The probabilistic minimum vertex covering problem

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Laboratoire d'Analyse et Modélisation de Systèmes pour

l'Aide à la Décision

CNRS UMR 7024

CAHIER DU LAMSADE

170

Juin 2000

The probabilistic minimum vertex covering problem

Cécile Murat, Vangelis Th. Paschos

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Résumé ii

Abstra t ii

1 Introdu tion 1

2 The strategies M1, M2 and M3 and a general preliminary result 2 2.1 Spe i ation of M1, M2and M3 . . . 2 2.2 Arst expressionfor the fun tionals . . . 2

3 The omplexity of

(H, C, M1, E(G

M1

C

), min)

3

4 The omplexity of

(H, C, M2, E(G

M2

C

), min)

4

5 The omplexity of

(H, C, M3, E(G

M3

C

), min)

4

5.1 Building

E(G

M3

C

)

. . . 4 5.2 Boundsfor

E(G

M3

C

)

. . . 6 5.3 Onthe quality ofthe boundsobtained . . . 7

6 Some simulation results 10

7 Con lusions 11

A Proof of expression (11) 13

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Résumé

Une instan eduproblèmedela ouverture desommetsprobabilisteestune paire(

G =

(V, E), Pr

)obtenueenasso iantà haquesommet

v

i

∈ V

uneprobabilité

((

d'o uren e

)) p

i

. Nous onsidéronsunestratégiedemodi ationMtransformantune ouverturedesommets

C

de

G

enune ouverturede sommets

C

I

pourlesous-graphede

G

induit parl'ensemblede sommets

I

⊆ V

. L'obje tif, pourla ouverture de sommetsprobabiliste,est de déterminer une ouverturedesommetsde

G

minimisantlasomme,sur touslessous-ensembles

I

⊆ V

, des produits: probabilité de

I

fois

C

I

. Dans et arti le, nous étudions la omplexité de résolutionoptimaleduproblèmede ouverturedesommetprobabiliste.

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

The probabilisti minimum vertex overing problem

Abstra t

An instan e of the probabilisti vertex- overing problem is a pair

(G = (V, E), Pr)

ob-tainedbyasso iatingwithea hvertex

v

i

∈ V

ano urren e probability

p

i

. We onsidera modi ationstrategy Mtransformingavertex over

C

for

G

intoavertex over

C

I

for the subgraph of

G

indu edby avertex-set

I

⊆ V

. Theobje tive for the probabilisti vertex- overingistodetermineavertex overof

G

minimizingthesum,overallsubsets

I

⊆ V

,of theprodu ts: probabilityof

I

times

C

I

. Inthispaper,westudythe omplexityofoptimally solvingprobabilisti vertex overing.

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Given a graph

G = (V, E)

, a vertex over is a set

C

⊆ V

su h that for every

v

i

v

j

∈ E

either

v

i

∈ C

, or

v

j

∈ C

. The vertex- overing problem (VC) onsists in nding a vertex over of minimum size. A naturalgeneralization ofVC,denotedbyWVCinthe sequel,istheone where positive weights areasso iated with the verti es of

V

. The obje tive for WVC is to ompute a vertex over forwhi hthe sumofthe weightsofitsverti es isthesmallest over allvertex overs of

G

.

Let

G

be the set of graphs. Given

G

∈ G

, denote by

C

(G)

the set of vertex overs of

G

. For any subset

I

⊆ V

we denote by

G[I]

the subgraph of

G

indu ed by

I

. The probabilisti vertex- overing, denoted by PVC in what follows, is aquintuple

(

H, C(G), M, E(G

M

C

), min)

su h that:

H

is the set of instan es of PVC dened as

H = {G = (G, Pr)}

, where

G

∈ G

and

Pr

is an

n

-ve tor of vertex-probabilities;

C

_(G)

is the setof vertex oversof

G

(

C

(G) = C(G)

);

M

is a modi ation strategy, i.e., an algorithm re eiving

C

∈ C(G)

and a

G[I]

,

I

⊆ V

as inputs and modifying

C

inorder to produ e avertex over for

G[I]

;

E(G

M

C

)

is the fun tional, or obje tive value of

C

; onsider

G

∈ H

,

C

∈ C(G)

,

I

⊆ V

, and let

p

i

= Pr[v

i

]

,

v

i

∈ V

, be the probability that vertex

v

i

is present; moreover, denote by

C

M

I

the solution omputed by

M

(C, G[I])

and by

|C

M

I

|

the ardinality of

C

M

I

; nally, denoteby

Pr[I] =

Q

i∈I

p

i

Q

i∈V \I

(1

−p

i

)

the probabilitythatvertex-subset

I

ispresent; then

E(G

M

C

) =

P

I⊆V

Pr[I]

|C

I

M

|

.

The omplexityofPVCisthe omplexityof omputing

C = argmin

ˆ

C∈C(

G

)

{E(G

M

C

)

}

. Using the terminology of [4, 5℄, we will all the solution

C

(in whi h the modi ation strategy isapplied) a priori solution.

Apriorioptimization,i.e., sear hingforoptimalapriorisolutionsofprobabilisti ombinato-rialoptimizationproblems,hasbeenstudiedinrestri tedversionsofroutingandnetwork-design probabilisti minimizationproblems([1 ,2 ,3,4,5,6,7 ,8℄), denedon ompletegraphs. Finally, a priori optimization has been used in [9℄ to study a probabilisti maximization problem, the longestpath.

One anremarkthatin theabove denitionofPVC, M ispartoftheinstan eandthis seems somewhat unusual withrespe ttostandard omplexitytheorywhereno algorithm intervenes in the denition of a problem. But M is absolutely not an algorithm for PVC, in the sense that it does not ompute

C

∈ C(G)

. It simply ts

C

(no matter how it has been omputed) to

I

.

Moreover,letusnotethatwhen hanging Mone hangesthedenitionofPVCitself. Stri tly speaking, the PVC-variants indu ed by the quintuples

Q1 = (

H, C, M

1 , E(G

M

1 C

), min)

and

Q2 =

(

_{H, C, M}

2 , E(G

M

_C

2 ), min)

aretwo distin t probabilisti ombinatorial optimization problems. Weusethreemodi ationstrategiesM1,M2andM3andstudytheir orrespondingfun tionals. For M1 and M2, we produ e expli it expressions for their fun tionals, expressions allowing us to ompletely hara terize the solutions minimizing these fun tionals. On the ontrary, the expressionforthefun tionalasso iatedwithM3isnotquiteexpli itinordertoallowa hievement of similar results. Therefore, we give boundsfor this fun tional and study their quality. Let us re allthat the threestrategies introdu e in fa tthree distin t probabilisti PVC-variants.

In what follows, given a graph

G = (V, E)

of order

n

, we denote by

C

a vertex over of

G

, by

τ

its ardinality and by

C

ˆ

i

,

i = 1, 2, 3

, the optimal PVC-solutions asso iated with

M1

,

M2

and

M3

, respe tively. Moreover, by

Γ(v

i

)

,

i = 1, . . . , n

, we denote the set of neighbors of the

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vertex

v

i

(thedegree of

v

i

). Wewill set

δ = min

v

i

∈V

{|Γ(v

i

)

|}

and

∆ = max

v

i

∈V

{|Γ(v

i

)

|}

. Given avertex over

C

of

G

andasubset

V

′

_{⊆ V}

,we willset

C[V

′

_{] = C}

_{∩ V}

′

and

¯

C[V

′

] = (V

\ C) ∩ V

′

. Finally, when dealing with WVC,we will denoteby

w

i

the weight of

v

i

∈ V

.

2 The strategies M1, M2 and M3 and a general preliminaryresult 2.1 Spe i ation of M1, M2 and M3

Given a vertex over

C

and a vertex-subset

I

⊆ V

,

M 1

onsists in simply moving verti es of

C

_{\ I}

(theabsent verti es of

C

) out of

C

andin retaining

C[I] = C

∩ I

asvertex over of

G[I]

. BEGIN (*M1(C,G[I℄)*)

C

[I]

← C ∩ I

; OUTPUT C[I℄; END. (*M1(C,G[I℄)*)

Itiseasy to seethat

C[I]

onstitutesa vertex over for

G[I]

. Infa t,in the opposite ase, there would be at leastan edge

v

i

v

j

of

G[I]

for whi h neither

v

i

, nor

v

j

would belong to

G[I]

. But, sin e

v

i

v

j

∈ E

and

C

is a vertex over for

G

, at least one of

v

i

,

v

j

, say

v

i

, belongs to

C

and, onsequently, to

C[I] = C

∩ I

. Therefore

v

i

ispartofthe vertex over for

G[I]

, ontradi ting so the statement thatedge

v

i

v

j

isnot overedin

G[I]

.

The se ond strategy studied in this paper is a slight improvement of M1 sin e one removes isolated verti es from

C[I]

.

BEGIN (*M2(C,G[I℄)*)

C

_[I]

_{← C ∩ I}

;

R

_{← {v}

i

∈ C[I] : Γ(v

i

) =

∅}

; OUTPUT

C2

[I]

← C[I] \ R

; END. (*M2(C,G[I℄)*)

Finally, strategy M3is a further improvement of M2sin e itremoves from

C2[I]

verti es all the neighbors of whi h belong to

C2[I]

. In the following spe i ation of M3, algorithm SORT sorts the verti es of

C2[I]

in in reasing order with respe tto theirdegrees.

BEGIN (*M3(C,G[I℄)*)

C2

[I]

← M2(C, G[I])

;

C3

_[I]

← SORT(C2[I])

; FOR

i

← 1

TO |C3[I℄| DO

IF

∀v

j

∈ Γ(v

i

), v

j

∈ C3[I]

THEN

C3

[I]

← C3[I] \ {v

i

}

FI OD

OUTPUT C3[I℄; END. (*M3(C,G[I℄)*)

Itiseasytoseethatthevertex over

C3[I]

omputedbystrategyM3isminimal(forthein lusion) for

G[I]

.

2.2 A rst expression for the fun tionals

Consider any strategy that starting from

C[I]

redu es it by removing some of its verti es (if possible) in order to obtain smaller feasible vertex overs. Clearly, M1, M2 and M3 are su h strategies. Then, the following proposition provides a rst expression for fun tionals

E(G

M1

C

)

,

E(G

M2

C

)

and

E(G

M3

C

)

.

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Proposition 1. Considera vertex over

C

of

G

andstrategies M1,M2andM3. Withea hvertex

v

i

∈ V

asso iate a probability

p

i

and a random variable

X

Mk,C

i

,

k = 1, 2, 3

, dened, for every

I

_{⊆ V}

,by

X

_i

Mk,C

=

1 v

i

∈ C

Mk

I

0

otherwise Then,

E(G

Mk

C

) =

P

v

i

∈C

Pr[X

Mk,C

i

= 1]

.

Proof. Bythe denition of

X

Mk,C

i

we have

|C

Mk

I

| =

P

n

i=1

X

Mk,C

i

. So,

E G

Mk

C

=

X

I⊆V

Pr[I]

C

_I

M

=

X

I⊆V

Pr[I]

n

X

i=1

X

_i

Mk,C

=

n

X

i=1

X

I⊆V

Pr[I]X

_i

Mk,C

=

n

X

i=1

E

X

_i

Mk,C

=

n

X

i=1

Pr

h

X

_i

Mk,C

= 1

i

=

n

X

i=1

Pr

h

X

_i

Mk,C

= 1

i

1 {v

i

∈C}

+ 1

{v

i

∈C}

/

=

n

X

i=1

Pr

h

X

_i

Mk,C

= 1

i

1 {v

i

∈C}

+

n

X

i=1

Pr

h

X

_i

Mk,C

= 1

i

1 {v

i

∈C}

/

.

But, if

v

i

∈ C

/

, then,

∀I ⊂ V

su h that

v

i

∈ I

,

X

Mk,C

i

= 0

. So,

Pr[X

Mk,C

i

= 1] = 0

,

∀v

i

∈ C

/

. Consequently,

E(G

Mk

C

) =

P

n

i=1

Pr[X

Mk,C

i

= 1]1

{v

i

∈C}

=

P

v

i

∈C

Pr[X

Mk,C

i

= 1]

.

3 The omplexity of

(H, C, M1, E(G

M1

C

), min)

Thefollowing theorem an bededu ed fromproposition 1. Theorem 1 .

E(G

M1

C

) =

P

v

i

∈C

p

i

and

C

ˆ

1 = argmin

C∈C(G)

{

P

v

i

∈C

p

i

}

. In other words

C

ˆ

1

is a minimum-weight vertex over of

G

where the verti es of

V

are weighted by their orresponding presen e-probabilities. Inthe asewhereallvertex-probabilitiesareidenti al,

E(G

M1

C

) = pτ

and

ˆ

C

1

isa minimum vertex over of

G

. Consequently,

(

H, C, M1, E(G

M1

C

), min)

isNP-hard.

Proof. By strategy M1, if

v

i

∈ C

, then

v

i

∈ C

M1

I

, for all subgraphs

G[I]

su h that

v

i

∈ I

. Consequently,

Pr[X

M1,C

i

= 1] = p

i

,

∀v

i

∈ C

andthe resultfollows from proposition1.

Let us now onsider

G

with its verti es weighted by their orresponding probabilities and denote the soobtained instan e of WVCby

G

w

. The total weight of every vertex over of

G

w

is

P

v

i

∈C

p

i

and the optimal weighted vertex over of

G

w

is the one for whi h the sum of the weightsof itsverti esisthesmallest over allvertex oversof

G

w

. Su havertex over minimizes also

E(G

M1

C

)

and, onsequently onstitutes anoptimal solution for

(

H, C, M1, E(G

M1

C

), min)

. If

p

i

= p

,

1 6 i 6 n

, then

P

v

i

∈C

p

i

=

P

v

i

∈C

p = pτ

and, onsequently, the vertex over minimizing this last expressionisthe one minimizing

τ

, a minimum vertex over of

G

.

Expression

P

v

i

∈C

p

i

(theobje tivevalueoftheproblem

(

H, C, M1, E(G

M1

C

), min)

) anbe om-putedin

O(n)

,therefore,

(

H, C, M1, E(G

M1

C

), min)

∈ NP

. Furthermore,bytheisomorphybetween thisproblemandWVC,the formerproblemishard forNPandthis ompletesthe proofof the-orem 1.

Let us revisit ase

p

i

= p

,

1 6 i 6 n

. Sin e

C

M

I

= C[I]

and

0 6

|C[I]| 6 τ

, we get:

|C

M

I

| = |C[I]|

P

τ

i=1

1 {|C[I]|=i}

and the fun tionalfor

M1

an be written as

E G

M1

C

=

X

I⊆V

Pr[I]

_|C[I]|

τ

X

i=1

1 {|C[I]|=i}

=

τ

X

i=1

i

X

I⊆V

Pr[I]1

{|C[I]|=i}

=

τ

X

i=1

i

τ

i

p

i

(1

− p)

τ −i

n−τ

X

j=0

n − τ

j

p

j

(1

− p)

n−τ −j

= pτ

(7)

where in thelast summation we ount allthe sub-graphs

G[I]

su hthat

|C[I]| = i

, weadd their probabilities and we take into a ount that

P

n−τ

j=0

C

j

n−τ

p

j

(1

− p)

n−τ −j

= 1

.

The above an be generalized in order to ompute every moment of any order for

M1

. For instan e,

E((C

M

I

)

2 ) =

P

I⊆V

Pr[I](

|C

I

M

|)

2 =

P

τ

i=1

i

2 C

i

τ

p

i

(1

− p)

τ −i

= pτ (pτ + 1

− p)

and, onse-quently,

Var G

M1

C

= E

G

M1

C

2 − E G

M1

C

2 = τ p(1

_{− p).}

So,for

M1

andfor the aseofidenti alvertex-probabilities,the randomvariablerepresentingthe size

τ

follows abinomial lawwith parameters

τ

and

p

.

4 The omplexity of

(H, C, M2, E(G

M2

C

), min)

We re all that strategy M2 onsists in removing the isolated verti es from

C[I]

. Then, the following theoremholds.

Theorem 2.

E G

M2

C

=

X

v

i

∈C

p

i



1

₋

Y

v

j

∈Γ(v

i

)

(1

_{− p}

j

)





and

C

ˆ

2

is a minimum-weightvertex over of

G

w

where, for every

v

i

∈ V

,

w

i

= p

i



1

−

Y

v

j

∈Γ(v

i

)

(1

− p

j

)





.

Consequently,

(

H, C, M2, E(G

M2

C

), min)

isNP-hard.

Proof. Remarkrst that

C

M2

I

= C2[I] = C[I]

\ (∪

{v

i

∈C:(Γ(v

i

)∩I)=∅}

v

i

)

. Starting fromproposition 1we get

E G

M2

C

=

X

v

i

∈C

Pr

h

X

_i

M2,C

= 1

i

=

X

v

i

∈C

X

I⊆V

vi∈I

Pr[I]1

{v

i

∈C2[I]}

=

X

v

i

∈C

X

I⊆V

vi∈I

Pr[I] 1

− 1

{v

i

∈C2[I]}

/

.

But, for any

v

i

∈ C

,

v

i

∈ C2[I] ⇐⇒ I ∩ Γ(v

/

i

) =

∅

; onsequently,

E G

M2

C

=

X

v

i

∈C

X

I⊆V

vi∈I

Pr[I]

₋

X

v

i

∈C

X

I⊆V

vi∈I

Pr[I]1

_{I∩Γ(v

_i

_)=∅}

=

X

v

i

∈C

p

i



1

₋

Y

v

j

∈Γ(v

i

)

(1

_{− p}

j

)





.

Itiseasyto seethat

E G

M2

C

anbe omputed in

O(n

2 ₎

; onsequently

(

H, C, M2, E(G

M2

C

), min)

∈

NP

. Witha reasoning ompletely similarto the oneof theorem 1 one an immediatelydedu e that

C

ˆ

2 = argmin

C∈C(G)

{

P

v

i

∈C

p

i

(1

−

Q

v

j

∈Γ(v

i

)

(1

− p

j

))

}

, i.e., aminimum-weight vertex over of

G

w

where,for

v

i

∈ V

,

w

i

= p

i

(1

−

Q

v

j

∈Γ(v

i

)

(1

− p

j

))

. 5 The omplexity of

(H, C, M3, E(G

M3

C

), min)

5.1 Building

E(G

M3

C

)

Re allthatM3 onsistsin removing from

C[I]

boththe isolatedverti es and the verti es all the neighborsofwhi hbelongin

C[I]

. Consequently, thesolution

C3[I]

omputedbyM3isminimal.

(8)

E G

M3

C

=

X

vi∈C

p

i



1

₋

Y

vj

∈Γ(vi)

(1

_{− p}

j

)





−

X

vi∈C

X

I⊆V

Pr[I]1{

vi

∈C

/

M3

I

}1

{I∩C[Γ(vi)]6=∅}

1{

I∩ ¯

C[Γ(vi)]=∅

}

(1)

Proof. Byproposition 1,

E(G

M3

C

) =

P

v

i

∈C

Pr[X

M3,C

i

= 1]

. Moreover,

Pr

h

X

_i

M3,C

= 1

i

=

X

I⊆V

Pr[I]1{

v

i

∈C

_I

M3

}.

Note that

Γ(v

i

) = C[Γ(v

i

)]

∪ ¯

C[Γ(v

i

)]

and that if

v

i

∈ C

, then

C[Γ(v

¯

i

)]

6= ∅

(be ause

C

is minimal).

Letnow

v

i

∈ C

and

I

i

=

{I ⊆ V : v

i

∈ I}

. Thesetof graphs

G[I]

,

I

∈ I

i

an be partitioned in the following four subsets,a graph

G[I]

belonging to onlyone of them:

(i) graphssu hthat

I

∩ Γ(v

i

) =

∅

;

(ii) graphssu hthat

I

∩ C[Γ(v

i

)]

6= ∅

and

I

∩ ¯

C[Γ(v

i

)]

6= ∅

; (iii) graphssu hthat

I

∩ C[Γ(v

i

)]

6= ∅

and

I

∩ ¯

C[Γ(v

i

)] =

∅

; (iv) graphssu hthat

I

∩ C[Γ(v

i

)] =

∅

and

I

∩ ¯

C[Γ(v

i

)]

6= ∅

. Letus analysestrategy M3in ases (i)to (iv).

In ase (i) sin e

v

i

is isolated,M3will remove itfrom

C[I]

. In ases (ii) and (iv)

v

i

will be in luded in

C

M3

I

(sin e

I

∩ ¯

C[Γ(v

i

)]

6= ∅

,

∃v

j

∈ I ∩ Γ(v

i

)

su h that

v

j

∈ C

/

).

In ase (iii) thefa tthat

v

i

will be in ludedin, orwill be removed from, the solutiondepends on the order on whi h it will be treated by M3 and on wether or not all its neighbors are partof

C[I]

.

Inall,

• v

i

∈ C

/

I

M3

forall graphs verifying ase(i) andfor a partofgraphs verifying ase(iii);

• v

i

∈ C

_I

M3

for all graphs verifying ases (ii) and (iv) and for the rest of graphs verifying ase(iii). We sohave:

Pr[X

_i

M3,C

= 1] =

X

I⊆V

Pr[I]1{

vi∈C

M3

I

} =

X

I⊆V

Pr[I]1{

vi∈C

M3

I

}

1 {I∩C[Γ(vi)]6=∅}

1{

I∩ ¯

C[Γ(vi)]6=∅

_}

+ 1

{I∩C[Γ(vi)]=∅}

1{

I∩ ¯

C[Γ(vi)]6=∅

} + 1

{I∩C[Γ(vi)]6=∅}

1{

I∩ ¯

C[Γ(vi)]=∅

}

=

X

I⊆V

Pr[I]1{

vi∈C

M3

I

}

1{

I∩ ¯

C[Γ(vi)]6=∅

} + 1

{I∩C[Γ(vi)]6=∅}

1{

I∩ ¯

C[Γ(vi)]=∅

}

=

X

I⊆V

Pr[I]1{

vi∈C

M3

I

}1{

I∩ ¯

C[Γ(vi)]6=∅

}

+

X

I⊆V

Pr[I]1{

vi∈C

M3

I

}1

{I∩C[Γ(vi)]6=∅}

1{

I∩ ¯

C[Γ(vi)]=∅

}

=

X

I⊆V

Pr[I]1{

vi∈C

M3

I

}

1 _{− 1{}

_{I∩ ¯}

_{C[Γ(vi)]=∅}

_}

(9)

+

X

I⊆V

Pr[I]1{

vi∈C

M3

I

}1

{I∩C[Γ(vi)]6=∅}

1{

I∩ ¯

C[Γ(vi)]=∅

}

= p

i



1

₋

Y

vj

∈ ¯

C[Γ(vi)]

(1

_{− p}

j

)





+

X

I⊆V

Pr[I]1{

vi∈C

M3

I

}1

{I∩C[Γ(vi)]6=∅}

1{

I∩ ¯

C[Γ(vi)]=∅

}

Pr[X

i

M3,C

= 0] =

X

I⊆V

vi∈I

Pr[I]1{

vi

∈C

/

M3

I

} +

X

I⊆V

vi /

∈I

Pr[I]1{

vi

∈C

/

M3

I

}

=

X

I⊆V

vi∈I

Pr[I]1{

vi

∈C

/

M3

I

}

1 {I∩Γ(vi)=∅}

+ 1

{I∩C[Γ(vi)]6=∅}

1{

I∩ ¯

C[Γ(vi)]=∅

}

+ 1

− p

i

=

X

I⊆V

vi∈I

Pr[I]1{

vi

∈C

/

M3

I

}1

{I∩Γ(vi)=∅}

+ 1

− p

i

+

X

I⊆V

vi∈I

Pr[I]1{

vi

∈C

/

M3

I

}1

{I∩C[Γ(vi)]6=∅}

1{

I∩ ¯

C[Γ(vi)]=∅

}

= p

i

Y

vj

∈Γ(vi)

(1

_{− p}

j

) + 1

− p

i

+

X

I⊆V

vi∈I

Pr[I]1{

vi

∈C

/

M3

I

}1

{I∩C[Γ(vi)]6=∅}

1{

I∩ ¯

C[Γ(vi)]=∅

}

It is easy to see, after some easy but tedious algebra, that

Pr[X

M3,C

i

= 1] + Pr[X

M3,C

i

= 0] = 1

. Then, the resultof proposition2 isget using

E G

M3

C

=

X

v

i

∈C

Pr

h

X

_i

M3,C

= 1

i

=

X

v

i

∈C

1 − Pr

h

X

_i

M3,C

= 0

i

.

5.2 Bounds for

E(G

M3

C

)

Letusrstnotethatfromtheorems1,2andproposition2,wehave

E(G

M3

C

) 6 E(G

M2

C

) 6 E(G

M1

C

)

. Sin etheresultofproposition2doesnotallowthea hievementofapre ise hara terizationof

C

ˆ

3

, we givein the following theoremboundsfor

E(G

M3

C

)

. Theorem 3.

X

v

i

∈C

p

i



1

−

Y

v

j

∈ ¯

C[Γ(v

i

)]

(1

− p

j

)



 6

E G

M3

_C

6 X

v

i

∈C

p

i



1

−

Y

v

j

∈Γ(v

i

)

(1

− p

j

)





.

If

p

i

= p

,

∀v

i

∈ V

and

δ

C

= min

v

i

∈C

{|Γ(v

i

)

|}

, then

p

τ

− max

n

τ (1

− p)

∆

, (1

− p)

δ

C

o

>

E G

M3

C

>

p

2 τ.

The upper bound for

E(G

M3

C

)

is attained for the bipartite graphs when onsidering the one of the olor lasses as a priori solution.

Proof. The upper bound is obvious sin e it is nothing else than the expression of theorem 2 for

E(G

M2

C

)

. Let

A

i

=

X

I⊆V

vi∈I

Pr[I]1{

v

i

∈C

/

I

M3

}1

{I∩C[Γ(v

i

)]6=∅}

1{

I∩ ¯

C[Γ(v

i

)]=∅

}

(10)

A

i

6 X

I⊆V

vi∈I

Pr[I]1

{I∩C[Γ(v

i

)]6=∅}

1{

I∩ ¯

C[Γ(v

i

)]=∅

}

6 X

I⊆V

vi∈I

Pr[I] 1

− 1

{I∩C[Γ(v

i

)]=∅}

1{

I∩ ¯

C[Γ(v

i

)]=∅

}

6 p

i





Y

v

j

∈ ¯

C[Γ(v

i

)]

(1

_{− p}

j

)

−

Y

v

j

∈ ¯

C[Γ(v

i

)]

(1

_{− p}

j

)

Y

v

j

∈C[Γ(v

i

)]

(1

_{− p}

j

)





6 p

i





Y

v

j

∈ ¯

C[Γ(v

i

)]

(1

_{− p}

j

)

−

Y

v

j

∈Γ(v

i

)

(1

_{− p}

j

)





.

Combiningtheexpressionfor

A

i

abovewiththeexpressionfor

E(G

M3

C

)

ofproposition2,weeasily getthe lower bound laimed.

Let usnowsupposethat

p

i

= p

,

∀v

i

∈ V

. Then the expressionfor

E(G

M3

C

)

be omes

E G

M3

C

= pτ − p

X

v

i

∈C

(1

− p)

|Γ(v

i

)|

₋

X

v

i

∈C

A

i

with

0 ≤ A

i

≤ p((1 − p)

| ¯

C[Γ(v

i

)]|

− (1 − p)

|Γ(v

i

)|

₎

. Sin e

A

i

>

0

we have

E G

M3

C

6 pτ

_{− p}

X

v

i

∈C

(1

_{− p)}

|Γ(v

i

)|

_.

Using,

∀i

,

|Γ(v

i

)

| ≤ ∆

,weget

E G

M3

C

6 p τ

_{− τ(1 − p)}

∆

.

(3) Remarkthat

p

X

v

i

∈C

(1

_{− p)}

|Γ(v

i

)|

_{= p(1}

_{− p)}

δ

C

X

v

i

∈C

(1

_{− p)}

|Γ(v

i

)|−δ

C

X

v

i

∈C

(1

_{− p)}

|Γ(v

i

)|−δ

C

_>

₁

be ausethere existsat leastavertex

v

i

0 ∈ C

with

|Γ(v

i

0 )

| = δ

C

. Consequently,

E G

M3

C

6 p

τ

_{− (1 − p)}

δ

C

.

(4)

Combining expressions (3)and(4) oneimmediately obtains the upper bound laimed. Foridenti alvertex-probabilities,thelowerboundfor

E(G

M3

C

)

be omes

pτ (1

−(1−p)

| ¯

C[Γ(v

i

)]|

₎

. Sin e

C

is minimal,

¯

C[Γ(v

i

)]

6= ∅

, i.e.,

| ¯

C[Γ(v

i

)]

| > 1

. Using the latter inequality in the former one we get

E(G

M3

C

) > pτ (1

− (1 − p)) = p

2 τ

, proving sothe lower bound laimed.

Considernowabipartitegraph

B = (V

1 , V

2 , E)

andoneofits olor lasses,say olor lass

V

1

, as a priori solution. Remark that,

∀v

i

∈ V

1

,

Γ(v

i

) = ¯

V

1 [Γ(v

i

)]

∈ V

2

. Consequently, the upper and lower boundsfor

E(B

M3

V

1 )

oin ide.

5.3 On the quality of the bounds obtained We prove in this se tionthatthe boundsof

E(G

M3

C

)

obtained in theorem3are quitetight . Let

G = (V, E)

be a graph onsisting of a lique

K

ℓ

(on

ℓ

verti es) and of an independent set

S

on

σ

verti es; moreover onsider that any vertex of

K

ℓ

is linked to anyvertexof

S

. Let

(11)

us suppose thatall the verti es of

G

have the same probability

p

and onsider

C = V (K

ℓ

)

, the vertex-set of

K

ℓ

, asa priorisolution. Finallyset

n = ℓ + σ

.

For

v

i

∈ C

wehave

C[Γ(v

i

)] = V (K

ℓ

)

\ {v

i

}

and

C[Γ(v

¯

i

)] = S

. Then, expression(1)be omes

E G

M3

C

=

X

v

i

∈C

p

₋

X

v

i

∈C

p

Y

v

j

∈Γ(v

i

)

(1

_{− p) −}

X

v

i

∈C

X

I⊆V

Pr[I]1{

v

i

∈C

/

_I

M3

}1

{I∩(V (K

ℓ

)\{v

i

})6=∅}

1 {I∩S=∅}

(5)

Let

V (K

ℓ

) =

{v

1 , v

2 , v

3 , . . . , v

ℓ

}

(the degrees of the verti es of

V (K

ℓ

)

are all equal) and let

S =

{v

ℓ+1

, v

ℓ+2

, v

ℓ+3

, . . . , v

n

}

. Instep

i

, strategy M3tests ifvertex

v

i

an be removed.

Revisit now the third term of expression (5) (the double sum). This term deals with in-stan es

G[I]

su hthat

I

⊆ V (K

ℓ

)

\ {v

i

}

, in otherwords, instan eswhere all present verti es are elementsof

V (K

ℓ

)

, i.e., they arealsopartof

C

. Sin e

v

i

∈ V (K

ℓ

)

,

Γ(v

i

)

∩ I ⊆ C ∩ I

andvertex

v

i

∈ C

an be removed only if, it isnot isolated and for

j < i

,

v

j

∈ I

/

(be ause,in the opposite ase,

v

j

would be removed at step

j < i

, in other words, before

v

i

and, sin e

v

i

v

j

∈ E(K

ℓ

)

,

v

i

shouldnotberemoved). Consequently,

1 {v

i

∈C

/

_I

M3

}

= 1

{I∩{v

1 ,v

2 ,v

3 ,...,v

i−1

}=∅}

andforaxedindex

i

, expression(2) be omes

A

i

=

X

I⊆V

Pr[I]1

{I∩{v

1 ,v

2 ,v

3 ,...,v

i−1

}=∅}

1 {I∩V (K

ℓ

)\{v

i

}6=∅}

1 {I∩S=∅}

=

X

I⊆V

Pr[I]1







I

_∩vi=vi

I∩

{

v1,v2,v3,...,vi−1

}

=∅

I∩

{

vi+1,vi+2,vi+3,...,vℓ

}

6=∅







1 _{I∩S=∅}

=

i−1

Y

j=1

(1

− p)p



1

−

ℓ

Y

j=i+1

(1

− p)





n

Y

j=ℓ+1

(1

− p)

= (1

_{− p)}

i−1

p

1 _{− (1 − p)}

ℓ−(i+1)+1

(1

_{− p)}

n−(ℓ+1)+1

= (1

_{− p)}

i−1

p

1 _{− (1 − p)}

ℓ−i

(1

_{− p)}

n−ℓ

.

We sohave,

X

v

i

∈C

A

i

=

ℓ

X

i=1

p

(1

− p)

n+i−ℓ−1

− (1 − p)

n−1

= p(1

− p)

n−ℓ−1

ℓ

X

i=1

(1

− p)

i

− pℓ(1 − p)

n−1

= (1

− p)

n−ℓ−1

1 − p − (1 − p)

ℓ+1

− pℓ(1 − p)

n−1

= (1

_{− p)}

n−ℓ

_{− (1 − p)}

n

_{− pℓ(1 − p)}

n−1

(6) Remark that omputation of

A

i

's in the expression above is polynomial and, onsequently, the whole omputation for

E(G

M3

C

)

is alsopolynomial. Combining nowexpressions (5)and (6),weget

E G

M3

C

=

ℓ

X

i=1

p

₋

ℓ

X

i=1

p

Y

vj

∈Γ(vi)

(1

_{− p) − (1 − p)}

n−ℓ

− (1 − p)

n

− pℓ(1 − p)

n−1

=

ℓp

− ℓp(1 − p)

n−1

− (1 − p)

n−ℓ

_{+ (1}

− p)

n

_{+ pℓ(1}

− p)

n−1

=

ℓp

_{− (1 − p)}

n−ℓ

1 _{− (1 − p)}

ℓ

.

(12)

e(ℓ, p, n)

def

=

ℓp

− (1 − p)

n−ℓ

1 − (1 − p)

ℓ

(7)

b(ℓ, p, n)

def

=

ℓp

1 _{− (1 − p)}

n−ℓ

(8)

B(ℓ, p, n)

def

=

ℓp 1

_{− (1 − p)}

n−1

(9) and remark thatexpressions (8)and (9) are, respe tively, the lower and upper boundsof theo-rem 3for

E(G

M3

C

)

.

Wenowstudy the dieren e

(B

− e)(ℓ, p, n)

(expressions(7) and(9)). We have,

(B

_{− e)(ℓ, p, n) = (1 − p)}

n−ℓ

_{− (1 − p)}

n

_{− ℓp(1 − p)}

n−1

6

1

(10) Startingfrom expression(10), we prove in appendix thatthe following hold:

0 6 D(ℓ, n)

def

= (B

_{− e)(ℓ, p, n) 6}

n − ℓ

n

− 1

n−ℓ

_ℓ−1

−

n − ℓ

n

− 1

n−1

_ℓ−1

(11)

Notethatthe upper boundofexpression(11)isquitetight for

D(ℓ, n)

. Letusnowestimatethis boundfor several values of

ℓ

:

•

if

ℓ = 1

, then

D(ℓ, n) = 0

;

•

if

1 < ℓ ≪ n

(e.g.,

ℓ = o(n)

), then

lim

n→∞

D(ℓ, n) = 0

;

•

if

ℓ = λn

, for axed

λ < 1

, then

lim

n→∞

D(λn, n) = e

1−λ

λ

ln(1−λ)

− e

ln(1−λ)

λ

and thislimit's valueis axedpositive onstant;

•

if,nally,

ℓ = n

− 1

, then

lim

n→∞

D(n

− 1, n) = 1

. Letus nowstudy the dieren e

(e

− b)(ℓ, p, n)

. We rsthave

(e

_{− b)(ℓ, p, n) = ℓp(1 − p)}

n−ℓ

_{− (1 − p)}

n−ℓ

+ (1

_{− p)}

n

6 (ℓ

_{− 1)}

(1

_{− p)}

n−ℓ

_{− (1 − p)}

n−ℓ+1

(12) one an renethe bound of expression(12)obtaining (seein the appendix for the proof)

0 6 d(ℓ, n)

def

= (e

_{− b)(ℓ, p, n) 6 (ℓ − 1)}

n

_{− ℓ}

n

− ℓ + 1

n−ℓ

1 ₋

n

− ℓ

n

− ℓ + 1

6 (ℓ

− 1)e

(n−ℓ) ln

(

1−

n−ℓ+1

1 )

1 n

− ℓ + 1

(13) Aspreviously, we estimatethis bound for severalvalues of

ℓ

:

•

if

ℓ = 1

, then

d(ℓ, n) = 0

;

•

if

ℓ = o(n)

, then

lim

n→∞

d(ℓ, n) = 0

;

•

if

ℓ = λn

, for axed

λ < 1

, then

d(λn, n)

≤

λn

− 1

(1

− λ)n + 1

e

n(1−λ) ln

1−

1 (1−λ)n+1

;

λ

1 − λ

e

−1

(13)

p = 0.1

p = 0.5

p = 0.9

p = 1/n

p = 1/

√

n

p = 1/ ln n

E(G

M3

C

∗

)/E(G

M1

_C

∗

)

0.34 0.82 0.98 0.26 0.62 0.74

n = 14

E(G

M3

C

∗

)/E(G

M2

_C

∗

)

0.73 0.88 0.98 0.73 0.79 0.84

E(G

opt

_)/E(G

M3

C

∗

)

0.97 0.96 0.99 0.98 0.95 0.95

E(G

M3

C

∗

)/E(G

M1

_C

∗

)

0.34 0.83 0.97 0.25 0.62 0.73

n = 15

E(G

M3

C

∗

)/E(G

M2

_C

∗

)

0.73 0.89 0.98 0.72 0.79 0.83

E(G

opt

)/E(G

M3

C

∗

)

0.97 0.95 0.99 0.97 0.95 0.95

E(G

M3

C

∗

)/E(G

M1

_C

∗

)

0.35 0.83 0.98 0.25 0.62 0.73

n = 16

E(G

M3

C

∗

)/E(G

M2

_C

∗

)

0.72 0.88 0.98 0.71 0.78 0.83

E(G

opt

_)/E(G

M3

C

∗

)

0.96 0.95 0.99 0.97 0.94 0.94

Table1: Relations between fun tionals.

p = 0.1

p = 0.5

p = 0.9

p = 1/n

p = 1/

√

n

p = 1/ ln n

n = 14

B

3 ℓ

/E(G

M3

C

∗

)

0.34 0.62 0.92 0.32 0.45 0.53

B

_u

3 /E(G

M3

C

∗

)

1.67 1.18 1.02 1.71 1.42 1.29

n = 15

B

_ℓ

3 /E(G

M3

C

∗

)

0.33 0.61 0.92 0.31 0.44 0.52

B

_u

3 /E(G

M3

C

∗

)

1.68 1.18 1.02 1.74 1.42 1.29

n = 16

B

_ℓ

3 /E(G

M3

C

∗

)

0.32 0.61 0.92 0.29 0.43 0.51

B

_u

3 /E(G

M3

C

∗

)

1.69 1.17 1.02 1.77 1.43 1.29

Table 2: On the qualityof the boundsfor

E(G

M3

C

∗

)

.

•

if,nally,

ℓ = n

− 1

, then

d(n

− 1, n) ∼ n/4

. One an remark that for

ℓ = o(n)

,

E(G

M3

C

)

tends (for

n

→ ∞

) to both

b(ℓ, n, p)

and

B(ℓ, n, p)

. Thisisdue to the fa tthat

o(n)/ lim

n→∞

(B

− b)(ℓ, n, p) ; 0

.

Weseefromtheabovedis ussionthatforthegraph andthe apriorisolution onsideredand for

n

→ ∞

, the distan e of

E(G

M3

C

)

fromthe bounds given in theorem 3 an take an innity of valuesbeingeither arbitrarily loseto,or arbitrarilyfarfromthem. Consequently,itseemsvery unlikely thatthe bounds omputed ouldbesubstantiallyimproved.

6 Some simulation results

Limited omputational experiments have been ondu ted in order to study the realisti per-forman es of the three strategies proposed. We have randomly generated graphs of orders

n

_{∈ {14, 15, 16}}

, fortygraphsperorder,inthefollowingway: forea hgraph,anumber

p

E

∈ [0, 1]

is randomly drawn; next,for ea h edge

e

, a randomly hosen probability

p

e

is asso iated with, and, if

p

e

≤ p

E

, then

e

is retained as an edge in the graph under onstru tion. For all the generatedinstan es,theminimum vertex-degreevariesbetween 0(graphnon- onne ted)and14, while the maximum one varies between 1 and 15. The ratio

|E|/n

varied between 0.06and 7.3 for all the graphs generated. Thea priorisolution onsidered wasthe minimum vertex over of ea hgraph, denotedby

C

∗

. We have supposedidenti alvertex-probabilitiesand have tested six probability-values: 0.1, 0.5,0.9,

1/ ln n

,

1/

√

n

and

1/n

. We have omputed quantities

E(G

M1

C

∗

)

,

E(G

M2

C

∗)

,

E(G

M3

C

∗)

and

E(G

opt

₎

. The last quantityis themathemati al expe tationofathought pro ess onsistinginndingtheoptimalvertex over

(14)

p = 0.1

p = 0.5

p = 0.9

p = 1/n

p = 1/

√

n

p = 1/ ln n

σ

M1

C

∗/E(G

M1

C

∗)

1.08 0.36 0.12 1.30 0.60 0.46

n = 14

σ

M3

C

∗/E(G

M3

C

∗)

2.26 0.46 0.13 3.08 0.90 0.63

σ

opt

_/E(G

opt

₎

2.23 0.46 0.14 3.05 0.89 0.63

σ

M1

C

∗/E(G

M1

C

∗)

1.03 0.34 0.11 1.29 0.58 0.45

n = 15

σ

M3

C

∗/E(G

M3

C

∗)

2.13 0.44 0.13 3.08 0.88 0.61

σ

opt

/E(G

opt

)

2.10 0.44 0.13 3.05 0.87 0.61

σ

M1

C

∗/E(G

M1

C

∗)

0.99 0.33 0.11 1.28 0.57 0.44

n = 16

σ

M3

C

∗/E(G

M3

C

∗)

2.03 0.42 0.12 3.13 0.87 0.61

σ

opt

_/E(G

opt

₎

2.01 0.42 0.13 3.09 0.86 0.60

Table3: Standarddeviations for M1,M3 andreoptimization.

for any subgraph of

G

, i.e.,

2 n

exe utions of bran h-and-bound. This is the reason for having restri tedourstudiesinsmallgraphs. Inthetable1wepresentthemeanratiosforthefun tionals of the dierent strategies. What we an see from this table is that the ratio

E(G

opt

_)/E(G

M3

C

∗

)

variesbetween0.94and0.99. Inotherwords,

C

∗

seemstobevery loseto

C

ˆ

3

. Intheoppositethe omputationstimesfor

E(G

M3

C

∗)

arequitelarge,mu hmore largethanthe onesfor both

E(G

M1

C

∗)

and

E(G

M2

C

∗)

, whose the ratios with

E(G

M3

C

∗)

arerelatively small. Another remark we an make from theresults of table1 is thatthe larger the vertex-probabilities, the larger ( loser to 1) the ratios.

Wealsohave onsideredthebounds

B

3 ℓ

and

B

3 u

for

E(G

M3

C

)

inthelastexpressionoftheorem3 and omputed their ratioswith

E(G

M3

C

∗)

. The results are presented in table 2. The mean value of ratio

B

3 ℓ

/E(G

M3

C

∗)

is 0.52, while for

B

3 u

/E(G

M3

C

∗)

this value is 1.39. So, although these values are fairly lose to 1, it seems to us that the bounds onsidered have to be improved. Also one an notethaton e more the larger the vertex-probabilities, the loserto 1 the ratios.

Finally,forstrategiesM1,M3wehave omputedtheirstandarddeviationsdened,for

k = 1, 3

, as

σ

Mk

C

∗

= [

P

I⊆V

Pr[I](

|C

I

Mk

|)

2 − (

P

I⊆V

Pr[I]

|C

I

Mk

|)

2 ]

1/2

and ompared theexpe tationswith the orresponding deviations. We also have done sofor

σ

opt

and

E(G

opt

₎

. The relative results are presented in table 3. As one an see entries of this table show important dispersions of the fun tionals around the orresponding mathemati al expe tations.

7 Con lusions

We have rstgiven aformaldenition for probabilisti vertex over andnext we haveanalyzed three modi ation strategies and studied the omplexities for the orresponding probabilisti vertex overing problems. Moreover we have shown thattwo of the three variants studied have natural interpretations asweighted vertex over versions.

The framework of our paper diersfor the one [1, 2, 3,4, 5 ,6, 7, 8℄where besides the fa t thatalltheproblems onsidered(even minimum spanningtreeand shortestpath)aredened in ompletegraphs, asinglemodi ationstrategy onsistingin droppingabsentverti es out ofthe apriori solutions (asour M1) isstudied.

Dealingwithourresults,furtherworkhastobemadeforstrategyM3inorderto(a) ompletely hara terize

E(G

M3

C

)

and(b)determine

C

ˆ

3

andthe omplexityofits omputation. Wethink that if one an answer point (a), then one will be able with very little more work to answer also point(b).

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Referen es

[1℄ I.Averbakh,O.Berman,andD.Sim hi-Levi.Probabilisti apriorirouting-lo ationproblems. Naval Res. Logisti s, 41:973989,1994.

[2℄ D. J.Bertsimas. Onprobabilisti traveling salesman fa ility lo ationproblems. T ransporta-tion S i.,3:184191, 1989.

[3℄ D. J. Bertsimas. The probabilisti minimum spanning tree problem. Networks, 20:245275, 1990.

[4℄ D.J.Bertsimas,P.Jaillet,andA.Odoni.Apriorioptimization.Oper.Res.,38(6):10191033, 1990.

[5℄ P. Jaillet. Probabilisti traveling salesman problem. Te hni al Report 185, Operations Re-sear h Center, MIT,Cambridge Mass.,USA, 1985.

[6℄ P. Jaillet. Apriorisolutionofa traveling salesmanproblemin whi harandom subset ofthe ustomers arevisited. Oper. Res.,36:929936, 1988.

[7℄ P. Jaillet. Shortest path problemswith node failures. Networks, 22:589605,1992.

[8℄ P.JailletandA.Odoni.Theprobabilisti vehi leroutingproblem. InB.L.GoldenandA.A. Assad, editors, Vehi le routing: methods andstudies. NorthHolland, Amsterdam,1988. [9℄ C. MuratandV.T.Pas hos. Theprobabilisti longestpath problem. Networks, 33:207219,

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Consider expression(10) andset

x = 1

− p

. Then, expression(10) be omes

(B

_{− e)(ℓ, x, n) = x}

n−ℓ

_{− x}

n

_{− ℓ(1 − x)x}

n−1

= x

n−ℓ

+ (ℓ

_{− 1)x}

n

_{− ℓx}

n−1

= x

n−ℓ

(1 + (ℓ

− 1)x

ℓ

− ℓx

ℓ−1

) 6 x

n−ℓ

− x

n−1

6

1

(A.1) Set

f (x) = 1 + (ℓ

− 1)x

ℓ

_{− ℓx}

ℓ−1

. Then,

f

′

_{(x) = ℓ(ℓ}

_{− 1)(x}

ℓ−1

_{− x}

ℓ−2

_{) < 0}

. Consequently,

f (x)

is de reasing in

x

∈ [0, 1]

, therefore

f (x) > f (1) = 0

and the lower bound of expression(11) is proved.

Revisitexpression(A.1)andset

g(x) = x

n−ℓ

_{− x}

n−1

in[0,1℄. Firstremarkthat

g(0) = g(1) =

0

. Moreover,

g

′

_{(x) = (n}

_{− ℓ)x}

n−ℓ−1

_{− (n − 1)x}

n−2

and

g

′

(x) = 0

_{⇐⇒ x}

0 =

n − ℓ

n

− 1

_ℓ−1

1 .

So,

g(x)

in reases, with respe tto

x

,in

[0, x

0 )

whileitis de reasing in

x

in

(x

0 , 1]

. Hen e,

(B

− e)(ℓ, x, n) 6 g(x) 6 g(x

0 ) =

n − ℓ

n

− 1

n−ℓ

_ℓ−1

−

n − ℓ

n

− 1

n−1

_ℓ−1

and the upper boundof expression(11) isalso proved.

B Proof of expression (13)

Revisitexpression(12) and,aspreviously, set

x = 1

− p

. Then, expression(12)be omes

(e

− b)(ℓ, x, n) = ℓ(1 − x)x

n−ℓ

− x

n−ℓ

+ x

n

= (ℓ

− 1)x

n−ℓ

− ℓx

n−ℓ+1

+ x

n

= x

n−ℓ

(x

ℓ

− ℓx + (ℓ − 1)) 6 (ℓ − 1)

x

n−ℓ

− x

n−ℓ+1

(B.1) Set

h(x) = x

ℓ

_{− ℓx + (ℓ − 1)}

. Then,

h

′

_{(x) = ℓ(x}

ℓ−1

_{− 1) < 0}

, so,

h(x)

isde reasing in

x

∈ [0, 1]

and, onsequently,

h(x) > h(1) = 0

, proving sothe lower bound ofexpression (13).

Revisitexpression (B.1)andset

ϕ(x) = x

n−ℓ

_{− x}

n−ℓ+1

. Then,

ϕ

′

_{(x) = (n}

_{− ℓ)x}

n−ℓ−1

_{− (n −}

ℓ + 1)x

n−ℓ

and

ϕ

′

(x) = 0

_{⇐⇒ x}

0 =

n

_{− ℓ}

n

− ℓ + 1

.

So,

ϕ(x)

in reases in

[0, x

0 )

andde reases in

(x

0 , 1]

. Hen e, in [0,1℄,

(e

− b)(ℓ, x, n) 6 (ℓ − 1)

n

− ℓ

n

_{− ℓ + 1}

n−ℓ

1 −

n

− ℓ

n

_{− ℓ + 1}

6 (ℓ

− 1)e

(n−ℓ) ln

(

1−

n−ℓ+1

1 )

1 n

_{− ℓ + 1}