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
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
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
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,
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,
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
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
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
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 ).
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
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.
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.
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
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
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
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
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 .
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
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 ),
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
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
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
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)
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
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
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
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)