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Optima locaux garantis pour l'approximation différentielle

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o ieghn u p v jie o iw x yf k zchcdl k eeidlpifhengl k jcpnqcf{gnfhpihcggnhl o k | } hidl k ip o~ cgghn{ k jcl k nd gnpdnj k cpi €  def k li m dnfe o } jndlhndeyf ~ fdqihlc k ddnj u hi o ighn u p v jieqnj u k dcln k hie u k idqnddfe m cggchl k iddidl‚qilliqpceei m lcd ok eyfi o~ cflhiecfee kuk idqnddfeyfipiegh } q } o idle m d ~  cggchl k iddidlgce €  d ƒ d m ide ~ cggfcdlefhqieh } efplcle m dnfe e k lfnde pcqpceeir st

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o cde pigceczi o ieqpceeie o~ cgghn{ k jc uk p k l } € „ … † _ ‡ ˆ ‰^Š cgghn{ k jcl k ndgnpdnj k cpi m qnjgpi{ k l } m ngl k j k ecl k ndpnqcpi m hiq ‹ ihq ‹ ipnqcpi € Œ _ † Ž ˆ †  i ƒ hel k dlhn o fqil ‹ iqpceer st

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β(G) > |V |/(B + 1)

¯ ¤ ¥ ¦§ ©© ª ¦ § ¨ ©  © ¤ © § ¬«¬ ¦ ±  < / " A?F #¡ /. #<

m(U, G) 6 B|V |/(B + 1)

/? # $£? "¢ . 

∆ 6 B

' =  # "" ?$ 9

ω(G) = |V |

¯ «¥ § ª ¤ ¦ § ¤ ¬ ¤ ¬ª § ª ¦  ¥ ¬  ¦ ¤ ª ªª ¤ ©© ¦ ª ± '  .   < #  < A.@ " //. <

δ(G, U ) = (n − |U |)/(n − β(G)) > 1/B

' E.$ # A  .$ # <  <"¢# $ < @  $? " <  < A ¢ ? !/ &   # / < # @./ "  <

K

1,B

' ? ./ < # ? " .@ " / . ? " $ ; . # $ # !$ G >   .   $$ < A  <   # /   "¢ $   " $ <   " A <  # "" 

B

# $$ ? A "   # / < # < # . < A # $£? "¢ ./ < # ?! " .    " $ < A.  / "¢ ? < $  " $ <   "  # $$?A "   # / < # < # .  < ;? < G < £? " / # $. " ? < # . $ < A.  / " $

B + 1

$. < $ ' E@ # <  #  " /?;A? / . #< A " /  . / .$ # < # . '        =  " /. #< GA? " G 9  )  3 3 3*3 >

B

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B

" $//. < $@ " $$ # £? < A #    < #  " $. < # A < # £?$ ' 8 <  < A.   ² £? 9 @. #" $ < A  . <   A$   9  )  3 3 3*3 > 

GLO

[ρ]

¯ @.$ # A   < " $;. # $ # !$ G >   .  $ ± 9 " /? ; A?/. # <  $ < @.@ " ? '        E.$ # A  .$? # $ < @

I(D, C)

A ) *3 )+  < @.$ < ? # $.$ ?  # $ < @

f (I) = G(V, E)

A )  3 3 3*3  $$.@ #  < ® @ &  £? $. ? $ > $   " 

C

j

A " ­ #"" 

C

? $. <

v

j

9  <  @   < ? < 

v

j

v

j

0

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C

j

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C

j

0

@.$/.A < $@./. < . < A$  "   < $@.?$³

f : I = (D, {C

1

, . . . , C

n

}) 7→ f (I) =

G(V, E)

 ; @

V = {v

1

, . . . , v

n

}

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E = {v

j

v

j

0

: j 6= j

0

, C

j

∩ C

j

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f (I)

/A? < /$?/ " ?$

n

2

m

2

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(C

j

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j

0

)

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I

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f (I)

. <"   $  " A$. " ? < # .$

{0, 1}

n

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s

$   # <  /  <  A " ­. $ ? # ;  <  ³ 

s

i

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i

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0

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I

A ) *3 )+ ´ 

s

i

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i

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f (I)

A )  3 3 3*3 ' 5 # $ # 9 " ­.@ < # .

g

$ < " ­.@ < # . # A < # <  ³

g =

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: {0, 1}

n

→ {0, 1}

n

9

s 7→ g(s) = s

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(( s

$ <   " A$

G ⇔ s

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C ))

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$ <   " A$

G ))

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(( ∀j < j

0

9

s

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# / "# £?

v

j

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0

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@£? # £? # ;? <

(( ∀j < j

0

9

s

j

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# / "# £?

C

j

∩ C

j

0

= ∅ ))

@£? ¢ ®$. < .?$ < £? # ; "  < ®

(( s

$. " ? < # . A ) *3 )+ A$

(C, D) ))

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$A?F # $ < @$. < A.@  $A$.? <# .$   # $ $ ' /?$ 9 < .? < $.? > < # .

s

   ; "  ? $ ? " $ # $ < @$

I

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f (I)

9 §  9

∀s ∈ {0, 1}

n

9

m(G, s) = m(C, s) =

P

n

i=1

s

i

' = @.$  £? < 9  ) *3 )+  <  )  3 3 3*3 $ . < £? # > //.F #   " $/.? " //. < A #   < #  " ¯ < /.? " //. < @ " $$ # £? ± ' 5 ?$$ # 9 #" $ < ­@ #" A;. # £? 9 / "   > A ? @ < # . £?  . ? $ ; .$AA  @  #   9 $ # " @  A #  " # <  A ?/ " ? $!  A$   " A " ­  # "" 

C

$ <  .  /

B

9  " .$ " A!  F # ?A

G

$ <   .  /

B

9  ! "  < ' 8² 9 $. # <

k

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k

> A # $ <  < $A ¢ ?$. " ? <# .

s

$. < " $  $$?

I

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f (I)

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0

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g(V

0

)

/ . ?  ) *3 )+ 9 . @.@ " ? ""¡ ! < £?$ #  )  3 3 3*3  >

B ∈ GLO[δ]

9  " .$  ) *3 )+ >

B ∈ GLO[δ]

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B

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B

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B

$ < /. <  9 /.?A$!/ & $@. F$ 9 ®

2/(B + 1)

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I

A. # < # < !?. # $

m/B

$. < $ / . ? @. ?; #  < . ? <

E

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B

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B

>   !? "# $ 9 " $ .   $

m

 <

n

A ¢   < $ < A$. < $$. < "# $/ "  "  < # . ³

2 × m = 2|E| =

P

n

j=1

d(v

j

) = B × n

9 .

d(v

j

)

$ < " A!  A?$. <

v

j

A$ " !/ &  ' 5 # $ # 9 A

β(I) > m/B

. A  A? # <

β(I) > n/2

9  < " //. < A #    < #  " A ¢ //. F #  < # . A ¢ ? @.?; < ? # #  " 

U

9 £? # $ << .?  .?$A <  # "" ?/ " ?$

n × B/(B + 1)

/ " A. # @A

U

9    "# $? //. < A #   < #  " A³

δ(I, U ) = (n − |U |)/(n − β(I)) > 2/(B + 1)

'                       

GLO

[δ]

:  / . / # < 

P

$ < $ §¦ « § $ #  ""  ;   # ²  9

∀X

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∀Y ⊆ X

9

P (X) ⇒ P (Y )

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X

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P

? /  . / # <  &    A # <  #   <

X

? $  "  ': 

P

> / < # < # . A

X

$ < ? $  " 

S = {V

1

, . . . , V

q

}

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X

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q

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9

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i

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j

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9

P (V

i

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Π

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NPO

A. <" $ # $ < @$$. <" A.  A ¢ ? $  " 

X

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p : X →

N

A$  "   < $A

X

´

Π

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I

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Π

(I) = min

(

q

X

i=1

α (V

i

) : S = {V

1

, . . . , V

q

}

$ < ? 

P

> /   < # < # . A

V

)

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|

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(9)

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Π

ª ¦ ¬¥ ¤  ©  ¥« ¦ § ¦ § ¤ ¬¬ © ¬ ¦ $  § ¦ « §   « « ¦ § ¤ ¬

α =|

« ¤ ª

Π ∈ GLO[δ]

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Π

? /.  " ¡ A/ <#<# .  < &    A #<  #  <

X = {x

1

, . . . , x

n

}

? $  " ®

n

"  < $ '  .B < ?/ < # < # . @.?  / < # < # . A$ "  < $A

X

$??/ " ?$

n

$  " $ 9 . /? < /  $ <  < .? <  $. " ? < # .

V

1

, V

2

, . . . , V

q

/? ;@ < ?

s ∈ {0, 1}

n

2

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i

j

= 1 ))

9 $ #  < $? "  < $ #

((

"¢  "   <

x

i

$ < A$ " $.?$ > $  " 

V

j

))

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∀i = 1, . . . , n

9

P

n

j=1

s

i

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= 1

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s

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∀j = q + 1, . . . , n

9

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s

i

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q

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α =|

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n

A "¢ # $ < @ 9 # " $ ¢  $ ? # < £? 

Π

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Π

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Π

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{V

1

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{V

1

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q

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{V

1

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k

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1

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k

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k

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P

<  < &    A # <  #  9 " $$. < $

v

1

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i

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Π

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β

Π

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V

k+1

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P

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q 6 (n + k)/2

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δ

Π

(X, {V

1

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q

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Π

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Π

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Π

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P

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X

n

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n

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n

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Π

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n

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n

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X

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b|X

n

|/2c

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(10)

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I(C, S)

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C = {c

1

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2

, . . . , c

m

}

A ¢  "   < $®@.?; #  < A ¢ ?­ #"" 

S =

{S

1

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2

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n

} ⊆ 2

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A$.?$ > $  " $A ¢  "   < $A

C

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S

i

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|S

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B

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1/(B + 1)

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˜

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(I, ˜

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S

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n

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˜

S

0

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˜

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˜

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1

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2

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p

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p

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c

j

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j

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∀j = 1, . . . , p

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c

i

j

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j

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∀j

0

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i

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/

j

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˜

S

9 " @.A # < # . ¯ ## ± $ # #  "# < ' = ¯ ## ± 9 . $ # < £? ¢ . /? < @.$ < ? # ? $  " 

˜

C = {c

i

1

, . . . , c

i

p

}

A

p

 "   < $A # $ <# @ < $¯?  "   < /$.?$ > $   " 

S

j

A " @. ?;   < ?  ± £? # ;   # ²  < ³

∀c

i

0

∈ ˜

C

9

∃!j ∈ {1, . . . , p}

<  " £? 

c

i

0

∈ S

j

' -#"¢ # $ < @ # # < #  " $ <<  "" £? < .? < "   <

c

i

∈ C

// < A$?. # $A?F$.?$ > $  " $

S

j

9  " .$ @ ¢ $ <  / <# @? " # ; # A$  "   < $

c

0

1

, . . . , c

0

p

0

´ @?F > @ # /.?; < 9 /@.$ < ?@ <# . 9 // <  #  ® ? $@.A$.?$ > $  " A

˜

S

9 #" $// # $$ < A.@A$ " $$.?$ > $  " $

S

p+1

, . . . , S

n

9 @ £? # /    < A ¢  <   " #  "¢ # @ " ? $ # .

˜

C ⊆ ∪

n

j=p+1

S

j

' 4 $$.?$ > $  " $

S

j

A

S\ ˜

S

<  < A <  #""    .  /

B

9 .  A  A? # < $?

p

"  "  < # . ³

p 6 B(n − p) ⇔ p 6 Bn/(B + 1)

9 @£? # .?$ ¡ ?//. < A/­.@³

δ(I, ˜

S) =

(ω(I) − p)/(ω(I) − β(I)) > (n − (Bn/(B + 1)))/n = (n/(B + 1))/n = 1/(B + 1)

'

-# ? @ <  #  "   < ®@.?; # 

c

i

¢ // < £?A$? $.?$ > $  " 

S

j

9 @ $.?$ > $  "  $@. < ?A$ < .? < $. " ? < # . ' 5 ?$$ # 9 /.?$?@$/  @  A < 9 $?  < > #" A ¢ // "# £? /   "   "    < ® "¢ # $ < @

I

" /  .@$$ ? $$ ? # ;  < £? # /    << A ¢ # $. " A <  " $$. ? $ > $   " $ < $ # $ # ?@$/  @  A < ³  /.$

S

0

= ∅

´

C

0

= C

´

S

0

= S

´

stop = faux

´  <  < £?

¬stop

­ # ³$ ¢#" F # $ < 

c ∈ C

0

;   #²  <

∃!j : (S

j

∈ S

0

) ∧ (c ∈ S

j

)

9   " .$/.$

S

0

= S

0

∪ {S

j

}

´

C

0

← C

0

\S

j

´

S

0

← S

0

\{S

j

}

´ /.? < .? <

S

k

∈ S

0

­ # ³

S

k

← S

k

\S

j

´  $ # . /.$

stop = vrai

´   < .?

[I

0

= (C

0

, S

0

), S

0

]

' 4 /  > <  # <  < @ # > A$$?$;. # ? $  " 

S

0

A$.?$ > $  " $@./ # $A$ < .? < $. " ? < # .    " # $  "  < ? # $ < @

I

0

A *3+3 >

B

/.? " £? ""  < .? < "   <

c

0

i

$ < @. < ? A$ ?  . # $A ? F$   " $

S

j

0

' 4 $ # $ < @$

I

 <

I

0

$. < <  . # <    < " # $/  " $ "  < # .$ $? # ; < $³

˜

S

0

∈ Sol(I

0

)

$ #  < $? "  < $ #

˜

S

0

∪ S

0

∈ Sol(I)

 <

m(I, ˜

S

0

∪ S

0

) = m(I

0

, ˜

S

0

) + |S

0

|

' 8"" $ # / " # £? <

δ(I, ˜

S

0

∪ S

0

) = δ(I

0

, ˜

S

0

)

 <" /?;A " /./.$ #<# . $ < @./ " ¡<  ' -# 9 /.?

j = 1, . . . , n

9 "  <  #"" A$$.?$ > $  " $

s

j

$ <  .  /

B

9 ?$. " ? < # . ./ < #  "  A;/A 9 /.?@.?; #  " $

m

 "   < $A

C

9 ?. # $

m/B

$.?$ > $  " $ ' =  #"" ?$ 9 $ # < .? < "   < //  < A$?/ " ?$

$.?$ > $  " $ 9 "  ­ # "" 

S

/? < A # $/.$A/ " ?$A

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