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Laboratoire d'Analyse et Modélisation de Systèmes pour

l'Aide à la Décision

CNRS UMR 7024

CAHIER DU LAMSADE

166

Janvier 2000

A priori optimization for the probabilistic maximum

independent set problem

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Résumé ii

Abstra t ii

1 Introdu tion 1

2 The modi ation strategies and a preliminary result 4

2.1 Strategy U1 . . . 4

2.2 Strategies U2and U3 . . . 4

2.3 Strategy U4 . . . 5

2.4 Strategy U5 . . . 5

2.5 Ageneral mathemati al formulation for the ve fun tionals . . . 5

3 The omplexity of PIS1 6 3.1 Computing optimala priorisolutions . . . 6

3.2 Approximatingoptimal solutions for PIS1 . . . 7

3.3 PIS1in bipartite graphs . . . 8

4 The omplexities of PIS2 and PIS3 8 4.1 Expressionsfor E U2 S and E U3 S . . . 8 4.2 Boundsfor E U2 S . . . 11

4.3 Approximatingoptimal solutions for PIS2 . . . 11

4.3.1 Using argmaxf P v i 2S p i g asapriorisolution . . . 11

4.3.2 Polynomial time approximations for PIS2 . . . 12

4.4 PIS2in bipartite graphs . . . 13

5 The omplexity of PIS4 14 5.1 An expressionfor E U4 S . . . 14 5.2 Using S  or argmaxf P v i 2S p i g asa priorisolutions . . . 15

5.3 PIS4in bipartite graphs . . . 17

6 The omplexity of PIS5 18 6.1 Ingeneral graphs . . . 18 6.2 Inbipartite graphs . . . 19 7 Con lusions 19 A Proof of proposition 1 22 A.1 Determining A  (G) . . . 22 A.2 Determining A  (G) 1 . . . 22

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Résumé

Nous ommençons pardonnerune dénition formelledu on ept relativement nouveau

desproblèmesd'optimisation ombinatoireprobabilistes(souslaméthodologiediteapriori

développéeparBertsimas,JailletetOdoni).Ensuite,nousétudionsla omplexitéderésoudre

optimalement le problème du stable maximum probabiliste sous diérentes stratégies de

modi ation, ainsi que la omplexité de solutions appro hées. Pour les diverses stratégies

étudiéesnousprésentonsaussidesrésultatssurlarestri tiondustablemaximumprobabiliste

surlesgraphesbiparties.

Mots- lé: problèmes ombinatoires, omplexité, stable,graphe

A priori optimization for the probabilisti maximum independent set

problem

Abstra t

We rstproposeaformal denition forthe on eptof probabilisti ombinatorial

opti-mizationproblem (undertheapriorimethod). Next,westudy the omplexityofoptimally

solvingprobabilisti maximumindependentsetproblem underseveralapriorioptimization

strategies as well as the omplexity of approximatingoptimal solutions. For thedierent

strategiesstudied, wepresentresultsabouttherestri tion ofprobabilisti independent set

onbipartitegraphs.

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In probabilisti ombinatorial optimization, the probabilities are asso iated with the data

de-s ribing an optimization problem (and not with the relations between data as, for example, in

random graph theory([4℄)). For aparti ular datum, we an see the probabilityasso iated with

it as a measure of how this datum is likely to be present in the instan e to be optimized, and

in this sense, probabilisti elements areexpli itly in luded in problem'sformulation. In su h a

formulation, the obje tive fun tionis akind of arefully dened mathemati al expe tationover

all the possible sub-instan esindu ed bythe initial instan e.

Thefa tthatintheframeworkofprobabilisti ombinatorialoptimizationproblems(PCOP),

the randomness lies in the presen e of the data, makes that the underlying models are very

adequatefor the modeling ofnatural problems, where randomness models un ertainty, or fuzzy

information, orhardness tofore ast phenomena,et .

Forinstan e,inatransportationnetworkwheneversituationsofseveraltypesof riseshaveto

bemodeled,wemeetPCOPslikeprobabilisti shortest(orlongest)path,probabilisti minimum

spanning tree, probabilisti lo ation, probabilisti traveling salesman problem (here we model

the fa tthatperhapssome itieswill notbevisited), et . Inindustrial automation, thesystems

for foreseeing workshops' produ tion give rise to probabilisti s heduling or probabilisti set

overings or pa kings. In omputers ien e, mainly in what on erns parallelism or distributed

omputer networks, PCOPshave to be solved;for example,modeling load balan ing with

non-uniform pro essors and failures possibility be omes a probabilisti graph partitioning problem;

also in network reliability theorymanyprobabilisti routing problems aremet ([3 ℄).

Denition 1. An NP optimization (NPO) problem  is ommonly dened as a four-tuple

(P;S;v P

;opt ) su h that:

 P isthe set ofinstan es of and it an bere ognized in polynomial time;

 given P 2 P (let n be the size of P), S(P) denotes the set of feasible solutions of P;

moreover,foreveryS 2S(P)(letjSjbethe sizeofS),jSjispolynomialin n;furthermore,

for any P and any S (with jSj a polynomial of n), one an de ide in polynomial time if

S 2S(P);

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

(S) denotes the value of S; v P

is polynomially omputable

and is ommonly alled obje tive fun tion;

 opt2fmax;ming.

Basedupondenition1,wewilltrytogiveinwhatfollowsaformaldenitionforPCOPs(under

the a priorithought pro ess).

In[2,3 ,11 ℄, athoughtpro ess alledin thesequela priorimethodology isadopted. Consider

an instan e P (where all the data are present) of an NPO problem , a sub-instan e I of P

and an algorithm U ( alled modi ation strategy), re eiving S 2 S(P) ( alled a priori solution)

asinput. Roughlyspeaking,theapriorimethodology onsistsin runningU inorder tomodifyS

andto produ e a new solution,dealing with the(present) sub-instan e I of P.

The a priori methodology together with the re-optimization one (an exhaustive

omputa-tional te hnique)arethe onlyonesuseduntil nowinthe study ofPCOPs. In[2,3,11℄,

parti u-larPCOPsarestudied,butnoformaldenitionofwhataPCOPis(intheaprioriframework)is

given. Inwhat follows in this se tion, we drawa formalframework for the apriori probabilisti

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P

modifying S and produ ing a feasible -solution for I. The probabilisti version P of  is a

quintuple(I;S;U;E P

(P U S

);opt ) su hthat:

 I is the set of instan es of P dened as I =fIP =(P; ~

Pr )g,where P 2P and ~ Pris an

n-ve tor ofo urren e-probabilities; I an be re ognizedin polynomialtime;

 givenIP =(P; ~

Pr)2I, S(IP)=S(P)

 U is a modi ation strategy, i.e., an algorithm whi h, given a solution S 2 S(IP) and a

sub-instan eI of P, re eivesS asinput and omputesa feasible-solutionfor I;

 given IP 2 I and S 2 S(IP), E P

(P U S

) denotes the value of S; if p i

= Pr[i℄, i 2 P,

is the o urren e probability of datum 1 i, F(I U S ) and F(I U S

) are the solution obtained

by running U(S) on I and its value, respe tively, Pr[I℄ = Q i2I p i Q i2PnI (1 p i ) is the

o urren e probability of I, then E P (P U S ) = P IP Pr[I℄F(I U S ); E P (P U S ) is ommonly alled fun tional;

 optisthe same asfor .

The omplexity of P is the omplexity of omputing

^ S = argopt fE P (P U S ) : S 2 S(IP)g.

Dealingwith denition 2,the following remarkmust be underlined.

Remark 1. In denition 2, U is part of the instan e and this seems somewhat unusual with

respe tto standard omplexitytheorywhereno algorithmintervenesin thedenitionofa

prob-lem. ButnotethatUisabsolutelynotanalgorithmforP,in thesensethatitdoesnot ompute

S2S(IP). It simplytsS (no matterhow S hasbeen omputed)to I.

Moreover, let us notethat hanging U one hanges the denitionof the problem itself. Inother

words, given a deterministi problem , the probabilisti problems indu ed by the quintuples

Q1 = (I;S;U1;E P

(P U1 S

);opt ) and Q2 = (I;S;U2;E P

(P U2 S

);opt ) are two distin t PCOPs.

Moreover,aswewillseeinthe sequel,the hoi eofthemodi ationstrategyplaysa ru ialrole

in the omplexity ofthe problem given thata strategy mayor maynot allowin lusion in NP.

A ommonthoughtpro essdealingwithaPCOPPisrsttoexpressE P

(P U S

)inanexpli it

way. Su h expression of the fun tional allows to pre isely hara terize the a priori solution ^ S

optimizing it and, onsequently, to de ide if ^

S an or annot be omputed in polynomial time.

Of ourse,if ^

S annotbe omputed inpolynomialtime,itisveryinterestingto de ideifat least

thefun tionalitself anbe omputed inpolynomialtime (inotherwords, iftheproblemathand

isor isnotin NP).Thisremark introdu esa(perhapsthe most)meaningfuldieren e between

probabilisti and deterministi problems. Thegeneral formulaof the fun tionalasit isgiven in

denition 2 (as well as in PCOP-literature) suggests that, in general, exhaustive omputation

of E P (P U S ) requires O(2 n

) distin t omputations. Consequently, a positive statement about

in lusion of P in NP is not immediate, at least for a great number of PCOPs. Of ourse, in

many ases, fun tional's omputation an be simplied and performedin polynomial time, but

aswe will seein the sequel,this isnot always the ase.

GivenagraphG=(V;E)ofordern,anindependentsetisasubsetV 0

V su hthatnotany

two verti es in V 0

arelinked byan edge in G, and the maximum independent set problem (IS)

is to nd an independent set of maximum size. A natural extension of IS whi h will be also

mentioned anddis ussedin thesequelisthe maximum-weight independentset(WIS).Here, the

1

Thisdatum anbeavertexifwedeal withgraph-problems,orasetifwedealwithoptimizationproblemin set-systems,et .

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maximize the sumof the weights ofthe verti es in an independent set.

An instan e of probabilisti independent set (PIS) is a pair (G; ~

Pr) and is obtained by

as-so iating with ea h v i

2 V an o urren e probability p i

and by onsidering a modi ation

strategy U transforming feasible IS-solution S of G into an independent set for the sub-graph

of G indu ed by a set I V. Asmentioned above, the obje tive for PIS is to determine the a

priorisolution ^

S maximizing the fun tionalE PIS

(G U S

).

Let us notethat PIS is quite dierent from PCOPs already studied ([2 , 3 , 11 ,12 ℄). There,

Eu lidean versions of minimization routing-problems (su h as the traveling salesman, or the

shortest path, or the minimum spanning tree), all of them dened in omplete graphs, are

onsidered. For all these problems only one modi ation strategy, onsisting, given an a priori

solution, in removing absent verti es (as our strategy U1 introdu ed in se tion 2) is onsidered.

Sin efor this kind ofproblems anyfeasible solutionis a onne ted subgraph ofthe input-graph

and sin e this latter graph is supposed omplete, there always exist a proper (and easy) way

to onne t the verti es of the survivingin order to onstru t afeasible solution for the present

sub-instan e. Su h strategies seem e ient for minimization problems. On the other hand,

for PIS, sin e it is a maximization problem, strategies as U1 are, as we will see later, rather

ine ient andlead to low-quality solutions. Finally, no parti ular restri tionsare imposed here

on the input-graphs.

Ex ept for its theoreti al interest, PIS has also on rete appli ations. In[7 ℄, we have

stud-ied some aspe ts of the satellite shots planning problem. We have proposed a graph-theoreti

modeling for thisproblemand we have proved that,via thismodeling, the solutionof the

prob-lem studied be ameexa tlythe omputation ofa maximum independent set in akind of graph

alled oni tgraph. However,wehavenottakenintoa ountthatshotsrealizedunderstrong

loud- overingarenotoperational. Consequently,itwouldbenaturaltoalsomodelweather

fore- asting. This an bedone byasso iatingprobabilityp i

with vertex v i

of the oni t graph; the

higher the vertex-probability, the more operational the shottaken. Su h a model for the

satel-lite shots planning problem allows, given an a priori IS-solution, omputation of the expe ted

number of operational shots.

There exist two interpretations of su h an approa h, ea h one hara terized by its proper

modi ation strategy:

 planisrstlyexe utedandone anknowonlyafterplan'sexe utionifashotisoperational;

in this ase,one retains onlythe operational onesamongthe shots realized; this, interms

ofPIS, amountsto appli ation ofstrategy U1introdu ed in se tion 2;

 weather fore asting be omesa ertitudejust beforeplan'sexe ution; in this ase, starting

froman a prioriIS-solution,one knows the verti es of this solution orresponding to

non-operational shots,onedis ardsthemfromtheapriorisolutionand,nally,onerenders the

survived solution maximal by ompleting it by new verti es orresponding to operational

shots;thisamounts toappli ation ofotherstrategies, forexample theonesdenotedbyU2,

U3, U4in the sequel andintrodu ed in se tion2.

Let us note that the probabilisti extension of the model of [7℄ an also be used to represent

another on ept, modeled in terms of PIS, where randomness on verti es represents this time

probabilities thatthe orrespondingshotsarerequested. Shot-probabilityequal to1means that

thisshothasalreadybeenrequested,whileshot-probabilityin[0,1℄meansthatthe orresponding

shot will eventually be requested just before its realization. The orresponding PIS an be

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In what follows we onsider maximal (although not ne essarily maximum) a priori

inde-pendent sets and use ve modi ation strategies, Ui, i=1;:::;5. For U1 and U5 we express

their fun tionalsin a losed form, we provethatthey are omputed in polynomial time,and we

determine the a priori solutions that maximizingthem. For U2and U3, the expressions for the

fun tionalsaremore ompli atedanditseemsthatthey annotbe omputedinpolynomialtime.

Dueto the ompli ated expressions for these fun tionals,we have not been able to hara terize

the a priorisolutions maximizing them. Finally, for U4, we prove that the fun tionalasso iated

anbe omputedin polynomialtime,butwe arenot abletopre isely hara terize theoptimala

priorisolution maximizingit. For allthe strategies studied we also study the omplexityof

ap-proximatingoptimalapriorisolutions. Letus,on e more,re allherethatthe strategiesstudied

introdu ein fa tvedistin t PCOPsdenotedinthesequelbyPIS1,PIS2,PIS3,PIS4andPIS5,

respe tively. Finally, we study the probabilisti version of a natural restri tion of IS, the one

where the inputgraph isbipartite.

In the sequel, given a graph G = (V;E) of order n, we sometimes denote by V(G) the

vertex-set ofG. WedenotebyS amaximalsolution ofISofG, by (G) its ardinality, byS 

a

maximumindependentsetofG, by



(G)its ardinality, by ^

S anoptimalPIS-solution (a priori

solution) and by (G)^ its ardinality. Moreover, by (v i

), i = 1;:::;n, we denote the set of

neighbors of the vertex v i

and the quantity j (v i )j is alled degree of v i ; by (V 0 ),V 0 V, we

denote the set [ v i 2V 0 (v i ); also, Æ G =min v i 2V fj (v i )jg,  G =max v i 2V fj (v i )jg and  G is the

average degree of G; nally, byPr[v i

℄=p i

, we denote the fa t thatthe presen e probability of

a vertex v i

2V equals p i

. Given a set I  V, we denote byG[I℄ = (I;E I

) the subgraph of G

indu ed by I (obviously, there are 2 n

su h graphs). Given a maximal solution S of IS (the a

priori solution) in G, we denote byS[I℄ the set S\I. For reasons of simpli ity, the fun tional

asso iatedwith PIS ongraph Gisdenoted byE U S (= P IV Pr[I℄F(I U S )).

2 The modi ation strategies and a preliminary result

In what follows we denote by GREEDY the lassi al greedy IS-algorithm. It works as follows:

it orders the verti es of V in in reasing degree-order, it in ludes the minimum-degree vertex

in the solution, it deletes it together with its neighbors (as well as all edges in ident to these

verti es) from V (E), itreorders the verti es ofthe surviving graph and soon,until all verti es

are removed. Moreover, we denote by SIMGREEDY, a simplied version of GREEDY where after

removingavertexanditsneighbors,the algorithm doesnotreorder theverti es ofthesurviving

graph.

2.1 Strategy U1

Given ana prioriIS-solutionS andapresent subsetI V, modi ationstrategy U1 onsistsin

simplymoving the absent verti es out ofS.

BEGIN (*U1(G,S)*) OUTPUT F(I U1 S ) S[I℄ S\I; END. (*U1*) 2.2 Strategies U2 and U3

Modi ationstrategy U2 isa two-step method: itrst appliesU1 to obtain S[I℄,it nextapplies

GREEDY onthe graph G[

~

I℄ =G[InfS[I℄[ (S[I℄)g℄ and, nally, itretains the union of the two

independent sets obtained asnalIS-solution for G[I℄.

2

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S[I℄ U1(G;S);

G[ ~

I℄ G[InfS[I℄[ (S[I℄)g℄ ;

S[ ~ I℄ GREEDY(G[ ~ I℄); /S[ ~ I℄ SIMGREEDY(G[ ~ I℄) ;/ OUTPUT F(I U2 S ) S[I℄[S[ ~ I℄ ; /OUTPUT F(I U3 S ) S[I℄[S[ ~ I℄ ;/

END. (*U2 /U3/*)

Finally, strategy U3 is identi al to U2 modulo the fa t that, instead of GREEDY, algorithm

SIMGREEDY isexe uted(instru tions between slashes above referto modi ation strategy U3).

2.3 Strategy U4

Strategy U4startsfrom S[I℄ and ompletes it with the isolatedverti es(verti es with no

neigh-bors)of the graph G[ ~ I℄. BEGIN (*U4(G,S)*) S[I℄ U1(G;S); G[ ~

I℄ G[InfS[I℄[ (S[I℄)g℄ ;

OUTPUT F(I U4 S ) S[I℄[fv i 2 ~ I: (v i )=;g ; END. (*U4*) 2.4 Strategy U5

Strategy U5 applies the natural relation between a minimal vertex over 3

and a maximal

in-dependent set in a graph, i.e., a minimal vertex over (resp., maximal independent set) is the

omplement,with respe tto thevertexsetofthe graph, ofa maximalindependent set(resp., a

minimalvertex over).

BEGIN (*U5(G,S)*) (1) C VnS; (2) C[I℄ C\I; (3) R fv i 2C[I℄: (v i )=;g ; (4) C[I℄ C[I℄nR; (5) OUTPUT F(I U5 S ) InC[I℄; END. (*U5*)

2.5 A general mathemati al formulation for the ve fun tionals

Theorem 1. ConsideranapriorisolutionSof ardinality (G)forG; onsiderstrategiesUk,k

= 1, ..., 5. Withea hvertexv i

2V weasso iateaprobabilityp i

andarandomvariableX Uk ;S i

,

k = 1, ..., 5, dened, forevery I V, by

X Uk ;S i =  1 v i 2F I Uk S  0 otherwise (1) Then E Uk S = X v i 2S p i + X v i 2(VnS) Pr h X Uk ;S i =1 i : (2)

In parti ular,if, for ea h vertexv i 2V, p i =p, then E Uk S =p (G)+ X v i 2(VnS) Pr h X Uk ;S i =1 i : (3) 3

Given agraphG=(V;E), avertex overofGis asetV 0

V su hthat for anyvivj 2E eithervi,or vj belongstoV

0 .

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Proof. Byexpression (1),F(I Uk S )= n i=1 X Uk ;S i . So, E Uk S = X IV Pr[I℄F I Uk S  = X IV Pr[I℄ n X i=1 X Uk ;S i = n X i=1 X IV Pr[I℄X Uk ;S i = n X i=1 E  X Uk ;S i  = n X i=1 Pr h X Uk ;S i =1 i = n X i=1 Pr h X Uk ;S i =1 i 1 fv i 2Sg +1 fv i = 2Sg  = n X i=1 Pr h X Uk ;S i =1 i 1 fv i 2Sg + n X i=1 Pr h X Uk ;S i =1 i 1 fv i = 2 Sg : But, if v i 2S, then ne essarily X Uk ;S i =1, 8I su h thatv i 2I; so, Pr[X Uk;S i =1℄= p i , v i 2S, and onsequently, E Uk S = X v i 2S p i + n X i=1 Pr h X Uk ;S i =1 i 1 fv i = 2Sg = X v i 2S p i + X vi2(VnS) Pr h X Uk ;S i =1 i : Ifp i =p,v i

2V,then theresultofexpression(3)isimmediatelyobtainedfromexpression(2).

Let us note that, asit an be easily dedu ed from the proof of theorem1, the above result

holds for any strategy whi h it rst determines S[I℄, it next omputes an independent set S[ ~ I℄

on G[

~

I℄, and itnally onsiders assolution forG[I℄ the set S[I℄[S[ ~ I℄.

3 The omplexity of PIS1

3.1 Computingoptimal a priorisolutions

From expression (1), we have X U1 ;S i = 0, 8v i = 2 S, and onsequently, Pr[X U1 ;S i = 1℄ = 0, for v i

2V nS; so,the following theoremisimmediately derived for strategy U1.

Theorem 2. GivenagraphG=(V;E),anapriorisolutionS andthemodi ationstrategyU1,

then E U1 S = P v i 2S p i

, and is omputed in O(n). Optimal PIS1-solution ^

S isa maximum-weight

independent set in a weighted version of G where verti es are weighted by the orresponding

probabilities. If p i =p, 8v i 2V, then E U1 S =p (G); in this ase, ^ S=S  and E U1 ^ S =p  (G). The hara terization of ^

S given in theorem 2 immediately introdu es the following omplexity

resultfor PIS1.

Theorem 3. PIS1is NP-hard.

We now show that for p i

=p, v i

2V, a mathemati al expression for E U1 S

an be built dire tly

without applying theorem 1 (used in next se tions for the analysis of other strategies). Given

that06jF(I U1 S )j def = F(I U1 S )=jS[I℄j6 (G), we get: F I U1 S  =jS[I℄j (G) X i=1 1 fjS[I℄j=ig :

So,the fun tional forU1 an be written as

E U1 S = X IV Pr[I℄jS[I℄j (G) X i=1 1 fjS[I℄j=ig = (G) X i=1 i X IV Pr[I℄1 fjS[I℄j=ig = (G) X i=1 i  (G) i  p i (1 p) (G) i n (G) X j=0  n (G) j  p j (1 p) n (G) j = p (G)

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their probabilities; also, P n (G) j=0 C n (G) j p j (1 p) n (G) j

=1. The above proof for E U1 S

an

begeneralized in order to ompute everymoment of any orderfor U1. For instan e,

E h G U1 S  2 i = X IV Pr[I℄F 2 I U1 S  = (G) X i=1 i 2  (G) i  p i (1 p) (G) i =p (G)(p (G)+1 p) and, onsequently, Var U1 S =E h G U1 S  2 i E U1 S  2 = (G)p(1 p):

So, for U1, the random variable representing the size (G) of the a priori solution follows a

binomial lawwith parameters (G) and p.

3.2 Approximating optimal solutions for PIS1

In this se tion we show how, even if one annot ompute the optimal a priori solution in

poly-nomialtime,one an ompute asub-optimalsolution,the value(expe tation)ofwhi hisalways

greaterthan afa tortimesthevalue(expe tation)oftheoptimalone. Forthis,weshallpropose

in what follows well-known (in the theory of polynomial approximation of NP- omplete

prob-lems) polynomial algorithms omputing good sub-optimal solutions, and will show that, also

in probabilisti ase, these algorithmsworkwell.

Let us rst re all that given an instan e P of an NP- omplete problem , a ommon way

toestimatethe apa ityofapolynomialapproximationalgorithmA inndinggoodsub-optimal

solutions for , is by evaluating its approximation ratio ([8℄), i.e., the ratio A(P)=OPT(P),

where A(P) denotes the value of the solution omputed by A on P and OPT(P) denotes the

value of the optimal solution of P. Then the approximation ratio of A for a maximization

problem  is the quantityinffA(P)=OPT (P): P instan eof g; the loser this ratio to 1, the

betterthe approximation algorithm.

Re allthataswehavealreadyseeninse tion3,PIS1isequivalent toaweighted IS-problem,

where ea h vertex is weighted by the orresponding probability. Consequently, the following

theoremholds immediately.

Theorem 4 . If there exists a polynomial time approximation algorithm A solving WIS within

approximation ratio , then A polynomially solves PIS1 withinthe same approximation ratio .

In [6℄, an algorithm is developed for WIS a hieving approximation ratio the minimum value

between logn=(3( G

+1)loglogn) and O(n 4=5

). Using this algorithm in theorem 4,one gets

the following orollary.

Corollary 1. PIS1 an be approximated within

min  logn 3( G +1)loglogn ;O  n 4=5 o 

The hara terizationofPIS1in termsofaweighted IS-problemdrawsnotonlyissuesfornding

reasonable a priorisub-optimal solutions but, unfortunately, limits the apa ityof the problem

to bewell-approximated sin e,via this hara terization,all thenegativeresults applyingtoIS

areimmediately transferred to PIS1also. So,PIS1 ishard to approximate within n 1 

, for any

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Theorem 5. Consider a bipartite graph B = (V 1 ;V 2 ;E B ). Then, E U1 S = P vi2S p i and is

omputed inpolynomialtime. Theoptimala priorisolution ^

S isamaximum-weight independent

setinB onsideringthat itsverti esare weightedbythe orresponding presen e probabilities, and

an be found in O(n 1=2

jE B

j). Consequently, in bipartite graphs, PIS12P.

Proof. Con erning the expression for E U1 S

and the omplexity of its omputation, the proof is

the same asthe one oftheorem2.

Determining anoptimal IS-solution ina bipartite graphis ofpolynomial omplexity inboth

weighted and unweighted ases (see [9℄for the unweighted ase; for the weighted one, we quote

here the result of [5℄ where the polynomiality of weighted IS in a lass of graphs in luding the

bipartite onesis proved).

4 The omplexities of PIS2 and PIS3

4.1 Expressions for E U2 S and E U3 S LetA i = P IV;jS\Ij=i Pr[I℄F(I U2 S ). Then, E U2 S

an be ni elywritten asfollows:

E U2 S = X IV Pr[I℄F I U2 S  = X IV 0  (G) X i=0 1 fjS\Ij=ig 1 A Pr[I℄F(I U2 S )= (G) X i=0 0 B  X IV jS\Ij=i Pr[I℄F I U2 S  1 C A = (G) X i=0 A i : Quantities A i , i=1;:::; (G) (A 0

=0) are very natural and interesting from both theoreti al

and pra ti al pointsof view. For instan e, formula for E U2 S

given by expressionabove holds for

every probability law; also, omputing analyti al expressions for A i

seems to be an interesting

problemin ombinatorial ountingofgraphs;moreover,thankstothesimplerelationbetweenE U2 S

and A

i

, i=1;:::; (G), analyti al expressions for the latterwouldprodu e expli it expressions

for theformer. Unfortunately, expressionabove for E U2 S

, even intuitive and smart,doesnotgive

anyhint allowing pre ise hara terization of ^ S.

Inproposition1,theproofofwhi hisgiveninappendixA,A  (G) andA  (G) 1 areexpli itly

omputed,forthe aseofidenti alvertex-probabilities. However,theexpli it omputationforA i

s

of lower index produ esvery longand non-intuitive expressions.

Proposition 1. Let 0 (v) = (v) n f (v)\ (S  n fvg)g, ` 1 = jfv 2 S  : 0 (v) = ;gj, ` 0 1 =  (G) ` 1 , andp i =p, 8v i 2V. Then, A  (G) =  (G)p  (G) A  (G) 1 = p  (G) 1 (1 p) 0 B   (G)` 1 ` 1 +  (G)` 0 1 X v2S  0 (v)6=; (1 p) j 0 (v)j 1 C A :

Weshallnowgiveanupperboundforthe omplexityof omputingE U2 S

. Forthis,wewillanalyze

(asan intermediatestep) strategy U3 introdu ed in se tion2.

Let G 0 = G[V nS℄ = (V 0 ;E 0 ), and let V 0 = fv 1 ;:::;v n (G)

g be the list of verti es of G 0

sorted in in reasing-degree order; let us denote by V i

the set of the i rst verti es of V 0 and let G 0 i = G 0 [V i

℄ (of ourse, for G 0 i

the verti es of V i

are not sorted in in reasing-degree order).

Let us denote by S i

the independent set found by U3 on (thepresent sub-instan e of) G 0 i

, and

by i

its ardinality, i=1;:::;n (G). Finally, let (G[I 0

℄)bethe ardinality ofthe solution

provided byU3 when appliedin graph G[I 0

℄,I 0

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E n (G)  = n (G) X i=1 p i Pr[v i = 2 (S i 1 )℄ E U3 S = E U1 S +E n (G)  : If we denote by T(E( n (G) )) and T(E U3 S

) the omputation times of E(

n (G)

) and E

U3 S

,

respe tively, then T(E( n (G) ))=O(2 n (G) ) andT(E U3 S )=O(2 n (G) ). Proof. E( i )=E( i jv i present )p i +E( i jv i absent )(1 p i

). Moreover,forstrategy U3wehave

the following relation, settingS 0 =;, (S 0 )=; and 0 =0: i =  i 1 (v i )\S i 1 6=; i 1 +1 otherwise So,E( i jv i present)=E( i 1 )+Pr[ (v i )\S i 1 =;℄; onsequently, E( i )=p i E( i 1 )+p i Pr[ v i = 2 (S i 1 )℄+(1 p i )E( i 1 ): Sin eE( 0 )=0, E( i ) = i X j=1 p j Pr[v j = 2 (S j 1 )℄ E n (G)  = n (G) X i=1 p i Pr[v i = 2 (S i 1 )℄: Now, let I 0

=f(I)=InfS[I℄[ (S[I℄)g. Thisset represents the subset of verti es of I whi h

arenot ontained neither in S, nor inthe neighbor-setof S[I℄. Consequently, U3will be applied

on G[I 0 ℄;so, F(I U3 S )=jS[I℄j+ (G[I 0 ℄) and onsequently E U3 S = X IV Pr[I℄jS[I℄j+ X IV Pr[I℄ G[I 0 ℄  = E U1 S + X IV Pr[I℄ G[I 0 ℄  = E U1 S + X IV 0  X I 0 V 1 ff(I)=I 0 g 1 A Pr[I℄ G[I 0 ℄  = E U1 S + X I 0 V X IV f(I)=I 0 Pr[I℄ G[I 0 ℄  : Sin ePr[I 0 ℄= P IV;f(I)=I 0 Pr[I℄,weget E U3 S =E U1 S + X I 0 V Pr  I 0  G[I 0 ℄  =E U1 S +E n (G)  :

Let us now introdu e the random variable C

i

representing the solution of PIS3 whenever we

onsider only the present verti es of V i

, i.e., when we apply the greedy algorithm implied by

strategy U3in thegraph indu edbythe present verti esof V i . C i = 8 < : C i 1 v i isabsent C i 1 v i ispresent and (v i )\C i 1 6=; C i 1 [fv i g v i ispresent and (v i )\C i 1 =; We then have E( n (G) )= P n (G) i=1 p i Pr[ (v i )\C i 1 =;℄.

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i i 1

isnot omputable in polynomial time.

Pr[ (v i )\C i 1 =;℄ = ( 1 p i 1 )Pr[ (v i )\C i 2 =;℄ +p i 1 Pr[ (v i )\C i 2 =;℄Pr[ (v i 1 )\C i 2 6=;℄ +p i 1 Pr[ (v i )\(C i 2 [f v i 1 g )=;℄Pr[ (v i 1 )\C i 2 =;℄ = ( 1 p i 1 )Pr[ (v i )\C i 2 =;℄ +p i 1 Pr[ (v i )\C i 2 =;℄(1 Pr[ (v i 1 )\C i 2 =;℄ ) +p i 1 Pr[ (v i )\(C i 2 [f v i 1 g )=;℄Pr[ (v i 1 )\C i 2 =;℄ = Pr[ ( v i )\C i 2 =;℄ p i 1 Pr[ (v i )\C i 2 =;℄Pr[ (v i 1 )\C i 2 =;℄ +p i 1 Pr[ (v i )\(C i 2 [f v i 1 g )=;℄Pr[ (v i 1 )\C i 2 =;℄:

Onthe other hand,

Pr[ (v i )\(C i 2 [fv i 1 g )=;℄ = Pr[( (v i )\C i 2 )[( (v i )\fv i 1 g)=;℄ = Pr[( (v i )\C i 2 =;)\( (v i )\fv i 1 g=;) ℄ = Pr[ (v i )\C i 2 =;℄1 f (v i ) \fv i 1 g=;g : Consequently, Pr[ (v i )\C i 1 =;℄ = Pr[ (v i )\C i 2 =;℄ p i 1 Pr[ (v i )\C i 2 =;℄Pr[ (v i 1 )\C i 2 =;℄ +p i 1 Pr[ (v i )\C i 2 =;℄Pr[ (v i 1 )\C i 2 =;℄1 f (vi)\fvi 1g=;g : Lett i 1 (v i

)bethe omputationaltimeofPr[ (v i

)\C i 1

=;℄. Bytheaboveequalities weeasily

dedu ethat t i 1 (v i )=t i 2 ( v i )+t i 2 (v i 1 ):

In order to ompute this re urren e relation, at ea h step we need to know two terms of the

pre edent step; sofor the omputation ofPr[ (v i )\C i 1 =;℄, we need 2 i 1 omputationsand expressionfor T(E U3 S ) immediatelyfollows.

AlgorithmU3is,asithasbeenalreadynoted, asimpliedversionofalgorithmU2. Moreover,

thereexistgraphswherethetwoalgorithmsgivethesameresultsbyperformingidenti al hoi es

and deletions of verti es (for example, onsider a graph on n isolated verti es). Consequently,

omputationtimeofU3isa(worst- ase)lowerboundfortheoneof U2andthefollowingtheorem

holds.

Theorem 7. Let T(E

U2 S

) be the omputationaltime of E U2 S . Then, T(E U2 S )=(T(E U3 S )).

The resultof theorem 6simply givesan upper bound on the omplexityof omputing E U3 S

and

does not prove that E U3 S

is not omputable in polynomial time (if this was true, it would be

a very interesting result sin e, in this ase, PIS2 and PIS3 would not belong to NP). In fa t,

the result of theorem 6 is based upon a parti ular re ursion-formula and a parti ular way for

omputing it. Inany ase, one an easily prove that PIS2 isintra table (following the notation

in the appendix of[8℄,PIS2 isa kindof starryproblem).

Indeed, ifone an polynomially determine an optimala priorisolution ^

S for PIS2,then one

an simply onsider an instan eof IS asa PIS2-instan e with p i

=1,8v i

2V. It iseasy to see

that, in this ase, ^

S =S



andthe following theoremimmediately holds.

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4.2 Bounds for E S

For la k of hara terizing the omplexity of omputing E U2 S

, we build in this paragraph upper

and lower boundsfor it.

Theorem 9. Let

~

 be the maximum degree of G[

~

I℄. Then, on the hypothesis of distin t

vertex-probabilities: X vi2S p i + X IV Y vi2I p i Y v i = 2I (1 p i ) ~ I ~ +1 6E U2 S 6 X vi2S p i + X IV Y vi2I p i Y v i = 2I (1 p i ) ~ I (4)

while,on the hypothesis of identi al vertex-probabilities:

p (G)+ X IV p jIj (1 p) n jIj ~ I ~ +1 6E U2 S 6p (G)+ X IV p jIj (1 p) n jIj ~ I (5)

Alwaysunder the latter hypothesis: pn=( G +1 )6E U2 S 6n( G +p) =( G +1 ).

Proof. Remark rst that jS[I℄j 6 F(I U2 S

) 6jS[I℄j+jI n(S[I℄)[( (S[I℄)\I)j = jS[I℄j+j ~ Ij; onsequently, E U2 S 6 X IV Pr[I℄  jS[I℄j+ ~ I  6E U1 S + X IV Pr[I℄ ~ I = X v i 2S p i + X IV Pr[I℄ ~ I : (6)

Ontheotherhand,Pr[I℄= Q v i 2I p i Q v i = 2I (1 p i

). Theupperboundresultsfromthe ombination

of expression(6)and the one forPr[I℄.

WenowprovethelowerboundforE U2 S

. Let (G[ ~

I℄)bethe ardinalityofthesolutionprovided

by U2when applied to G[ ~

I℄; we then have

E U2 S = X vi2S p i + X IV Pr[I℄  G h ~ I i = X vi2S p i + X IV Y vi2I p i Y v i = 2I (1 p i )  G h ~ I i (7) For (G[ ~

I℄), sin e the greedy algorithm implied by U2 provides a maximal independent set,

the following holds ([1 ℄): (G[ ~ I℄)> j

~ Ij=(

~

+1). By substituting the expression for (G[ ~ I℄) in

expression(7), we obtain the lower bound laimed.

In the ase where all the verti es have the same presen e probability p, P v i 2S p i = p (G) and Pr[I℄=p jIj (1 p) n jIj

, and expression(5)follows immediately.

In order to obtain bounds implied by the last expression of the theorem (always

assum-ing identi al o urren e probabilities), we use inequality (G) > n=( G + 1). Moreover, P IV p jIj (1 p) n jIj =1(so, P IV p jIj (1 p) n jIj >0),and ~ I

=jInfS[I℄[ (S[I℄)gj6jInS[I℄j=jIn(S\I)j6jV n(S\V)j=jV nSj=n (G):

So, P IV p jIj (1 p) n jIj j ~ Ij6(n (G)) P IV p jIj (1 p) n jIj

=n (G) and ombining the

above inequalities, we obtain the laimedbounds.

4.3 Approximating optimal solutions for PIS2

4.3.1 Using argmaxf P v i 2S p i g as a priorisolution Set  S = argmax f P vi2S p i

: S independent setofGg and suppose that it is used as a priori

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Theorem 10. Approximation of ^ S by  S=argmax f v i 2S p i

: S independentset of Bg

guar-antees for PIS2 approximation ratio

max  p min 1+p max ; 1  G +1  :

Proof. Byexpression (7)(proof oftheorem 9),and sin e (G[ ~ I℄)6  (G),we get X vi2S p i 6E U2 S = X vi2S p i + X IV Pr[I℄  G[ ~ I℄  6  (G) 0  p max + X IV Pr[I℄ 1 A =  (G)( 1+p max ) (8) Sin e,  S=argmax f P v i 2S p i

: S independent setof Gg,then

E U2  S > X v i 2  S p i > X v i 2S  p i >p min  (G) (9)

Remark now that expression (8) holds also for E U2 ^ S

; onsequently, ombining expressions (8)

and (9)we obtain E U2  S E U2 ^ S > p min  (G) (1+p max )  (G) = p min 1+p max (10)

Onthe other hand,remarkthat fromexpression(2)

E U2 ^ S 6 X v i 2V p i (11)

and fromthe lefthand-sideof expression(4)

E U2  S > X vi2  S p i (12)

Combination of expressions (11)and (12)gives

E U2  S E U2 ^ S > P v i 2  S p i P v i 2V p i > 1  G +1 (13)

where the last inequality (remark that 

S is maximal) is the weighted version of Turàn's

theo-rem ([16℄).

Expressions (10)and (13) on lude the theorem.

4.3.2 Polynomial time approximations forPIS2

Theset 

S onsideredinthepreviousparagraph annotbe omputedinpolynomialtime. Instead,

suppose that one uses a polynomial time approximation algorithm A (a hieving approximation

ratio ) for (unweighted) IS in order to ompute a solution SOL (obviously, we an suppose

thatSOL ismaximal) on Gwhere vertex-probabilities areomitted. Then, expression(9) in the

proofof theorem10 be omes

E U2 SOL >p min jSOLj>p min   (G)

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approximation ratio , then algorithm A polynomially solving PIS2withinapproximation ratio max  1  G +1 ;  p min 1+p max    :

If A is the algorithmof [6℄, then:

 in the ase of xed vertex-probabilities, PIS2 an be approximately solved in polynomial

time withinratio

min  O  logn 3( G +1)loglogn  ;O  n 4=5  

 inthe ase where probabilities dependonn,PIS2ispolynomiallyapproximable withinratio

max  1  G +1 ;  p min 1+p max  min  O  logn 3( G +1)loglogn  ;O  n 4=5   :

4.4 PIS2in bipartite graphs

Theorem 12. Consider a bipartite graphB =(V

1 ;V 2 ;E B ). Then, E U2 V1 = X v i 2V 1 p i + X v i 2V 2 p i Y v j 2 ( v i ) ( 1 p j )

and an be omputed in polynomial time. Consequently, in bipartite graphs, whenever

olor- lass V 1

(or V 2

)is onsidered as a priori solution,PIS22NP.

Proof. Letus rstnotethe following:

 ifall the verti es ofV 1

areabsent,then the solutionprovided byU2is exa tlythe present

verti es ofthe olor- lass V 2

;

 ifall theverti es ofV 1

arepresent,then,despitethe stateofthe setV 2

, the solutionofthe

present sub-instan eofB isexa tlythe olor- lass V 1

;

 in the ase where a part of the verti es of V 1

is present, the nal solution for B[I℄ will

eventually in lude someverti esof V 2

.

Applying theresult oftheorem 1,we get:

E U2 V1 = X vi2V1 p i + X vi2V2 Pr h X U2 ;V1 i =1 i (14) (re all that Pr[X U2 ;S i

= 1℄ represents the probability that vertex v i

=

2 S will be hosen when

applyingU2).

Note also thatPr[X U2 ;V 1 i =1℄, v i 2V 2

, dependsonly on the present verti es of (v i

);

onse-quently, it does not depend on the other elements of V 2

. Hen eforth, insertion of the elements

ofV 2

is performedindependently the onesfrom the othersand v i

2V

2

willbe introdu ed in the

solution forB[I℄ onlyif (v i )\V 1 [I℄=;. So,Pr[X U2 ;V 1 i =1℄=p i Q v j 2 (v i ) (1 p j ).

Repla ing thisexpressionfor Pr[X U2;V

1 i

=1℄inexpression (14), weobtain the result laimed

for E U2 V

1

. One an see that this expression implies the omputation of E U2 V 1 in at most O(n 2 ) steps.

Fromtheproofoftheorem12,one anseehowthe parti ularstru tureofthebipartite graph

intervenesinasigni ant waytosimplifytheexpressionforthe fun tionaland, onsequently, its

omputation. Expression(14)holdsthankstothefa tthatthevertexsetofB anbepartitioned

(17)

i i 1 2 1 2 1 2

respe tively,and suppose thatn 1 >n 2 . Then,E U2 V1 =pn 1 +p P vi2V2 (1 p) j (v i )j . Naturally, E U2 V1 is omputed in O(n 2 ).

From orollary 2 we an obtain the following framing of E U2 V 1 by E U1 V 1

for the ase of identi al

vertex-probabilities: E U1 V 1 +n 2 p(1 p)  B 6E U2 V 1 6E U1 V 1 +n 2 p(1 p) Æ B

. So, in regular bipartite

graphs (i.e., the oneswhere  B =Æ B =): E U2 V1 =E U1 V1 +n 2 p(1 p)  . Consider set  S = argmax f P v i 2S p i

: S independent set ofBg as a priori solution. Then,

onsidering vertex-probabilitiesasvertex-weightsandusingthe resultof[5℄, 

S anbe omputed

in polynomial time. Moreover,

E U2  S > X v i 2  S p i > P vi2V p i 2 :

Usingtheexpressionabovetogetherwithexpression(11),approximationratio1/2isimmediately

yielded. Proposition 2.  S =argmax f P vi2S p i

: S independent set of Bg is a polynomial time

approx-imationof ^

S guaranteeing approximation ratio1/2 for PIS2 in bipartite graphs.

Obviously, from theorem12andthe dis ussionjustabove,the sameapproximationratio an be

yielded ifoneuses argmax f P v i 2V 1 p i ; P v i 2V 2 p i gasa priorisolution.

5 The omplexity of PIS4

5.1 An expression for E

U4 S

Re all that strategy U4 starts from S[I℄ and ompletes it with the isolated verti es of the

graph G[ ~ I℄.

Proposition 3. Givena graph G=(V;E), an a priori independentset S andthemodi ation

strategy U4,then

E G U4 S  = X v i 2S p i + X v i 2(VnS) p i Y v j 2 S (v i ) (1 p j )  Y v k 2 VnS (v i ) 0  (1 p k )+p k 0  1 Y v l 2 S (v k ) (1 p l ) 1 A 1 A (15) where S (v i )= (v i )\S and VnS (v i )= (v i )\(V nS). E(G U4 S

) an be omputed inpolynomial

time. If p i =p, for all v i 2V, then E G U4 S  =p (G)+ X v i 2(VnS) p(1 p) j S (vi)j Y v k 2 VnS (v i )  1 p(1 p) j S (v k )j  (16) Proof. Set B i =Pr[X U4;S i =1℄. Then B i = X IV v i 2I Pr[I℄1 fvi2F(I U4 S )g Letv i

be anyvertexofV nS and letI be anysubset of V ontaining v i . Obviously, v i 2F(I U4 S ) () v i isolated in G[ ~ I℄ () (v i 2G[ ~ I℄) ^ (v i

haslost all itsneighborsin ~ I):

(18)

Sin ev i

= 2S, v

i

2I onlyiftheredoesnotbelongto theneighborhoodofanyvertexin S[I℄,i.e.,

v i 2G[ ~ I℄ () S (v i )\I =; (17)

Onthe other hand,allthe neighbors ofv i

in G[ ~

I℄ have been removed i VnS (v i )\ ~ I =;. This

last onditionis satisedonly ifevery vertexof VnS

(v i

) iseither absent or (being present) has

beenremoved from

~

I be ause itbelonged to the neighborhood of avertex inS[I℄. Inall,

VnS (v i )\ ~ I =; () 8v j 2 VnS (v i )( (v j is absent )_ ( (v j is present )^( 9v k 2 (v j )\S su h thatv k is present )) ) (18)

From relations (17)and (18)and the dis ussionabove we get(v i 2V nS): B i = X IV v i 2I Pr[I℄1 fv i 2F(I U4 S )g = X IV v i 2I Pr[I℄1  v i 2G[ ~ I℄ VnS (v i )\~ I=;  = X IV v i 2I Pr[I℄1 fv i 2G[ ~ I℄g 1 f VnS (v i )\ ~ I=;g = X IV Pr[I℄1 n fv i g\I=fv i g S (v i )\I=; o 1 f VnS (v i )\ ~ I=;g = X IV Pr[I℄1 ffv i g\I=fv i gg 1 f S (v i )\I=;g Y vj2 VnS (vi) 1 fI\fv j g=;g +1 fI\fv j g=fv j gg 1 fI\ S (v j )6=;g  = p i Y v k 2 S (v i ) (1 p k ) Y v j 2 VnS (v i ) 0  (1 p j )+p j 0  1 Y v l 2 S (v j ) (1 p l ) 1 A 1 A (19)

Repla ingthe se ondtermof expression(2)byexpression(19)we easilyobtain expression(15).

Moreover, one an see that omputation of E(G

U4 S

) in expression (15) takes at most n 2

multipli ations. Also,setting p i

=p,8v i

2V, we immediately obtain expression(16).

5.2 UsingS  or argmaxf P v i 2S p i g as a priori solutions

Expression (15) although polynomial does not allow pre ise hara terization of the optimal a

priori solution ^

S asso iated with U4. In theorem 13 below, we restri t ourselves to the ase of

identi al vertex probabilities and suppose that 

(G), a maximum-size independent set of Gis

usedasan apriori solution. Ourobje tive is toestimatethe ratioE U4 S =E U4 ^ S .

Theorem 13. Under identi al vertex-probabilities:

E U4 S  E U4 ^ S >  (G) n : The ratio 

(G)=nisalways bounded belowby 1=( G +1). Proof. Set j S (v i )j=k i , j VnS (v i )j=j (v i )j k i

. Then expression(16) be omes

E G U4 S  =p (G)+ X v i 2(VnS) p(1 p) ki Y v k 2 VnS (v i )  1 p(1 p) k k  :

Also,notethat, 8k, k k 6 G 1(v i andv k arein V nSand v i v k 2E) andE(G U4 S ) isin reasing in k k . Then, E  G U4 ^ S  6 p (G)^ + X v i 2(Vn ^ S) p(1 p) k i Y v k 2 Vn ^ S (v i ) 1 p(1 p)  G 1 

(19)

6 p (G)^ + v i 2(Vn ^ S) p(1 p) k i 1 p(1 p)  G 1  j (v i )j k i 1 p61 p(1 p)  G 1 6 p (G)^ + X v i 2(Vn ^ S) p 1 p(1 p)  G 1  j (v i )j j (v i )j>1 6 p (G)^ +(n (G))^ p 1 p(1 p)  G 1  6 pn (20)

Onthe other hand,ifwe onsiderS 

asana priorisolution, we get

E G U4 S   = p  (G)+ X v i 2(VnS  ) p(1 p) k i Y v k 2 VnS (v i )  1 p(1 p) k k  1 p(1 p) k k >1 p > p  (G)+ X vi2(VnS  ) p(1 p) k i Y v k 2 VnS (vi) (1 p) = p  (G)+ X v i 2(VnS  ) p(1 p) ki (1 p) j (vi)j ki = p  (G)+ X v i 2(VnS  ) p(1 p) j (v i )j > p  (G)+(n  (G))p(1 p)  G = p  (G) 1 (1 p)  G  +pn(1 p)  G (21)

Combining expressions (20)and(21), we have

E G U4 S   E  G U4 ^ S  >  (G) n 1 (1 p)  G  +(1 p)  G >  (G) n > 1  G +1 :

For instan e, ifGis ubi , i.e.,  G =3, then  (G)>n=4 then, E(G U4 S  )=E(G U4 ^ S )>((1=4)(1 (1 p) 3 ))+(1 p) 3

>1=4. If, in addition p=1=2, then E(G U4 S )=E(G U4 ^ S )>11=32.

The result of theorem 13 an be easily extended in the ase where vertex-probabilities are

distin tand  S =argmax f P v i 2S p i

: S independent setof Ggisusedasapriorisolution.

More-over,without lossof generality, one ansuppose that 

S ismaximal. Then, fromexpression(15)

one easily gets:

E  G U4 ^ S  6 X v i 2 ^ S p i + X v i 2(Vn ^ S) p i = X vi2V p i (22) E G U4  S  > X v i 2  S p i (23)

Combining the above expressions weobtain

E G U4  S  E  G U4 ^ S  > P v i 2  S p i P v i 2V p i > 1  G +1

where the last inequalityis the weighted versionofTuràn's theorem.

Finally, let us note thatthe same approximation ratio an be obtained if one treats vertex

(20)

weight oververtex-degree andeliminatesitsneighbors. Inthis ase, ifS 0

isthe independentset

omputed we have([15 ℄) E G U4 S 0  > X v i 2S 0 p i > P v i 2V p i  G +1

and using expression (22), approximation ratio bounded below by 1=( G

+1) is immediately

on luded.

5.3 PIS4in bipartite graphs

Theparti ularstru tureofabipartitegraph,denotedaspreviouslybyB =(V 1

;V 2

;E),doesnot

allowrenementof theresultof proposition3in ordertoobtain abetter hara terization ofthe

apriori solutionmaximizing E(G U4 S

) and ofthe omplexityof its omputation.

However,ifargmax fjV 1

j;jV 2

jgisusedasapriorisolution, thenexpression(15) anbe

simpli-ed. Plainly,letusrevisititandsuppose,withoutlossofgenerality,thatV 1 =argmax fjV 1 j;jV 2 jg. So, we have S =V 1 and V nS =V 2 . Consequently, for v i 2V 2 , (v i )\V 2 = VnS (v i )=; and theterm Q v k 2 VnS (v i ) ((1 p k )+p k (1 Q v l 2 S (v k ) (1 p l

)))(thelastprodu tofexpression(15)),

omputed on an empty set takes, by onvention, value 1. So, dealing with bipartite graphs,

expression(15) be omes E B U4 V1  = X v i 2V 1 p i + X v i 2V 2 p i Y v j 2 V 1 (v i ) (1 p j ) (24)

In what follows, we will prove that if we use the olor- lass argmax f P v i 2V1 p i ; P v i 2V2 p i g as

a priori solution for PIS4, then it is solved within approximation ratio 1/2. In fa t, let V 1 = argmax f P v i 2V 1 p i ; P v i 2V 2 p i

g. Then, byexpression (24), we get

E B U4 V 1  > X v i 2V 1 p i > P v i 2V 1 [V 2 p i 2 (25)

Combining expression(25)with expression(22)we obtain

E B U4 V 1  E  B U4 ^ S  > P v i 2V 1 [V 2 p i 2 P v i 2V 1 [V 2 p i = 1 2 :

Of ourse, the same worst ase approximation ratio is also a hieved if one sees probabilities

as weights and onsiders the maximum-weight independent set (of total weight at least equal

to P v i 2V1 p i

;thisset anbefoundinpolynomialtime([5℄))asapriorisolutionandthefollowing

theorem on ludes the dis ussionabove.

Theorem 14. Sets argmaxf

P v i 2V1 p i ; P v i 2V2 p i g, and argmax SindependentsetofB f P v i 2S p i g

are polynomialapproximationsofPIS4inbipartitegraphs a hievingapproximationratiobounded

below by 1/2.

Finallyremark thatfor the ase of identi alvertex-probabilities, the result oftheorem 14 ould

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6.1 In general graphs

Were allthatstrategyU5 onsiderstherestri tionC[I℄ofanapriorivertex over inthepresent

subgraph G[I℄ of G, it removes the isolated verti es (if any) from C[I℄ and it nally takes the

omplement,with respe tto I,of the resulting set.

Theorem 15 . Given a graph G=(V;E), an a priori independent set S and the modi ation

strategy U5,then

E G U5 S  = X v i 2S p i + X v i 2(VnS) p i Y v j 2 (v i ) ( 1 p j ): (26) E(G U5 S

) is omputable in polynomialtime.

Proof. Lines (1) to (4) of modi ation strategy U5 in se tion 2 onstitute a modi ation

strategy, denoted by U in what follows, for probabilisti vertex over problem. For an a priori

vertex over C, the fun tional asso iated with U is(see [14℄ for adetailed omputation):

E U C = X v i 2C p i X v i 2C p i Y v j 2 (v i ) (1 p j ): (27)

Using expression(27), we havethe following for E(G U5 S ): E G U5 S  = X IV Pr[I℄F G[I℄ U5 S  = X IV Pr[I℄  jIj F  G[I℄ U VnS  = X IV Pr[I℄jIj X IV Pr[I℄F  G[I℄ U VnS  = X IV Pr[I℄ X vi2V 1 fvi2Ig E(G U VnS ) = X vi2V X IV Pr[I℄1 fv i 2Ig E  G U3 VnS  = X vi2V p i X v i 2(VnS) p i + X v i 2(VnS) p i Y v j 2 (v i ) ( 1 p j ) = X v i 2S p i + X vi2(VnS) p i Y vj2 (vi) (1 p j ):

Expression(26) anberewritten as

E G U5 S  = X vi2V p i X v i 2(VnS) p i + X v i 2(VnS) p i Y v j 2 (v i ) (1 p j ) = X v i 2V p i X v i 2(VnS) p i 0  1 Y v j 2 (v i ) ( 1 p j ) 1 A :

Sin e, given a graph G, the quantity P v i 2V p i

is onstant, maximization of E(G U5 S

) be omes

equivalent to the minimization of P vi2(VnS) p i (1 Q vj2 (vi) (1 p j

)). But S being a maximal

independentset,VnSisaminimalvertex overingofGandinordertondthevertex overingC

minimizing thequantity P v i 2C p i (1 Q v j 2 (v i ) (1 p j

)),onehassimplyto onsiderea hvertex

v i

2V asweighted by the weight w i =p i (1 Q v j 2 (v i ) (1 p j

)) and to sear h for a

minimum-weight vertex over. Consequently the following theorem hara terizes the a priori solution

maximizingE(G

U5 S

).

Theorem 16. The a priori solution

^

S maximizing E(G

U5 S

) is the omplement, with respe t

to V, of a minimum-weight vertex over of G where every vertex v i is weighted by a weight p i (1 Q v j 2 (v i ) (1 p j

)). Consequently, PIS5 isNP-hard.

Inotherwords, theorem16establishes that,asinthe aseofPIS1,PIS5isequivalentto aWIS.

(22)

Sin e maximum-weight independent is polynomial in bipartite graphs ([5 ℄), so does

minimum-weight vertex overing. Sothe following theorem immediatelyholds.

Theorem 17. The a priori solution

^

S maximizing E(B

U5 S

) is the omplement, with respe t

toV 1

[V 2

,of a minimum-weight vertex over ofB where every vertexv i isweighted by a weight p i (1 Q v j 2 (v i ) (1 p j

)). Consequently, PIS5 ispolynomial for bipartite graphs.

7 Con lusions

We have drawn a formal framework for the study of PCOPs and studied ve variants of the

probabilisti maximum independent set, dened with respe tto natural modi ationstrategies

usedto adapt an apriori solutionto the present sub-instan e.

Table 1: Complexities of omputing fun tionals and

hara teriza-tions and omplexities of omputing a priori solutions for several

variants ofprobabilisti independent set.

PIS1 PIS2 PIS4 PIS5

Generalgraphs T(E) O(n) O 2 n (G)  O n 2  O n 2  ^ S WIS(G w p ) hard ? WIS(G w p 0 )

Complexity NP-hard ? ? NP-hard

Bipartitegraphs T(E) O(n) ? ( ) O n 2  O n 2  ^ S WIS(B w p ) ? ? WIS(B w p 0 ) Complexity P ? ? P ( ) Polynomialifargmax fjV 1 j;jV 2

jgisusedasa priorisolution.

Dealing with the quality of the solutions obtained we have the following relation for the

same apriori solutionS (letus note that,until now, we have not been ableto ompare E(G U2 S ) with E(G U3 S )): E(G U1 S )6E(G U5 S )6E(G U4 S )6E(G U2 S ) (28)

First inequalityin expression (28)isobviousand followsfrom theorem2 andexpression (26)of

theorem15. In orderto prove the se ondinequality, remark that(expression (15))

p k  0  1 Y v l 2 S (v k ) ( 1 p l ) 1 A 0 and onsequently, E G U4 S  > X v i 2S p i + X v i 2(VnS) p i  Y v j 2 S (v i ) (1 p j ) Y v k 2 VnS ( v i ) ( 1 p k ) > X vi2S p i + X v i 2(VnS) p i  Y v j 2 ( S ( v i ) [ VnS (v i ) ) (1 p j ) = X v i 2S p i + X v i 2(VnS) p i  Y v j 2 (v i ) (1 p j ) > E G U5 S  :

(23)

S Approximationratio

Generalgraphs

PIS1 Theone omputed in [6 ℄ r

Theone omputed in [6 ℄ max

n p min 1+pmax  r; 1  G +1 o (a) PIS2 argmax f P vi2S p i g max n p min 1+pmax ; 1  G +1 o argmax f P v i 2S p i g 1  G +1 (b) PIS4

Theoutput ofthe greedy algorithm

1 

G +1

PIS5 Theone omputed in [6 ℄ r

Bipartitegraphs PIS2 argmax f P v i 2S p i g 1 2 ( ) PIS4 argmax f P v i 2S p i g 1 2 (d) (a) minfO(logn=(3( G

+1)loglogn));O(n 4=5 )g if vertex-probabilities independent ofn. (b)  (G)=nwhenever S =S 

and vertex-probabilitiesareidenti al. ( )

The same ratio is a hieved onsidering argmax f P v i 2V 1 p i ; P v i 2V 2 p i g asa priorisolution. (d)

The same ratio is a hieved onsidering argmax f P vi2V1 p i ; P vi2V2 p i g asa priorisolution.

Lastinequalityisduetothefa tthatfor everysubgraphG[I℄,the ardinalityofthesolution

omputed applying U2will be greater than the one of the solution omputed applying U4 sin e

GREEDY( alled byU2) will always addin thesolution, at leastthe isolatedverti es of G[ ~ I℄.

Table1 summarizes the main resultsof this paper about the omplexities of omputing the

fun tionals and the onesof omputingthe apriori solutions maximizingthem. Inthis table we

denotebyG

w p

,agraphGwhoseverti esareweightedbytheir orrespondingprobabilities,byG w p

0

agraphwhosevertexv i

isweighted bythequantityp i (1 Q v j 2 (v i ) (1 p j )),16i6n,byT(E)

the time needed for the omputation of the fun tional E and by WIS(G

w p ) (resp., WIS(G w p 0)),

the fa tthat the a priorisolution maximizingthe fun tional isa maximum-weight independent

setin G w p (resp., G w p 0

). Finally, G andB denotegeneral and bipartite graphs, respe tively.

Intable2,wherewedenotebyrthe quantityminfO(logn=(3( G

+1)loglogn));O(n 4=5

)g,

asummary ofthe main approximationresults ispresented. Letus notethatthe approximation

ratioin thelineforPIS5in generalgraphsoftable2isdire tlyobtained withargumentsexa tly

analogous to the ones of orollary 1 in se tion 3.2, onsidering p i (1 Q v j 2 (v i ) (1 p j )) as vertex-weight for v i , i=1;:::;n.

(24)

[1℄ C. Berge. Graphs and hypergraphs. NorthHolland, Amsterdam,1973.

[2℄ D.J.Bertsimas. Probabilisti ombinatorial optimization problems. PhDthesis, Operations

Resear h Center, MIT, Cambridge Mass.,USA,1988.

[3℄ D. J. Bertsimas, P. Jaillet, and A. Odoni. A priori optimization. Oper. Res., 38(6):1019

1033, 1990.

[4℄ B. Bollobás. Random graphs. A ademi Press, London,1985.

[5℄ J.-M. Bourjolly, P. L.Hammer, and B. Simeone. Node-weighted graphshaving the

König-Egervaryproperty. Math.ProgrammingStud., 22:4463,1984.

[6℄ M.DemangeandV.T.Pas hos.Maximum-weightindependentsetisaswell-approximated

astheunweighted one.Te hni al Report163,LAMSADE,UniversitéParis-Dauphine,1999.

[7℄ V. Gabrel, A. Moulet, C. Murat, and V. T. Pas hos. A new single model and derived

algorithms forthe satelliteshotplanning problemusing graphtheory on epts. Ann.Oper.

Res., 69:115134,1997.

[8℄ M. R. Garey and D. S. Johnson. Computers and intra tability. A guide to the theory of

NP- ompleteness. W. H.Freeman, SanFran is o, 1979.

[9℄ F. Harary. Graphtheory. Addison-Wesley, Reading, MA,1969.

[10℄ J. Håstad. Cliqueishard to approximate within n 1 

. Toappearin A taMathemati a.

[11℄ P. Jaillet. Probabilisti traveling salesman problem. Te hni al Report 185, Operations

Resear h Center, MIT,Cambridge Mass., USA,1985.

[12℄ P. Jaillet. Shortestpath problemswith node failures. Networks, 22:589605,1992.

[13℄ C.Murat.Probabilisti ombinatorialoptimizationgraph-problems.PhDthesis,LAMSADE,

Université Paris-Dauphine, 1997. InFren h.

[14℄ C. Murat and V. T. Pas hos. The probabilisti minimum vertex overing problem.

Manus ript, LAMSADE, UniversitéParis-Dauphine, Paris,Fran e.

[15℄ V.T.Pas hos.Asurveyabouthowoptimalsolutionstosome overingandpa kingproblems

an beapproximated. ACM Comput. Surveys, 29(2):171209,1997.

[16℄ P. Turán. On an extremal problem in graph theory (in Hungarian). Mat. Fiz. Lapok,

(25)

A.1 Determining A

 (G)

Letus rst prove a small preliminarylemma.

Lemma 1. If SI, then F(I

U2 S

)= (G).

Proof. Infa t,bydenitionofU2,F(I U2 S

)=jS[I℄j+jS[ ~

I℄j. ButifS I, thenS[I℄=S andthis

implies S[I℄[ (S[I℄)= S[ (S). Moreover, the maximality of S implies that S[ (S)= V;

and onsequently, S[ ~ I℄=;. So, F(I U2 S )=jS[I℄j= (G). For A  (G) wehave jS  \Ij=  (G)and,bylemma1,F(I U2 S  )=  (G); therefore, A  (G) =  (G) 0 B  X IV jS  \Ij=  (G) Pr[I℄ 1 C A =  (G)p  (G) n  (G) X i=0 n  (G) i ! p i (1 p) n  (G) i =  (G)p  (G)

where,in theabove expression, thetermp

 (G)

representsthe fa tthatthe 

(G)verti esofS 

areallpresent in I (S 

\I =S 

) andthe term P n  (G) i=0 C n  (G) i p i (1 p) n  (G) i stands for

allpossible hoi esfortherestoftheelementsofV (aswehavealreadyseen,thistermequals1).

A.2 Determining A  (G) 1 ConsideranelementvofS 

andnotethat,sin eS 

isamaximumindependentset, 0

(v)iseither

empty, or a lique on at least one vertex(if not, S  ould beaugmented to (S  nfvg)[ 0 (v)).

Consequently, by onsidering thatv isthe element of S 

whi h isabsentfrom I, we have:

 if I \ 0

(v) = ;, then the independent set S 

nfvg remains a maximal one for G[I℄, so

F(I U2 S  )=  (G) 1;  ifI\ 0

(v)6=;,thenthe independent setS 

nfvg(in ludedin G[I℄) anbeaugmented by

exa tlyone element of 0 (v),soF(I U2 S )=  (G).

We nowstudy the twofollowing ases with respe tto 0 (v),namely 0 (v)=; and 0 (v)6=;. 0 (v)=;. F(I U2 S )=  (G) 1 and X IV (S  nfv g)I Pr[I℄F I U2 S   =(  (G) 1) X IV (S  nfv g)I Pr[I℄ X IV (S  nfv g)I Pr[I℄=p  (G) 1 (1 p) n  (G) X i=0  n  (G) i  p i (1 p) n  (G) i =p  (G) 1 (1 p) Consequently, X IV (S  nfvg)I Pr[I℄F I U2 S   =(  (G) 1)p  (G) 1 (1 p): 0 (v)6=;.

Here, we study the following two sub- ases,namely, I\ 0

(v)=; andI \ 0

(26)

Sub aseI\ (v)=;: here F(I U2 S  )= 

(G) 1 and, moreover, no vertex of

0 (v) is ontained in I; so, we get X IV (S  nfvg)I I\ 0 (v)=; Pr[I℄=p  (G) 1 (1 p)(1 p) j 0 (v)j =p  (G) 1 (1 p) j 0 (v)j+1 : (A.1) Sub aseI\ 0 (v)6=;: F(I U2 S  )= 

(G)andI ontains at leasta vertexof 0

(v);so,weget forthis ase:

X IV (S  nfvg)I I\ 0 (v)6=; Pr[I℄=p  (G) 1 (1 p) h 1 (1 p) j 0 (v)j i ; (A.2)

Consequently, for ase 0

(v) 6=; we have ombining expressions (A.1) and (A.2) together

with expressions for F(I U2 S



)of the two sub- ases:

X IV (S  nfvg)I Pr[I℄F I U2 S   =p  (G) 1 (1 p) h  (G) (1 p) j 0 (v)j i :

This on ludesthe study of ase 0

(v) 6=;.

By summingthe aboveexpressions over all elementsof S  , we get: A  (G) 1 = X v2S  X IV Pr[I℄F I U2 S   1 f(S  nfvg)Ig = X v2S  X IV Pr[I℄F I U2 S   1 f(S  nfvg)Ig 1 f 0 (v)=;g +1 f 0 (v)6=;g  = ` 1 (  (G) 1)p  (G) 1 (1 p)+ X v2S  0 (v)6=; p  (G) 1 (1 p)   (G) (1 p) j 0 (v)j  = ` 1 (  (G) 1)p  (G) 1 (1 p)+(  (G) ` 1 )p  (G) 1 (1 p)  (G) p  (G) 1 (1 p) X v2S  0 (v)6=; (1 p) j 0 (v)j = p  (G) 1 (1 p) 0 B ` 1  (G) ` 1 +  (G) 2 ` 1  (G) X v2S  0 (v)6=; (1 p) j 0 (v)j 1 C A = p  (G) 1 (1 p) 0 B  ` 1 +  (G)(` 1 +` 0 1 ) X v2S  0 (v)6=; (1 p) j 0 (v)j 1 C A :

Figure

Table 1 summarizes the main results of this paper about the 
omplexities of 
omputing the

Références

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