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
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
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.
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
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 .
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
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
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 .
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)
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
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
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 =;℄.
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.
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
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)
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
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):
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
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
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
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.
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 :
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.
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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
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 :