distan esareonly1or2(TSP12).

3 Notations

Given a problem dened on a graph G=(V;E), andits weighted versionW, we denoteby

~ w2IN

jVj

theve toroftheweights, byw v

theweight ofv2V,andbyw max

(G)andw

min

(G)the

largestandthesmallestvertex-weights,respe tively. Moreover,weadoptthefollowingnotations:

n: the order ofG, i.e., n=jVj;

(v): the neighborhoodof v2V; Æ S (v): the quantityj (v)\Sj,S V, v2V nS; w(V 0

): the total weight ofV

0

V, i.e.,the quantity P v2V 0 w v ; w

(G): the valueof an optimalsolution for W;

0 w

(G): the weight ofan approximated W-solution of G

~ d: the ve tor (d 1 ;:::;d n ), where d i

denotes the degreeof vertexv i

2V;

(G): the maximum degree ofG, i.e., (G)=max

i fd

i g;

(G): the average degreeof G, i.e.,(G)=(

P i d i )=n; w (G): the quantity P v2V w( (v))=w(V)

(G): the hromati number of G (the minimum number of olors with whi h one an

feasibly olor the verti es ofG);

G: the omplement ofG denedby

G=(V; E)with E=fij 2V V;i6=j;ij 2= Eg (obviously, G=G); G[V 0

℄: the subgraph ofG indu edbyV

0 V;

n 0

: the order ofthe graph G[V

0 ℄, V 0 V, i.e., jV 0 j=n 0 ; S (G[V 0

℄): an optimal (maximum-size)-solutionin G[V 0 ℄, V 0 V. 1 Anorderinghv 1 ;v 2 ;:::;v n ioftheverti esofK n su hthatv n v 1 2E(K n )and,for16i<n,v i v i+1 2E(K n ).

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independent set by (G), the value of a maximum-weight independent set by w

(G) and the

valueofanapproximatedmax_WIS-solutionby 0 w

(G). Moreover,whennoambiguity ano ur,

we will use,, w , w max andw min instead of(G),(G), w (G),w max (G)and w min (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

transposeofB, respe tively. Finally, givena ve tor ~u,we denotebyj~u j its Eu lidean norm.

4 The s ope of the paper and its main ontributions

Inwhatfollows,werstgeneralizethenotionoftheapproximationalgorithmbyintrodu ingthe

one ofthe approximation hain. Thisgeneralization allows us,for example,to expressin proper

termsalgorithms, oftime- omplexityO(f(jIj) k

)(wheref isapolynomialofjIj),a hievingratios

of the form '(I) '

0

(k) o('(I)), where '(I) denotes an approximation ratio depending on

parametersofI and' 0

isanintegerde reasingfun tionofk 2IN. Thiskindofdoublyasymptoti

approximation ratiomeans thatone an pre-xa value for k and nextthe smallerk's valuethe

loserto '(I) o('(I)) theratio (and, of ourse,the heavierthe algorithm's omplexity). This

re allsthe notionofthe polynomialtime approximations hema(without beingasu hone). On

theotherhand,ifthe term' 0

(k)missed,then'(I) o('(I))wouldbevery lose tothe lassi al

notion ofasymptoti approximationratio.

Next,weproposeanatural wayfor lassifyingNPOproblems following their

approximabil-ity behavior, by introdu ing the notion of the approximation level; it informally asso iates an

approximation result with a family of approximation ratios. Premises of su h a notion an be

found in the denition ofthe lass F-APXin [35℄. However, the authors of [35 ℄ restrain

them-selves to problem- lassi ations with respe t to orders of possible approximation ratios. Su h

a lassi ation does not allow distin tion between ratios of the same order, or between ratios

whose values are polynomially related. For example, based upon [35 ℄, one annot distinguish

approximation lasses indu ed by ratios O(jIj k

), where k is a xed onstant, for dierent

val-uesof k. However, numerous re ent approximation results are basedupon su h distin tions, in

parti ularwhendealing withinapproximabilityresultsbringingto theforehardnessfa tors,i.e.,

values of k for whi h approximation withinbetter than jIj k

ishard. For example, for max_IS,

approximation ratio 1=n is guaranteed by any approximation algorithm, while no PTAA an

guarantee approximationratio (1=n) 1

, for any>0, where nisthe orderof the input-graph,

unlessNP=ZPP([31 ℄). Anotherweakpointof[35 ℄isthattheirproblem- lassi ation isdened

with respe t to jIj. Certainly, this parameter is among the most interesting ones for

express-ing non- onstant approximation ratios, but is not the only one. For many NPO problems

max_IS, min_C, min_SC, et ., are some notable examples approximation results are

ob-tained as fun tions of dierent instan e-parameters. For max_IS, for example, there exist two

popular typesof positive approximation results, the ones expressed in terms of n and the ones

expressed in terms of graph-degree (maximum or average); moreover no links are known until

now between these two types. The latter family of results (ratios, fun tions of degree) annot

be expressed in terms of the lass F-APX. But even ifone tried to me hani ally rewrite

de-nitions of [35℄ in order to take into a ount ratios fun tions of graph-degrees, then, in the ase

ofmax_IS,onewouldhave tofa eanotherobsta le. Thebestknownratiosex lusivelyinterms

of(G)formax_ISare:=(G),foreveryxed onstant 2IN,forunweighted max_IS([18℄)

and 3=((G)+2) for the weighted ase. Following the lassi ation indu ed by [35℄, max_IS

and max_WIS are of the same hardness, O(1=(G)), with respe t to their approximability.

Thismathemati allyisnotveryfairsin e theimprovement ofthe terminthe numeratorbyjust

one unit uses very ompli ated ombinatorial arguments and mathemati al te hniques. On the

other hand, the best approximation ratios ex lusively fun tions of n are O(log 2

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for max_IS and O(log n=(nloglogn)) for max_WIS(see theorem3 of se tion 10.1, page 15).

Consequently, ifone hooses n to express ratios, unweighted and weighted max_ISare not (at

the urrentstate of knowledge) ofthe same hardnessregardingtheir approximability. Ase ond

example justifying the real need of using lasses of approximation ratios larger than these of

onstant ratios or of ratiosdepending on jIj an be taken from [12℄. It is proved there that for

theminimum-weight maximalindependentset, noSPTAA an a hieveapproximationratio that

doesnot depend on the vertex-weights. In other words, su h a problem(in some sense among

the hardest ones regardingits approximation) annotbein luded into the lassi ation of [35℄.

Thesame holdsfor min_TSP whi h annot be approximatedwithin betterthan 2

P(n)

where P

isa polynomial ofn (thisfollows byasimple remark fromthe result of[44℄).

Next, we introdu e a kind of approximation-preserving redu tion, alled FP-redu tion, and

showthatmanyoftheredu tionsknown(forexample theonesof[41 ,46 ℄) anbeseenasspe ial

ases of FP-redu tions. An interesting feature of FP-redu tion is that it an link, for a given

problem, its approximation behaviors in standard anddierential approximations.

InpartIII,weshowthatmanyknownapproximationresults anbeexpressed verynaturally

in the framework proposed in part II and also that this framework is very suitable for the

a hievement of newones.

More pre isely, in se tion 9, we devise an FP-redu tion from weighted hereditary

indu ed-subgraph problems to unweighted ones transforming any approximation ratio for the latters

into approximationratio O(=logn)for the formers.

Su h a ratio an be improved when dealing with parti ular problems. For example, it

be- omes O(=loglogn) when dealing with the pair max_ISmax_WIS (se tion 10.1). Based

upon this redu tion we draw rst improvements for the ratio of max_WIS. In se tion 10.3,

we propose new improved approximation results for max_IS. Next, always based upon the

redu tion of se tion 9, we further improve approximations for max_WIS and obtain an

ap-proximation ratio for max_WIS with value greater than the minimum between O(log

k n=n)

andO(log(G)=((G)log 2

log(G)). Thisisanimportant improvement sin eifthe

max_WIS-ratio obtained is O(log k

n=n), itouter-performs O(log 2

n=n),the best-knownratio for max_IS.

On the other hand, if the ratio obtained is O(log(G)=((G)log 2

log(G))), then we a hieve

the rst (1=(G)) ratio for max_WIS. Letus note that this is the rst time that non-trivial

results for max_WISare produ ed by a redu tion to max_IS. In se tion 10.4, we propose

an-other PTAA for max_WIS whi h further improves (although in reasing the time- omplexity)

the results of se tion 10.3. We generalize a result of [3 ℄, by linking, devising another

FP-redu tion, the approximation of the lass max_WIS

k

of max_WIS-instan es with weighted

independen e number greater than w(G)=k to the approximation ofa lassG `

ofgraph- oloring

instan es in luding the `- olorable graphs. Combining this result with re ent works of [34 , 38℄

about G

`

, we obtain an approximation ratio for max_WIS of value greater than the

mini-mum between O(n

4=5

) and O(loglog(G)=(G)). Consequently, ex eptfor the very unlikely

ase where max_WIS an be approximated within O(n

4=5

) (re all that approximation of IS

within n 1

is hard for every > 0,unless NP=ZPP ([31 ℄)), our algorithm is the rst PTAA

a hievingratio (loglog(G)=(G)) for max_WIS.

Inse tion11,wedeviseanewFP-redu tionbetween max_KLandmax_WKL.Based upon

thisredu tionandusingourresultsonmax_WIS,wededu etherst(1=(G))approximation

ratio for the maximum-size and maximum-weight lique problems. We note thatthe results of

this se tionwork even for unbounded valuesof (G).

Inse tion12,usinganotherFP-redu tionfrommax_WIS,weobtainimprovementsofre ent

approximationresultsformin_WC`,whileinse tion13wedeviseanFP-redu tionguaranteeing

thatmin_WCHS isequi-approximable (up to amultipli ative onstant)with max_WIS.

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standard-approximationratiominfO(n );O(logn=((G)loglogn))g. Intheunlikely ase

whe-re the minimum is realized by the quantity O(n

) (in [21 ℄ it is proved that for any

posi-tive , min_C annotbe solved within standard-approximation ratio n 1

, unless NP=ZPP),

it outer-performs all the ratios, fun tions of n, while if the minimum is realized by the

quan-tity O(logn=((G)loglogn)), it outer-performs the best known ratio, fun tion of (G), and

onstitutesthe rst(1=(G))ratio for minimum oloring.

Finally, in se tion 15 we study FP-redu tions between standard and dierential

approxi-mations. We rst give su ient onditions for transferring results between the two thought

pro esses. Next, we mention results dealing with su h FP-redu tions for a number of NPO

problems.

Part II

Roughing out a new approximation framework

5 Approximation hains: a generalization of the approximation algorithms

We rst dene what in the sequel we will all an approximation hain whi h isa generalization

of the notionof approximationalgorithm.

Denition 1. Approximation hain.

Consider an NPO problem = (I;S;v

I

;opt ) and let : I IN ! [0;1℄, (I;k) 7! (I;k),

be a mapping in reasing in k. An approximation hain with ratio for is a sequen e of

algorithms (A k ) k2IN , indexedbyk 2IN (byA k

we denote the kth algorithm of the hain), su h

that, for allk, A k

isan approximationalgorithm guaranteeing ratio at least(I;k).

Let T((A k ) k2IN ) be the time- omplexity of (A k ) k2IN . If T((A k ) k2IN

) is polynomial in n (the size

ofI) butexponential ink,then (A k

) k2IN

is alled polynomialtime approximation hain(PTAC),

while ifT((A k

) k2IN

) ispolynomialin both nand k, then (A k

) k2IN

is alled fullypolynomial time

approximation hain (FPTAC). Su h hains will be denoted by SPTAC and SFPTAC (resp.,

DPTAC and DFPTAC) when dealing with standard (resp., dierential) approximation. When

ratiosin two thought pro esses oin ide, we will simplyusetermsPTAC andFPTAC.

The useofapproximation hains suggests toasso iate approximabilityresults not only with

ratiosbut rather with sequen es of ratios. Thisallowsus, for example,to asso iate an

approxi-mation levelwiththe existen eofaPTASwhi hisimpossiblebymeansof F-APX.Ingeneral,

the useofthe lassi ation of[35℄ doesnotallowus to distinguish,evenwithin S-APX,

prob-lemsapproximated within onstant ratios (for onstants stri tly smallerthan 1) from problems

approximated within ratio 1 ,for every>0.

6 Approximation level

Thefollowing denition2 generalizesthe notionof lassesF-APX.

Denition 2. Approximation level.

Consider an NPO problem, letP

=f: IIN ![0;1℄;(I;k) 7! (I;k)g and let P 22 P

.

We say that P is min-invariant if 8(; 0 ) 2P P, minf; 0 g 2P. An approximation levelis a set P 2 2 P

min-invariant. A problem is approximable on level P if it an be solved by

aPTAC a hievingratio in P.

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if we denote by P

the approximation level f : I ! [0;1℄;I 7! (I)g, i.e., the set of

ratiosindependenton k,then aPTACwith ratioin ~ P

orrespondstothe lassi alnotion

ofPTAA for ;

onsiderlevel ^ P

=f:(I;k)![0;1℄;(I;k)7!(k)g,i.e.,the lassofapproximationratios

independent on I; then, bysimple fun tional analysisarguments, 9 2[0;1℄, (I;k) !

whenk !1;if=1,then our PTAC(A k

) k2IN

isnothing elsethan a PTAS for ;

if we onsider level P = ~ P \ ^ P

, then a PTAC with ratio in P

is exa tly the very

well-known onstant-ratio PTAA; in otherwords, 2APX.

Let us note that one an asso iate with an approximation level (a set of ratios) P the set of

problems approximately solved by algorithms guaranteeing ratios in P. For example, P

(the

setof onstant independent on both I and k ratios) anbeseenasthe lassAPX. Inthe

sequel,when no onfusion arises,we will indierently useP to denote either an approximation

level,orthesetofproblemswithratiosinP. Forexample,followingthis onvention,APXmeans

either the lass of problems, or the lass of xed onstant ratios admitted by these problems.

Thesame holds alsofor PTAS.

7 Convergen e and hardness threshold

We now introdu e and dis uss the on ept of the onvergen e of the ratio of a PTAC, whi h

naturally followsthe notionof PTAC.

Denition 3. Convergen e (limit with respe t to k).

Given a problem and~2 ~ P

, the approximation ratio of(A k

) k2IN

onverges to ~if,

8>0;9;8k>;8I 2I; (I;k)>(I~ )(1 ):

For instan e, aPTAS isa PTAC, the ratio of whi h onverges to 1. Fromthe above denition,

one an easily see that (I;k), seen as I-depending fun tion-sequen e, is uniformly equivalent

to(I)~ whenk!1. Thisremarkimpliesthe existen eofanother weaker onvergen ereferring

to the lassi al notion of weak equivalen e of fun tion-sequen es. Under the same notations as

in denition3,a PTAC(A k

) k2IN

admitsapproximation ratio weakly onverging to ~if,

8>0;8I 2I;9;8k>;(I;k)>(I~ )(1 ):

For instan e, for a problem , an approximation ratio of the form p(jIj) 1=k

, where p is a

polynomialof jIj,is anexample ofweak onvergen e to 1.

Letusnowintrodu ethese ondtypeofratio- onvergen e,the onvergen ewithrespe ttoI.

Thisgeneralizeswhat in polynomialapproximationis ommonly alled asymptoti

approxima-tion ratio. In general, in either standard or dierential approximation, in order to dene the

asymptoti ratio, a set of instan es, alled also interesting instan es, is used. Many riteria

areused to express theseinstan es as, for example, the optimal value (see se tion1.2), the size

of the instan e or, for graph-problems, the maximum degree, or even (espe ially in dierential

approximation) the number of feasiblevalues of aninstan e ([19 ℄). Using asymptoti ratio

on-sists of dropping the non-interesting instan es out and of studying the approximation behavior

of an algorithm only on the interesting ones. Interesting or not-interesting instan es in whi h

sense? A verypopulargeneral riterion for delimitinginteresting fromnon-interesting instan es

of aproblem istheir omputationalhardness. Given an NP- ompleteproblem ,a sub- lassC

of I is onsidered asinteresting, or hard, if is better approximable (or even optimally solved

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notionofinstan e-hardnesstogetherwiththeoneofinstan etendingto1haveprodu edseveral

more spe i riteria about whi h instan es are onsidered as interesting or not. For example,

in [4℄a notionof problem'ssimpli ity, alled AAP-simpli ityin the sequel,isdened asfollows:

aproblem is AAP-simple ifits restri tion to the setof instan es

J M

=f I 2I :j!(I) (I)j6M;M anyxed onstantg

is in P. Remark that many NP- omplete problems, for example max_IS, min_SC, max_C`,

min_C,BP,et .,areAAP-simple. Ontheotherhand,in[43℄anothernotionofsimpli ity, alled

PM-simpli ity in what followsis dened: a problem isPM-simple ifits restri tion to the set

of instan es

K M

=fI 2I :(I)6M;M anyxed onstantg

is in P. Under this denition of simpli ity, max_IS, min_SC, or, nally max_C` are

PM-simple, while min_C, or BP are not. Both notions delimit the interesting instan es from the

non-interestingonesinafairlyintuitiveway. ForexampleforanAAP-simpleproblem,interesting

instan es are the ones in I nJ M

, while for a PM-simple problem the interesting instan es are

the ones in I nK M

. Another instan e-interest riterion is the one dened in [19 ℄ relying upon

the notion of the radial problem. Informally, a problem (with integer obje tive values) is

radial if, given an instan e I of and a feasible solution S 2 S(I), one an, in polynomial

time, on the one hand deteriorate S as mu h as one wants (up to nally obtain a worst-value

solution) and, on the other hand, one an greedily improve S in order to obtain (always in

polynomialtime)a sub-optimalsolution(eventuallythe optimal one). Thisdenitiongenerates

anotherboundarybetweeninterestingandnon-interestinginstan es,sin e,asitisproved in[19℄,

denoting byjv(S(I))j the number ofthe feasible-solution values ofI and setting

L M

=fI 2I :jv(S(I))j6M;M anyxed onstant g

then the restri tion of a radial problem (with integer obje tive values) to instan es in J M

, or

in L M

is in P. Here, in order to unify the dierent riteria upon whi h the several denitions

of interesting instan es are based, we propose the following denition of what we all hardness

threshold.

Denition 4. Hardness threshold.

Consider =(I;S;v I

;opt), and anapproximation level

P P

su hthat, under a omplexity

theoryhypothesis (forexampleP6=NP) isnot approximable within

P. Then,h:I !IN isa

hardness threshold withrespe t to

P if, 8M 2IN, the restri tion of to the instan e-set

fI 2I;h(I)6Mg

admitsa PTACwith ratio in P.

For example,under the hypothesis P6=NP:

h(I)=nisa hardnessthreshold forevery NP-hard problem,with respe tto the

approxi-mation level

P =f1g (theexa t solution);

h(I) = log(max(I)), where max(I) is the largest number of the instan e, is a hardness

thresholdfor the weakly NP-hard problems ([23 ℄)with respe tto

P =f1g;

h(I)=(I) isa hardnessthresholdfor the PM-simple NP-hard problemswith respe tto

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respe t to

P =f1g;

h(I) = jv(S(I))j and h(I) =!(I) (I) are hardnessthresholds for the radial NP-hard

problems (withinteger obje tive values)with respe tto

P =f1g;

h(I)=(G) isa hardnessthreshold formax_IS with respe tto P =APX; h(I)=jfx i :x i

61=3gj isa hardnessthreshold forBP with respe tto

P =f1g ([33 ℄).

Denition 5. Asymptoti approximation ratio (limitwith respe t to I).

Given a problem and a hardness threshold h with respe t to an approximation level

P, hain(A k ) k2IN

hasasymptoti ratio 0 2P if 8>0;8k2IN;9H;8I 2I;h(I)>H; (I;k)> 0 (I;k)(1 ):

Finally, letusremarkthatone an ombinedenitions3and5to obtainthefollowingdenition

of asymptoti onvergen e.

Denition 6. Asymptoti onvergen e.

Under the hypotheses of denition 5, (A k

) k2IN

admits approximation ratio asymptoti ally

on-verging to ~2 ~ P

if

8>0;92IN;8k>;9H;8I 2I;h(I)>H; (I;k)>(I~ )(1 ):

Let us now show that many results an very naturally (and quite elegantly) be expressed by

means of hains onvergen e.

There exists an O(n

k

) PTAC for max_IS a hieving approximation ratio asymptoti ally

onverging to6=(G) ([30 ℄).

There exists an O(njEj) FPTAC for max_IS guaranteeing asymptoti approximation

ra-tio6=(G) ([18 ℄).

ThereexistsaSPTACformin_3-SCwithapproximationratio onvergingto=5=7([28 ℄).

There exists an O(nlogn) DFPTAC with ratio asymptoti ally onverging to Æ = 2=3

for BP ([19 ℄).

8 Fun tional approximation-preserving redu tions

Weproposeinwhatfollowsanewpolynomialredu tion,thefun tionalapproximation-preserving

redu tion (FP-redu tion). This new redu tion en apsulates several approximation-preserving

redu tions,for example the onesof [10,40, 46 ℄.

Denition 7. FP-redu tion. Let =(I ;S ;v I ;opt ) and 0 = (I 0;S 0;v I 0 ;opt

0) be two NP optimization problems.

AFP-redu tionfromto 0

withexpansiong, denotedby g

0

, isatriple(f;h;g)su hthat:

1. f :I

!I

0 and,for anyI 2I

, f(I)2I

0 is omputable in time polynomial in jIj;

2. h : I S 0 ! S

and, for any I 2 I

and for any S 0 2 S 0 (f(I)), h(I;S 0 ) 2 S is

omputable in time polynomial in max fjIj;jS 0

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3. g:P

0 !P

isa mappingsu hthat, 8I 2I

andfor every PTAC(A k

) k2IN

guaranteeing

approximationratio:(f(I);k)7!(f(I);k),for 0 , algorithmA k =hÆA 0 k Æf guarantees

approximationratio g()for .

IfP P and P 0 P 0

are approximation levels for and 0

, respe tively, su hthat g(P 0

)

P, then the FP-redu tion transforms P 0

into P. In parti ular, if P = P 0

, then FP-redu tion

preserves level P andwill be alled level-preserving redu tion.

Proposition 1. FP-redu tions are transitive and ompose.

TheL-redu tion of [41℄ orrespondsto FP-redu tionfor whi h

g:( f(I);k)7!1 (ab( 1 ( f(I);k) ))

where a and b (a:b < 1) are onstants appearing in the original denition of [41℄. Here,

g(PTAS) PTAS , i.e., L-redu tion preserves the level of the approximation by polynomial

time approximation s hemata. For minimization problems, g(APX) APX while, for

maxi-mization ones, g(APX)6APX ,unlessP=NP\ o NP([9 ℄).

On the other hand, the ontinuous redu tion of [46℄, is an FP-redu tion with fun tional

expansion g: 7!e: 2P 7!0 otherwise

Here, g(APX )APX .

In the literature ([10 , 40, 41, 46℄), most of the known approximation-preserving redu tions

are more restri tive than the one of denition 7, sin e they impose onditions on how optimal

and approximated values are transformed. The value of the expansion (and, onsequently, of

the new ratio) is a onsequen e of these onditions. Moreover, several existing redu tions use

the very strong underlying hypothesis that 0

is solved by (xed) onstant-ratio approximation

algorithms. Aswe will seein the sequel,su h hypothesis isnot usedbyFP-redu tion.

Averyinterestingproblemdealing withapproximation-preserving redu tionsishow

approx-imation algorithms devised for unweighted problems an be transformed to e iently work for

their weighted versions, in parti ular when these algorithmsdo not guarantee onstant

approx-imation ratios. There exist, to our knowledge, very few redu tions between weighted and

un-weighted versionsofthe sameproblemindu ing expansionsleadingto interestingapproximation

ratios for the latters. Unfortunately, works as the ones of [11, 42 ℄, et ., despite their interest,

neither produ e satisfying results, nor propose general tools for the design of su h redu tions.

We think thatFP-redu tions an ontributeto thedesign ofsu h tools. In whatfollows,we

de-s ribesomeredu tionsguaranteeingexpansionsthat ontributetothea hievementofnon-trivial

approximation ratiosfor anumber of NP-hard weighted problems.

Another equally interesting domain of FP-redu tions is the investigation, for a given

NP-hard problem, of theapproximation links between itsstandard and dierential approximations.

InpartIII,wegivesu ient onditionsfortransferringresultsbetween standardanddierential

approximation. Moreover we present results dealing with FP-redu tions linking standard and

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A hieving approximation results in the new

framework

9 Indu ed hereditary subgraph maximization problems

Consider a hereditary property , an indu ed subgraph problem stated with respe t to

and the weighted version W of (we supposethat weights arepositive). We proposein this

se tionan FP-redu tionbetween max_ andmax_W. Theunderlying ideaof thisredu tion

is the following. Suppose, without loss of generality, that the subgraphs indu ed by all the

singletonsof verti es (every su h subgraph isredu ed to asinglevertex)verify. Partition the

input-graph G into lusters G[V (i)

℄, V (i)

V, and,by ommitting the vertex-weights, ompute

the solution of an unweighted max_ on any su h luster. Let S (i) be a solution for G[V (i) ℄. Then, G[V (i) ℄[S (i) ℄ = G[S (i) ℄ veries; hen e S (i)

is feasible for G. Next, by re onsidering the

vertex-weights, hoose the heaviest among the solutions obtained and argmax vi2V

fw i

g as the

nalsolution for max_W.

Theorem 1. max_W

g

max_, withg:7!O(=logn).

Proof. FixM >2 and set, fori=1;::: ,

V (i) = v j 2V : w max (G) M i <w j 6 w max (G) M i 1 x = sup `: w G [ 16i6` V (i) < w (G) 2 G x = G [ 16i6x V (i) G x+1 = G [ 16i6x+1 V (i) G d = G V n [ 16i6x V (i) : Of ourse, w (G d )> w (G)=2and w (G x+1 )> w (G)=2.

Lemma 1. There exists a PTAA for max_W a hieving approximation ratio M

x =(2n).

Proof of lemma 1. Thealgorithm laimed onsistsin simplytaking v 2argmax vi2V fw i g as

max_ W-solution. Then, 0 w (G)=w max (G)and w (G)62 w (G d )62j S (G d ) jw max (G d )62jS (G d ) j w max (G) M x : Consequently, 0 w (G) w (G) > M x 2j S (G d )j > M x 2n =) max_W (G)> M x 2n (1)

Remark 1. Foreveryi>1,theweightofanymax_-solutionS (i)

ofG[V (i)

℄liesintheinterval

[jS (i) jw max (G)=M i ;jS (i) jw max (G)=M i 1 ℄: it is at least jS (i) jw min (G[V (i) ℄) > jS (i) jw max (G)=M i and atmost jS (i) jw max (G[V (i) ℄)6jS (i) jw max (G)=M i 1 .

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1. Let w (G)=2 > w (G x ) > ((M 2)=2M) w (G) and p 2 argmax 16i6x f w (G[V (i) ℄)g.

If max_ is approximable on level max_

(G[V (p)

℄) in G[V (p)

℄, then max_W is

ap-proximable on level ((M 2)=2xM 2 ) max_ (G). 2. Let w (G x )6((M 2)=2M) w

(G). Ifmax_isapproximablewithin max_ (G[V (x+1) ℄) in G[V (x+1)

℄, then max_W is approximable on level (1=M 2

) max_

(G).

Proof of item 1. Obviously,

w (G x )6x w G[V (p) ℄ (remark 1) 6 x S G h V (p) i w max (G) M p 1

and,bythe hypothesis of theitem,

w (G)6 2M M 2 w (G x )6 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 max_ (G)< 1 for max_in G[V (p) ℄ onstru ts a solutionS (p)

of max_Wof weight at least

S (p) w min G h V (p) i > S (p) w max (G) M p : Notethat S (p)

ismax_-feasible forG. Moreover, starting fromthis solution, one an greedily

augment it in order to nally produ e a maximal max_W-solution for G. This nalsolution

veries 0 w (G)>jS (p) jw max (G)=M p .

Combination ofthe above expressions for 0 w (G)and w (G)yields 0 w (G) w (G) > M 2 2xM 2 S (p) S G V (p) ! > M 2 2xM 2 max_ G h V (p) i > M 2 2xM 2 max_ (G) and, onsequently, max_W (G)> M 2 2xM 2 max_ (G): (2)

This on ludes theproof ofitem 1.

Proof of item 2. We now suppose that

w (G

x

)6((M 2)=2M)

w

(G). Note thatsin e x is

the largest `for whi h w (G[[ 16i6` V (i) ℄)< w (G)=2, setV (x+1) isnon-empty. LetS opt (G x+1 )beanoptimalmax_W-solutioninG x+1 (i.e., w (G x+1 )=w(S opt (G x+1 ))). LetS(G x )=S opt (G x+1 )\V(G x )(whereV(G x

)denotesthevertex-setofG x )andS(G[V (x+1) ℄)= S opt (G x+1 )\V (x+1)

(inotherwords,fS(G x );S(G[V (x+1) ℄)gisapartitionofS opt (G x+1 )). Sin e is hereditary, sets S(G x ) and S(G[V (x+1) ℄), being subsets of S opt (G x+1

), also verify (and,

onsequently theyarefeasiblemax_W-solutionsfor G x andG[V (x+1) ℄,respe tively). We then have: w(S(G x ) ) 6 w (G x ) w S G h V (x+1) i 6 w G h V (x+1) i w (G x+1 ) = w( S(G x ))+w S G h V (x+1) i 6 w ( G x )+ w G h V (x+1) i 6 M 2 2M w (G)+ w G h V (x+1) i

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and also w (G x+1 ) > w

(G) =2. It follows from the above expressions that w

(G[V ℄) >

w

(G)=M and this together with remark 1yield, aftersome easy algebra,

w (G)6 S G h V (x+1) i w max (G) M x 1 : (3)

As previously, suppose that a PTAA provides a solution S (x+1)

for max_ in G[V

(x+1) ℄, the

ardinalityof whi h isat least max_ (G[V (x+1) ℄)jS (G[V (x+1) ℄)j. Then, 0 w (G)> S (x+1) w min G[V (x+1) ℄ > S (x+1) w max (G) M x+1 (4)

Combination of expressions (3)and(4) yields

0 w (G) w (G) > 1 M 2 S (x+1) S G V (x+1) ! > 1 M 2 max_ G h V (x+1) i > 1 M 2 max_ (G): Therefore, max_W (G)> 1 M 2 max_ (G) (5)

and this on ludes the proofof item 2and of the lemma.

Remark 2 . For the asewhere x=0, i.e., w

(G[V (1)

℄)> w

(G)=2, arguments similarto the

onesoftheproofofitem2inlemma2leadto

max_W (G)= 0 w (G)= w (G)> max_ (G)=2M,

betterthan the one of expression(5).

Considernowthe followingalgorithm wherewetakeuptheideasoflemmata1and2andwhere,

for a graph G 0

, we denotebyA(G 0

) the solution-set provided bythe exe ution of the -PTAA A

on the unweighted versionof G 0 . BEGIN (*WA*) fix a onstant M>2; partition V in sets V (i) fv k :w max =M i <w k 6w max =M i 1 g; S (0) v 2fargmax v i 2V fw i gg; OUTPUT argmaxfw(S (0) );w(A(G[V (i) ℄));i=1;::: g ; END. (*WA*)

Revisitexpressions (1),(2) and(5). It iseasy to seethat

max_ W (G)> WA (G)>max M x 2n ;min M 2 2M 2 x max_ (G); 1 M 2 max_ (G) (6)

Byexpression(1)andbythefa tthattheapproximationratio ofanyPTAAformax_Wmust

be less than 1 (max_W being a maximization problem), x 6 O(log

M

n). Taking this value

for x into a ount in expression 6, on ludes the proof of the theorem whi h, obviously, works

also in the asewhere weightsareexponential in n.

10 Maximum independent set

10.1 A rst improvement of the approximation of the maximum-weight

indepen-dent set via theorem 1

Itiswell-known ([46℄)that,8k>1,thegeneral weighted independent setproblempolynomially

redu es to max_PWIS(k) (the max_WIS-subproblem where the weights are bounded by n

k ),

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the ratios for max_WIS and max_PWIS(k), 8k > 1, and works also for instan e-depending

ratios. On the other hand, the following approximation preserving redu tion from max_PWIS

to max_ISworking onlyfor onstant ratios is establishedin [46℄.

Denition 8 . 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 = f(u;i):u2V;i2f1;:::;w u gg E w = f(u;i)(v;j): i2f1;:::;w u g; j 2f 1;:::;w v g; u6=v; uv 2Eg:

Inother words, everyvertex uof V isrepla ed byan independent set W u ofsize w u in G w and

every edge uv of E orrespondsin G w

to a omplete bipartite graph between W u

and W

v .

One an easilyshowthatevery independent setS ofGoftotal weight w(S) indu es,in G w

, the

independent setf(s;i): s2S;i2f1;:::;w s

gg of size w(S), and onversely, for every

indepen-dent set S w of G w , the set S = fu 2 V : 9i 2 f1;:::;w u g;(u;i) 2 S w g is an independent set of weight w(S) >jS w j. Consequently, w (G)=(G w

) and,byapplying a (G)-approximation

max_IS-algorithm to G

w

, one an derive an approximated max_WIS-solution of (G;w)~

guar-anteeing ratio(G w

).

By the above redu tion, a ratio (n;(G)), non-in reasing in , for max_IStransforms to

a ratio (w(V);w max (G)(G w ))>(w(V);w max

(G)(G)) for max_WIS, i.e., ex eptfrom the

aseof onstantapproximationratios,theredu tionaboveresultsinmax_WIS-ratiosdepending

on the weights. Morepre isely, the following result an be easilyproved.

Proposition 2. For every > 0 and every onstant k > 0, there exists an FP-redu tion

from max_WIS to max_IS transforming approximation level (n;;) for the latter into

ap-proximation level onverging to (n 1+ w(V)=w max (G);n 1+ ; w ) > (n 2+ ;n 1+ ;) for the former.

Unfortunately, the approximationresults known for max_ISdo not allowa hievement of

inter-estingapproximation ratiosfor max_WISusingproposition 2.

Revisit expression (6) in the proof of theorem 1 and let be max_IS. Set, for every k,

M =6. Then:

ifx>kloglogn=logM, then

max_WIS

(G)>log k

n=2n;

ifx6kloglogn=logM, then

max_WIS

(G)>0:099

max_IS

(G)=(kloglogn)

and the following theorem holds.

Theorem 2. For every xed `, every approximation level

max_IS

(G) for max_IS an be

transformed intoapproximation level

max_ WIS (G)>min ( log ` n 2n ; 0:099 IS (G) `loglogn ) :

Intermsofn,thebest-knownapproximationratiofor max_ISis,toour knowledge,O(log 2

n=n)

a hievedbythe max_IS-PTAA of[7 ℄. Embeddingitin max_IS

(G)-expressionoftheorem2,we

obtain the following on luding theorem.

Theorem 3. max_WIS (G)>O log 2 n nloglogn :

Theabove resultimprovesbyafa torO(loglogn) thebest-knownapproximation ratiofun tion

of nfor max_WIS(O(log

2

n=(nlog 2

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graph-degree

10.2.1 An approximation hain with ratio fun tion of (G)

The main result of this se tion is based upon the following fa ts; fa t1 is proved in [18 ℄ while

fa t2 isproved in [1,30 ℄.

Fa t 1. If there exists an algorithm guaranteeing, for every ` 2 IN and every graph without

lique of order `, an approximation ratio `

for max_IS, then, for every graph G, for

every > 0 and for every > 0, there exists an algorithm guaranteeing for max_IS

approximationratio bounded below by

min ; 0 (1 ) ` ; 2(` )(1 ) (G)+2 where 0 issu hthat1=(` )=(1=`)+ 0 .

Fa t 2. Thereexistsaxed onstant su hthat,forevery onstant`,thereexistsapolynomial

time approximation algorithm L_FREE(`;G) su h that, for every graph G of order n

without liquesoforder`, itprovidesanindependent setof ardinalitygreater than,or

equal to, n[log [(log(G))=`℄℄=(G).

Consider now the following algorithm of time- omplexity O(max fnjEj;T(`;n);(G) ` 2

jEjg)

where by T(`;n)wedenote the omplexity of L_FREE.

BEGIN (*MDCHAIN*)

S any non-empty independent set;

REPEAT b1 FALSE; b2 FALSE; IF9v2VnS: (v)\S=; THEN S S[fvg ; ELSE b1 TRUE; FI IF 9fu;vgVnS:( (u)[ (v))\S=fsg;uv2= E THEN S (Snfsg)[fu;vg ; ELSE b2 TRUE; FI UNTIL b1 AND b2; ~ V fv2VnS:Æ S (v)>2g[S;

ompute a maximal olle tion C

` of disjoint `- liques in G[ ~ V℄; X ` fv2 ~ V:v2=[ C2C ` Cg; S ` L_FREE(`;G[X ` ℄) ; OUTPUT S argmaxfjSj;jS ` jg ; END (*MDCHAIN*)

Consider a onstant and set ` = d(=2)e+1. Then, using fa ts 1 and 2 and appropriately

hoosing and ,the following theoremholds.

Theorem 4. ([18℄) For every xed integer onstant , algorithm MDCHAIN is a PTAC

guar-anteeing asymptoti (with respe t to hardness threshold (G)) approximation ratio =(G)

for max_ ISin time O(n d=2e

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We rst re allthe well-knowngreedy max_IS-algorithm, alled GREEDYin the sequel. BEGIN (*GREEDY*) S ; ; REPEAT v argmin vi2V fÆ(v i )g ; S S[fvg ; V Vn(fvg[ (v)); G G[V℄; UNTILV=;; END. (*GREEDY*)

The improvement of the result of theorem 4 is based upon the following fa ts, the rst proved

in [18 ℄ andthe se ond in [47 ℄.

Fa t 4. Consider,for everyxedinteger onstant `,the simultaneousexisten eofL_FREE(`;G)

and of an algorithm omputing, for every graph G, a maximalindependent set of size

atleastnf(G). Then, thereexistsaPTACsolvingmax_ISwithinapproximationratio

boundedbelowbyminf

0

`

;(` )f(G)g,where `

istheapproximationratioof L_FREE

and 0

issu h that1=(` )=(1=`)+ 0

.

Fa t 5. GREEDY omputes,inagraphG,amaximalindependentsetofsizeatleastn=((G)+1).

BEGIN (*ADCHAIN*)

S GREEDY(G);

ompute a maximal olle tion C

` of disjoint `- liques in G; X ` fv2V:v2=[ C2C ` Cg; S ` L_FREE(`;G[X ` ℄) ; OUTPUT S argmaxfjSj;jS ` jg ; END. (*ADCHAIN*)

Using fa ts4 and5 with `= and=1,the following theoremholds.

Theorem 5. ([18℄) For everyxed integer onstant , ADCHAIN guarantees, in O(n

),

asymp-toti (with respe t to hardness threshold (G)) approximation ratiobounded belowby

min (G) ; 0 loglog(G) (G) where 0 = 0

=(( 1)), for a xed onstant 0

.

Theresult oftheorem 5further improves(sometimesquitelargely) the resultof theorem4.

10.3 Towards (1=) -approximations

Inthisse tionwe showhowone anusethe lique-removalmethodof[7 ℄to obtain,for every,

(1=)-approximations for max_IS and max_WIS. The authors of [7℄ repeatedly all a

pro- edure omputing either a k- lique ( lique of order k), or an independent set of expe tive size.

At mostn=k liques an sobe dete ted, whileat 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 essarilydete ted during one exe ution of the pro edure. In all, the

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omputing, for every graph G of order nwith (G)>n=k+m and k su h that 26k62logn,

an independent set of sizeO(m 1=(k 1)

).

In[7℄kisnotsupposedtobe onstant;ratiolog 2

n=nforLARGEISisobtainedfork=O(logn). If

wesetk =(logn=(`(n)loglogn))+1,where`isamappingsu hthat,8x>0,0<`(x)6loglogx

and m=n=k

2

, then an analysissimilarto the one for theorem6 leadsto the following.

Theorem 7. For every fun tion ` with 0< `(x)6 loglogx, 8x>0, there exist onstants C

andK su hthatalgorithmLARGEIS omputes,forevery graphGoforder n>C,anindependent

set S su h that if (G)>`(n)nloglogn=logn, then jSj >Klog `(n)

n.

Let `, C and K be as in theorem 7, denote by EXHAUST an exhaustive-sear h algorithm for

maximum independent set, and onsiderthe following algorithm.

BEGIN (*STABLE*)

IFn6C

THEN OUTPUT S EXHAUST(G);

ELSE OUTPUT S argmaxfLARGEIS(G);GREEDY(G)g;

FI

END.(*STABLE*)

If n 6 C, then EXHAUST omputes in onstant time a maximum independent set for G. So,

supposen>Cand onsider ases(G)>`(n)nloglogn=lognand(G)<`(n)nloglogn=logn.

If (G)>`(n)nloglogn=logn, then, by theorem 7,STABLE guarantees jSj >Klog `(n)

n;

sin e (G)6n,the approximation ratio obtained isat leastKlog `(n)

n=n;

if (G)< `(n)nloglogn=logn, then by Turán's theorem ([47℄) jSj > n=(+1), and the

approximationratio obtained islogn=(`(n)(+1)loglogn).

Sin eGREEDYisoftimelinearinjEj,STABLEhasthesameworst- asetime- omplexityasLARGEIS.

So,the following theorem on ludesthe above dis ussion.

Theorem 8. Consider `su h that, 8x>0, 0<`(x)6loglogx. ThenSTABLE a hieves ratios

on approximation level max_ IS (G)>min ( K log `(n) n n ; logn `(n)(+1)loglogn ) :

Ifloglogn>`>2(inparti ularif`is onstant),eitherlog `(n)

n=n,orlogn=(`(n)(+1)loglogn)

orresponds to a signi ant improvement with respe t to O((log 2

n)=n) and k=, respe tively.

In terms of graph-degree, this is, to our knowledge, the rst (1=) approximation result

for max_IS. Also,

logn `(n)(+1)loglogn > log (+1)log 2 log > loglog :

Ratio loglog= for max_ISwas, up to now, the only known (1=) ratio for max_IS

(pre-sentedin[30℄). Butthealgorithmof[30 ℄hasthedrawba ktobepolynomialonlyifisbounded

above. Moreover,the omplexityofalgorithmSTABLEisO(nm),whereastheO(k=)ratioof[18℄

isobtainedbyaPTAA of omplexityO(n k

). Of ourse,our resultdoesnot guaranteearatio

al-waysboundedbelowbylogn=(`(n)( + 1)loglogn). Iftheminimumintheorem8isO(log `(n)

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stan e, onsiderthe ase`(n)=loglogn). Infa t,ifKlog `(n)

n=n6logn=(`(n)(+1)loglogn)

and n > C 0

, for a xed C 0

, then n= > K`(n)(loglogn)log `(n) 1

n and in this ase GREEDY

guarantees ratioboundedbelowbylog `(n) 1

n=n. Inall,the following orollary anbededu ed.

Corollary 1. Given a graph G and ` su h that, 8n, 2 <`(n) 6loglogn, at least one of the

two following onditions holds:

1. GREEDY guarantees ratio bounded below by log `(n) 1

n=n (improving the best known

(n)-ratio if`>3);

2. STABLE guarantees ratio bounded below by logn=(`(n)(+1)loglogn) (a hieving so

ra-tio(1=)).

The dis ussion above draws an interesting remark about the instan e-parameters expressing

non- onstantapproximationratios. Untilnow,studiesabouttheapproximationofmax_ISwere

limitedin expressingratiousingonlyoneinstan e-parameter(eitherthesizeorthedegree). The

results ofthis se tionshow that onsidering both parameters,it ispossibleto rea h tighterand

more interesting approximationratios.

Combination oftheorems8and2allowsthe a hievementofimportantapproximationresults

also formax_WIS.

Theorem 9. For every onstant k,max_WIS an be approximated withinratios on the level

min ( log k n n ; 0:099logn k(k+1)(+1)log 2 logn ) >min ( log k n n ; 0:099log k(k+1)(+1)log 2 log ) :

Inotherwords,theorem 9guarantees formax_WISthe existen eofPTAAsa hievingeither

n-dependingratios mu h better than the ones knownfor max_IS (and omparable with the ones

of orollary 1), or the rst (1=) ratiosfor max_WIS (re all that the best-known ()-ratio

for max_WIS without restri tions on was, until now, the one of [27℄, bounded below

by 3=(+2)).

10.4 Further improvements

In this se tion, we propose a polynomial approximation result for max_WIS not dedu ed by

redu tion from the unweighted ase. It improves all the approximation results of se tion 10.3,

for both weighted and unweighted ases, butthe orresponding omplexity ishigher.

10.4.1 A weighted versionof Turán's theorem

Re allthat w (G)= P v2V w( (v))=w(V)(notethat w

(G)6(G))and onsiderthefollowing

algorithm for max_WIS.

BEGIN (*WGREEDY*) S ; ; WHILEV6=; DO v argmin V fw( (v))=w(v)g; S S[fvg ; V Vn(fvg[ (v)); update E; OD

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Turán's theorem).

Theorem 10. For everyweighted-graph withmaximumdegree ,algorithmWGREEDY omputes

an independent set of weight atleast w(V) =( w

(G)+1) >w(V) =(+1 ).

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

i2 f1;:::tg let us denote by G i =(V i ;E i

) the surviving graph at the beginning of iteration i,

by v i

the vertex sele ted during the ith exe ution and by i (v) the neighborhood of v in G i . Then, 8i2f1;:::tg, X v2(fv i g[ (v i )) w( (v))> X v2(fv i g[ i (v i )) w( i (v))> w( i (v i )) w v i (w v i +w( i (v i ) )) (7)

and, onsequently, byaddingside-by-side the expressions (7) above,for i=1:::t, we get(note

thatV =[ i=1:::t ffv i g[ i (v i )g), w(V) w > t X i=1 w( i (v i )) w v i ( w v i +w( i (v i ))): (8)

Onthe other hand,

w(V)= t X i=1 ( w v i +w( i (v i ) )) (9)

and usingexpressions (8)and (9), we getbythe Cau hy-S hwarz inequality

w(V)( w +1)> t X i=1 (w v i +w( i (v i ) )) 2 w vi > (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 orollaryof theorem 10 is thatwhenever w

(G)6w(V)=k, then WGREEDY a hieves

inGratiok=(+1)formax_WIS.Consequently, inordertodeviseO(k=)-approximationsfor

every graph,one an fo ushim/herself onthe approximationof max_WISin graphswith large

weighted independen enumber. Foranintegerfun tionk =k(n)6n,wedenotebymax_WIS k

the lassofgraphswith w

(G)>w(V)=k; forreasonsofsimpli ityweassimilatethis lasswith

the orresponding max_WIS-subproblem. The work of [7℄ and its improvement by [3 ℄ show

how max_IS

k

(the unweighted version of max_WIS

k

) an be approximated within O(n

k 1 ) where k

depends only on k (note that using the redu tion of [46℄, one immediately gets a

ratio O(w(V)

k 1

) for max_WIS).

10.4.2 Graph oloring and the approximation of max_WIS

k

In this se tion, we adapt the method of [3℄ (originally devised for the unweighted max_IS) to

theweighted aseandrelatethe approximationofmax_WIS k

tothe oloringofa lassofgraphs

( alled G k+1

in the sequel) ontaining the (k+1)- olorable graphs. We rst re all two notions

very loselyrelated, the Lovász -fun tion andthe orthonormal representationof a graph.

Denition 9. ([37℄)Consider a graph G=(V;E), V =f1;:::ng:

theLovász-fun tion(G)of Gisthemaximumvalueof

P n i;j=1 b ij whereB=(b ij ) i;j=1:::n

rangesover allpositive semidenitesymmetri matri eswith tra e1andsu h thatb ij

=0

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i i=1:::n

dimensional Eu lidean spa e, su h that for every (i;j) su h that ij 2= E, u~ i

and u~ j

are

orthogonal.

Proposition 3. ([37 ℄) Given a graph G, the following holds:

given an orthonormal representation (u~ i

) i=1:::n

of G, and a unit ve tor ~ d, there existsi2 f1;:::ng su h that (G)61=( ~ du~ i ) 2 ; (G)6(G)6( G).

For an integer fun tion ` = `(n) 6 n, set G `

= fG : (

G) 6 `g. By proposition 3, every

`- olorable graph belongs to G `

. In [34℄ it is shown that every graph in G `

with order n and

maximum degree an be olored with O(minf

1 2=` log 3=2 n;n 1 3=(`+1) log 1=2 ng) olors in

randomized polynomial time ([34 ℄). The algorithm of [34℄hasbeen derandomized later in [38 ℄.

Theorem 11. Let k = k(n) 6 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 2 G k+1

, jVj 6 n, a f k

(n;)- oloring, then there exists

a O(max fn

3

;T(n)g) PTAA for max_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 nand G w

be asin denition 8 of

se tion9. Sin e the weights aresupposedto be integral and w (G)>w(V)=k, we have w (G)=( G w )> w(V) k+1 + w(V) k(k+1) + 1 k : In[25℄ itis shownthat: (G w

) isequal to the maximum valueof P n i;j=1 p w i w j b ij , where b ij areasindenition 9;

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 p w i w j b ij >(G w ) 1 2k >0 ij2= E b ij =0 ij2E (10)

Given that B is symmetri and positive semidenite, there exist n ve tors u^ i 2 IR n , i 2 f1;:::ng (IR n

being seen as Eu lidean spa e) su h that, 8(i;j) 2 f1;:::ng 2 , u^ i u^ j = b ij . In parti ular,we have b ii

>0. For our purposewe justneed to ompute n ve torsu~ i

satisfying

the following expression(11) for=1=(2knw(V)):

~ u i u~ j =b ij ; i6=j ju~ i j 2 =b ii +>0 n P i=1 ju~ i j 2 =Tr(B)+n=1+n n P i=1 p w i ~ u i 2 > n P i=1 n P j=1 p w i w j b ij (11)

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

, i = 1:::n. The system of

ve -tors (u~ i

) i=1:::n

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positive denite matrix B + I (where I is the identity matrix); one gets (in O(n )) an

n-dimensional triangular matrix U su h that B = t

UU I. Let then (u~ i

) i=1:::n

be the olumns

of U; they learly satisfyexpression(11).

Set, for i2f1;:::ng, ~ d = n P i=1 p w i ~ u i n P i=1 p w i ~ u i ~ z i = ~ u i ju~ i j

and notethatz~ i

onstitutes anorthonormal representation of

G. Withoutlossof generality, we

an assumethat( ~ dz~ 1 ) 2 >( ~ dz~ 2 ) 2 >:::( ~ dz~ n ) 2 . Lemma 3. Set j = max ( i2f1;:::;ng: i 1 X l =1 w l 6 w(V) k(k+1) ) (12) K = G[f v i :i=1;:::;jg℄: Then,( ~ dz~ j ) 2 >1=(k+1) whi himplies K2G k+1 .

Proof of lemma 3. ByCau hy-S hwarzinequalityand expressions(10) and(11), we get:

(1+n) n X i=1 w i ~ dz~ i 2 = n X i=1 ju~ i j 2 ! n X i=1 w i ~ dz~ i 2 > n X i=1 ~ d p w i ~ u i ! 2 = n X i=1 p w i ~ u i 2 > n X i=1 n X j=1 p w i w j b ij > (G w ) 1 2k > ( G w ) 1 2k > w(V) k+1 + w(V) k(k+1) + 1 2k (13)

On the other hand, sin e for i2 f1;:::ng, ~ dz~ i 6 1,(re all that ~ d and z~ i , i=1:::n areunit ve tors), (1+n) n X i=1 w i ~ dz~ i 2 6 n X i=1 w i ~ dz~ i 2 + 1 2k : (14)

Consequently, ombining expressions (13)and (14), we get

n X i=1 w i ~ dz~ i 2 > w(V) k+1 + w(V) k(k+1) : (15) Re all that ( ~ dz~ 1 ) 2 >( ~ dz~ 2 ) 2 >:::( ~ dz~ n ) 2 . We have ( ~ dz~ j ) 2 > 1=(k+1) >0 sin e, in the opposite ase, n X i=1 w i ~ dz~ i 2 6 j 1 X i=1 w i + n X i=j w i ~ dz~ j 2 < w(V) k+1 + w(V) k(k+1)