! " # $% & '( )*+,*
∗
-./ & # 0 121*3 456 -578 9: # ;$ # <= # $ > 7 ?/ &# = " @A ? 6 @ & " 7 4 << A%$$ # ! B 9C D CC D= # $EA FG H 9I @ JKLLK M N OPQR S KQ N M KT U KTQV WU PJQP X V Y X PTO SZ LV Y [ \ ] ^ _`a ^ b cdefd ghij k ihlijge m dnfe k dlhn o f k endepcqpceeir st[δ]
o ieghn u p v jie o iw x yf k zchcdl k eeidlpifhengl k jcpnqcf{gnfhpihcggnhl o k | } hidl k ip o~ cgghn{ k jcl k nd gnpdnj k cpi def k li m dnfe o } jndlhndeyf ~ fdqihlc k ddnj u hi o ighn u p v jieqnj u k dcln k hie u k idqnddfe m cggchl k iddidlqilliqpceei m lcd ok eyfi o~ cflhiecfee kuk idqnddfeyfipiegh } q } o idle m d ~ cggchl k iddidlgce d d m ide ~ cggfcdlefhqieh } efplcle m dnfe e k lfnde pcqpceeir st[δ]
o cde pigceczi o ieqpceeie o~ cgghn{ k jc uk p k l } _ ^ cgghn{ k jcl k ndgnpdnj k cpi m qnjgpi{ k l } m ngl k j k ecl k ndpnqcpi m hiq ihq ipnqcpi _ i hel k dlhn o fqil iqpceer st[δ]
k l k dqpf o iengl k j k cl k ndghn u pijel clzfchcdlii l iyfcp k ln l i k hpnqcpngl k jc m k l hiegiqllnl i ok| ihidl k cpcgghn{ k jcl k ndhcl k n i{l m ie n l clcqihlc k ddfj u ihn ipp dn dqnj uk dclnh k cpghn u pije u ipndzlnl k eqpcee m k pinl ihndie o ndnl k dcpp m u cei o fgndl ihiefplen u lc k di o m igpcqir st[δ]
k dl i eqidin l icgghn{ k jc uk p k lqpceeie _ gnpdnj k cpcgghn{ k jcl k nd m qnjgpi{ k l m pnqcpngl k j k cl k nd m pnqcpeichq 4 $/. " ¡ $A ¢ ./ < # # $ < # . ?F£? " $. $ ¢ << @ & A$@ < < # @ " 9 $. < " /A <¤ ¥ ¦ § ¨ © § ª« ¦ § ¤ ¬A$/. " ¡ $AA @ # $ # . ANP
': /. " ¡Π
$ < @@ < # $ / "¢ $ "I
Π
A$$ # $ < @$ 9 "¢ ? # .Sol
Π
= ∪
I∈I
Π
Sol
Π
(I)
A$$ " $A$$. " ? <# .$ " # $ " $$$.@ # $ ?F # $ < @$
I
9 A " .@ < # . ®./ < # # $m
Π
$?@$$ " $ < A?$$opt
Π
A "¢ ./ < # # > $ < # . ¯©«° § © § ª« ¦ § ¤ ¬.?© § ¬ § © § ª« ¦ § ¤ ¬ ± ' = " $? # < 9 . . <|I|
" < #"" A "¢# $ < @I
': /. " ¡Π
// < < ® " @ " $$NPO
$ < ? £?A?/ " <(I
Π
, Sol
Π
, m
Π
, opt
Π
)
£? # ; # ² ³ ¯ # ±I
Π
$ < @. # $$ " < /$/. " B . # "|I|
´ ¯ ## ±∃ p
Π
/ . " B < ' £ '∀I ∈ I
Π
9Sol
Π
(I) ⊆ {0, 1}
p
Π
(|I|)
´ ¯ ### ±∀I
9∀s
9 . $ # < A @ # A < /$/. " B. # "|I|
$ #s
$ < "# $ " /.?I
´ A/ " ?$ 9∀I ∈ I
Π
9 . $ # < A < # ?$. " ? < # .triv
Π
(I)
< /$/. " B. # "|I|
´ ¯ # ; ±m
Π
: I
Π
× Sol
Π
→
N
$ < @ " @? " " < /$/. " B. # "|I|
´ ¯ ; ±opt
Π
∈ {min, max}
' 8 < < A. ? # $ < @I
A ¢ ? / . " ¡ 9 A ?F; " ? $ / < # @ ? " #¡ $ . ? $ # < $$ < / . ? " $. " ? < # . AI
³@ "" $A " #"" ? < A " / # $. " ? < # . ' -. # <Π
? /. "¡ ANPO
<I
? # $ < @AΠ
9 . A ²# < " $; " ?$β
Π
(I)
<ω
Π
(I)
/β
Π
(I) = opt
Π
{m
Π
(I, s) : s ∈ Sol
Π
(I)}
∗
µ¶ · ¸¶¹º»¶¹ · »¶ · ¸¼»¹¹¸½º»¾ ¿ À¾¼¸¼»¾¼¸¹¾À¾ ·Á  À¹ · à ¼ Á ¸ Ä · Å Á ÆÇÈ ÉÊÊ È ËÊËÌÍ ÄÎ ÀÏ ÇÈ ÉÊÊ È ËÊÉÐÌ Æ<
ω
Π
(I) = opt
Π
{m
Π
(I, s) : s ∈ Sol
Π
(I)}
9 .∀Π ∈ NPO
9opt
Π
= max ⇒ opt
Π
= min
<
opt
Π
= min ⇒ opt
Π
= max
' 4 $.?$ > @ " $$A$/. " ¡ $£? # .?$ # < $$ < / " ?$/ < # @? "# ¡ < A$@ << < # @ " $ <" @ " $$ A$/. " ¡ $/. " B. # " < . $ANP
' E << @ " $$ # < .A? # < A$ GG $ < " @ " $$A$ / . " ¡ $ / . ? " $ £? " $ # " F # $ < ? / . " Bq
Π
< " £?∀I ∈ I
Π
9β
Π
(I) 6 q
Π
(|I|)
' @ # 9 .?$! "# $.$ " A ²# < # . AGG / < @./ < A$ .$ $? "¢ $ " A$; " ?$ " # $ " $ 9 @£? # $ < A? # /? .$? " ; " ?A " / # $. " ? < # . ' 7 $@£ ? # $ ? # < 9 ? / . " ¡Π ∈ NPO
// < # < ® ª § ¦ ª © ¬ ¦ ª §§ ° § ª ¦ ¬¥ ¤ ¬ ©q
Π
¦∀I ∈ I
Π
max{ω
Π
(I), β
Π
(I)} 6 q
Π
(|I|)
' 4¢# A " $ # < A$;. # $.?A " $/. " ¡ $A < /$/. " B. # " ´ "& ??$ > < 9 " / $. / < # . $ < . < A/$ £?P
6= NP
.?@.PO
6= NPO
. 9 ; $ # . ./ < # # $ < # . A " @ " $$ 9 A $ # ! "¢ $ " A$/. " ¡ $A £? "¢ . $ # < $.?A < /$/. " B. # " ' E ¢ $ < /.?£?. # . $$ # A ¢ //.@ & $? " < 9 # $? # ?F 9 @$/. " ¡ $ A # @ # " $ ' 4¢ //. F # < # . /. " B . # " /.? # $$ # . A $.?A " $/. " ¡ $$ " . " $A? F F # !@$³¯ # ± < A < /$ 9 A! < # A ¢ ?F @? < # . / # A¯@./ " F # < /. " B. # " A$ " !. # < & $ ± 9 ¯ ## ± < A/.@ 9 A! < # A ¢ ? @ < # # ;?A ¢ //.F # < # . ': " !. # < & // .@ & ¢ $ < # / " ? $ # . # $£ ? ¢ ? " !. # < & £ ? # A. ?$. " ? < # . " # $ " ? /. "¡ /.$ ': < " " !. # < & $ < /. " B. # " $ ¢#" $A .? " /.? < .? < # $ < @ ? < /$ /. " B. # " " < #"" A "¢# $ < @ ' -. # <Π
? /. " ¡ ANPO
< ? " !. # < & //.@ & /.?Π
9 . . < $/@ < # ; < /.?? # $ < @I
A@/. " ¡λ
A
(I)
9β
Π
(I)
<ω
Π
(I)
" $; " ?$A " $. " ? < # . A. / "¢ " !. # < & 9 A "¢ ./ < # ? < A " / # $. " ? < # . $?I
' 4¢ $ < # < # . A " £? "# < A ¢ ? " !. # < & $ < # < ® "¢ # AA//. < $A ¢ //.F # < # . ' 4 //. < @ " $$ # £? 9 £? # $ < @ " ? # $? " £? " $.A "¢ $$ < # " A " < & . # A "¢ //. F # > < # . /. " B. # " 9 $ < $$ # < ? # < # ³ #" @.$ # $ < ®@./ " ; " ?A " $. " ? < # . A. / "¢ " !. # < & ® " ; " ?A "¢ ./ < # ? ' 4 £? "# < A ¢ ?$. " ? < # . //.@ & /.?I
9 //.F # > < # . @ " $$ # £? 9 $ # < A.@A. / "¢ $ < # < # . A?//. <ρ
A
Π
(I) =
λ
A
(I)
β
Π
(I)
Π
$ < ? / . " ¡ A F # # $ < # .β
Π
(I)
λ
A
(I)
Π
$ < ? / . " ¡ A # # # $ < # . < " / . @A / . ?Π
/ ³ρ
A
Π
= inf
I∈I
Π
{ρ
A
Π
(I)}
' 7 $@£ ? # $ ? # < 9 . ? $ // " .$[ρ]
" @ " $$A$/. " ¡ $A $. " ? " $ < /$/. " B. # " /A$ " !. # < & $//.@ & $ ! < # $$ < ? //. <ρ
@.$ < < ¯ # A /A < A$/ ¡< $A "¢# $ < @A?/. " ¡ ± ' = # "" ? $ 9 . ? $ // " .$[ρ]
" @ " $$A$/ . " ¡ $A $. " ? " $ < / $/. " B . # " /A$ " !. # < & $//.@ & $! < # $$ < " //. <ρ > 1 −
9 /.? < .? < @.$ < <> 0
' 4 //. < A # < # " 9 . # $@.? < ?$ # < # $£? # A ®A. "# ?®A$ $? " < < $ @. ; # @ < $ 9 $ ¡ ®A? F /. # < $A/ ¡ ³ "¢ ./ < # ? 9 # $?$$ # " / # A$$. " ? < # .$ ' 8 A # < # " 9 #" $ << .? < ?$$ ## /. < < A$ ¢ " . # !A " ; " ?A ¢ ?/ # $. " ? < # . £?A $ ¢ //.@ & A " ; " ?A ¢ ?$. " ? < # . ./ < # " ' -?? # $ < @I
#" $ < A. /δ
A
Π
(I) =
|ω
Π
(I) − λ
A
(I)|
|ω
Π
(I) − β
Π
(I)|
<" /.@A /.?Π
/³δ
A
Π
= inf
I∈I
Π
{δ
Π
A
(I)}
' 7 . " .!? .?$A ²# $$.$ " $ @ " $$$[δ]
<[δ]
' .?$ ; .$ " $ " < # .$A ¢ # @ " ?$ # . $ < # @ < ¯®. # $£?P
= NP
±PTAS
[ρ] ⊂ APX[ρ]
<PTAS
[δ] ⊂ APX[δ]
' 4 ? < A@ < < # @ " $ < A. < £?/.?? " !@ " $$A/. " ¡ $ # @. ?$ A @. < ?A$[δ]
< .? < $ " $$. " ? < # .$./ < # " $ " .@ " < $. <((
/.@ & $))
A$ $. " ? < # .$¯! " . " < ± ./ < # " $ ' =" ?$/ @ # $ < 9 .?$A ² # $$.$?@ " $$A ¢ //.F # > #"# < £? ¢ . // ""[δ]
< .?$ " $/. " ¡ $A " £? "" ; #² < " /./ # < $? # ; < ³«¥¥ ¤¦§ ¬ ¦§ ¦¤ ¦ ª ¤ ¦§¤ ¬ ¤ ¥ ¦§ ©« ¤ « © ¬ ¦ ¥« «¥¥ ¤¦ ¬ ¤§ ª § ¬« ¥ ¤ ¥ © ¬ ¦ ¬ §¦ ¥ ª ¤ § ¬ ª ¦ ¤ ¬ § ¬ § © ¬ ¦ ¥« ¬ ¤ ¬ª ¦ «¬ ¦ § ¬ ¥ ¬ «¬¥««© ¦ § ¬ª ¦ «¬ ¥ ¤ © ¤ ¬ ¤ ¬ª § ' . < .$£? "¢ £? # ; " < A@ < ; #" /.? " //. < @ " $$ # £?¯ # ' '9 " A ²# < # . A " @ " $$
[ρ]
± / $ < A$ 9 $ << A?A$G 9 ' E ¢ $ < ® . < @. # $$@ " / #¡ . # $ £? " $ < & < # £? $A " @ & @ & < A "¢ . / < # # $ < # . " .@ " $. < $B$ < < # £? << # < $ //.F # < # . @ " $$ # £? ' . < < ; #" $ < "¢ F < $ # . A "¢ < ?AA$ <# @ " $ 9 A$ " @AA?//. < A # <# " ' 4 $ A?F@A$ 9 " ! " ?. > < A ¢ < ? A@. ? 9 " < & . # A "¢ // . F # < # . / . " B . # " 9 A # ¡ < A # @ " << < / //. < ® " ? " .! # £? < ?F < & .A. " .! # $@. # < . # $ # / "# £? $ 9 £? ¢ ?F $? " < < $. < ?$ A$@ & @? A ¢ ?F ' = @.$ £? < 9 " A ²# < # . <"¢ < ?AA " @ " $$[δ]
# A? # $ < " ?$ / . / $ " B $$ < $ ? " < < $£ ? # A # ¡ < 9 @. " " @ < ?/ ? < " @.$ < < 9 A@ ?F/ . / .$ $ A$ 9 ' 5 ; < A / " A ¢ . / < # " .@ ?F 9 # " ? < A ² # ? . < # . A/ . F # # < ³ " $.@ < # .$;. # $ # ! .?$/ << < A " # ' . "" < 9 $. # <Π
? /. " ¡ A ¢ ./ < # # $ < # . 9 ? ;. # $ # !V
/.?Π
$ < ? .@ < # . £? # ® < .? < $. " ? < # .s
A < .? < # $ < @I
AΠ
$$.@ # ? $.?$ > $ "V(I, s)
A$. " ? < # .$? < .?As
´ " $ " < $AV(I, s)\{s}
$. < // " $$. " ? < # .$ ; . # $ # $ As
' < < A. ? ;. # $ # !V
$?Π
9 ? ./ < # ? " .@ " A ¢ ? # $ < @I
¢ $ < ? < £? ¢ ? $. " ? < # . ?. # $?$$ # . £?$$;. # $ # $ ' 5 << < # . 9 . @ & @ & /$?$. " ? < # . ./ < # " $?? $.?$ > $ " $ < < # £? 9 # $?$. " ? < # . ./ < # " $?? $ " ¬ ¦ « ¦ ¤ ¦¦ ª ¤ ¦ § ¤ ¬³ #" $ ¢ ! # < A.@A ¢ ? $.?$ > $ " ;. " ? < # .@ < # . A " $. " ? < # . @.? < < "¢ ./ < # "# < " .@ " $A ²# < /$®/ < # A ¢ ? $.?$ > $ " A ²# / # . # # $®/ < # A " $. " ? < # . @.$ # A < A$;. # $ # $£? ¢ "" A $ # ! 9 /.??$ < ?@ < ?A;. # $ # !@.$ # A ' 5² A ¢ ; # < A ¢ ;. # ®/ @ # $ " $$$A ¢ ./ < # # $ < # . A$/. "¡ $ # /? " $ 9 .?$? < #"# $.$ / " $? # < " $$ # !$ ¯$/ '9 ± /.?$ # ! #² £? ¢ ?$. " ? < # . $ < ?. # $?$$ # . ¯$/ '9 $ < # @ < < # "" ? ± £? ¢ ?? < ' 4 $. " ? < # .˜
s
$ < . / < # ? " .@ " AI
" < # ; < ®V
$ # < $ ? " < $ #∀s ∈
V(I, ˜
s)
9m
Π
(I, ˜
s) m
Π
(I, s)
' 7# $/.$ < A ¢ ? /. " ¡ < A ¢ ? ;. # $ # ! 9 #" $ < @ #" AA < # ? ./ < # ? " .@ " ' = < < A ¢ ?$. " ? < # . 9 . F / " .$. ;. # $ # !® " @ & @ & A ¢ ? ; < ? "" # "" ?$. " ? < # . 9 < # $ # A$? # < ?$£? ¢ ®@£? " $. " ? < # . @.? < ¢ # < /$A # "" ?;. # $ # ³@ ¢ $ < ? ./ < # ? " .@ " ' E << @ & @ & $ < . "# $ $.?$ " .A ! ¯ " ¤ « # « $% ¤§ ¦ $ © ± ³ " !. # < & A @ & @ & " .@ " ' < < A. $? /. " ¡Π
® $.?A 9 ? ;. # $ # !V
/.?Π
< ? " !. #< & /. " B. # " //.@ &A
Π
/.?Π
9 ? ! $A .? " @.$? # < ³ ¯ ± /.$s
1
= A
Π
(I)
´s = s
1
´ok = faux
´ ¯ ± < < £?¬ok
# ³ ¯G ± A < #V(I, s)
´ ¯ ± $ ¢ # " F # $ <s
0
∈ V(I, s)
< "" £ ?m
Π
(I, s
0
) m
Π
(I, s)
9 " . $s = s
0
9 $ # .ok = vrai
´ ¯@ ± < .?$ ' 4 $. " ? <# . # #<# "s
1
$ < . < ? < /$p(|I|)
'& "¢ #< # ?A " .?@ "((
< < £?))
¯ #< ¯ ±± £? # @.$ # $ < " @.$ < ?@ < # . A?;. # $ # ! 9 @$$ # < ? < /$A "¢ .AA|V(I, s)| × t
V
(I)
$ #t
V
(I)
/ $ < " < /$F # ?A/$$!A ¢ ?$. " ? < # . ®?$. " ? < # . ;. # $ # /.? " ;. # $ # !V
$? "¢# $ < @I
' 4¢ ; " ? < # . A$;. # $ # $AA? < /$?/ " ?$|V(I, s)| ×
t
m
(I)
$ #t
m
(I)
$ < " < / $A ¢ ; " ? < # . A ¢ ? $. " ? < # . $ ? "¢ # $ < @I
9 $ ?// .$ / . " B . # " / É Æopt
Π
= min ⇒
” ”
=
” 6 ”
” ”
=
” < ”
opt
Π
= max ⇒
” ”
=
” > ”
” ”
=
” > ”
ÆA ² #<# . A
NPO
' 8² 9 " .?@ "((
< < £?))
$ <#< ?/ " ?$|ω
Π
(I) − β
Π
(I)|
. # $³ " $ @. @ # < $A " .@ < # . . @ < # # $ # £? " $; # " $ < < ®; " ? $ < #¡ $ 9 " ; " ? A " $. " ? < # . ?! < $ #opt
Π
= max
9 A # # ?$ # . 9 A ¢ ?. # $?? # < ®@ & £? # < < # . ' ? " £?$. < $$? " @./ " F # < A "¢ " !. # < & ! ' " $ < @$$ # /.? " /@.?$A? ; . # $ # !A " $. " ? < # . @. ? < £? @ " ? # > @ # $. # < A < # "" / . " B . # " ´ # " ? < A / " ? $ £? " @.$ < ?@ < # . A < .? < ;. # $ # ®/ < # A " $. " ? < # . @.? < $. # < "" > /. " B. # " ' < A? #< @$F # !@$/ " $A?F@.A #<# .$ @$$ # $³¯ # ± # " F # $ < ? /. " Bp
1
< " £?∀I
9∀s ∈ Sol
Π
(I)
9|V(I, s)| 6 p
1
(|I|)
< ¯ ## ± # " F # $ < ? / . " Bp
2
< " £?∀I t
V
(I) 6 p
2
(|I|)
' ? < ? . A ¢# < < # .$A " .?@ "((
< < £?))
A "¢ " !. # < & ! 9 . /? < $ ¢ $$?A$ /. " B. # "# < £? ¢ . < " . A ¢ < /$£? @$$ # < # < " /$$!A ¢ ?$. " ? < # . A ; " ?m
Π
(I, A
Π
(I))
®? . / < # ? A; " ?β
Π
(I)
': @.A # < # . $ ? $ < /. ?£? "¢ # < ¯ ± $. # < /. " B. # " $ < A ¢ ;. # £? ¢ ? . /. " B. # " A; " ?$/.$$ # " $/.? " $$. " ? < # .$ A?/. " ¡ < " $; " ?$m
Π
(I, A
Π
(I))
<β
Π
(I)
9 @ ' > ® > A '9 #" F # $ < ? /. " Bp
£? # ; #² /.? < .? < # $ < @I
" " < # . $? # ; < ³$ #α
I
= m
Π
(I, s
1
)
<n(α
I
) = |{m
Π
(I, s) α
I
, s ∈
Sol
Π
(I)}|
9 " .$n(α
I
) 6 p(|I|)
' 6 "& ??$ < 9 @ << @.A # < # . ¢ $ < /$; # ² " / # . # < /$/. " B. # " ' . ? $ A # $ < # ! ? .$A ? F.$A " ; # ² ³$. # < " / . " ¡ " ? # > ¢ A < £? ¢ ? . /. " B. # " A; " ?$A # $ < # @ < $/.? < .? < # $ < @¯@ ¢ $ < . < < ; # $ #" /. "¡ $ < /. " B. # " < . ± 9 $. # < @ ¢ $ < ® .?$£? ¢#" ; # < A$//.@ & A " ; " ?./ < # " / < < A ¢ ? . $. " ? < # .A
Π
(I)
' 8 @ < # ; < 9 $ #" ; " ?A ¢ ?$. " ? < # . ./ < # " $ < . /? /. " B ³¯ §∀I ∈ I
Π
9|β
Π
(I)| 6 p
β
(|I|)
± 9 # " $? # < " .$/.?A " @ & @ & " .@ " /. " B. # " A/ < # A ¢ ?$. " ? < # . # # < # "((
«ªª ¥ ¤ $ ¤ ¥ ¦ § ©©))
9 $. # < AA # $/.$ A ¢ ? " !. # < &A
Π
//.@ & ® //. < @.$ < <r
/.? " //. < @ " $$ # £?A$ " $$. 9 $ #opt
Π
= max
¯ " @$opt
Π
= min
$ < # < F@ < < A " . ± 9 " .$λ
A
(I) > rβ
Π
(I)
@ £? ## / "# £?|λ
A
(I) − β
Π
(I)| 6 |1 − r|β
Π
(I) 6 |1 − r|p
β
(|I|)
' E << A # $@?$$ # . .?$ ¡ ?FA?F @.A # < # .$$? $ < $$? # ; < $@.@ < " /. " B . # " # < A ¢ ? " !. # < & ! ³ ¯ # ± # " F # $ < ? / . " Bp
< " £?∀I
9 " . A$$. " ? < # .$ " # $ " $AI
$ <# # ? . ? ! " ®p(|I|)
< ¯ ## ±Π ∈ APX[ρ]
<β
Π
(I)
$ < . /? /. " B ' 4 @.@ " ?$ # . $ < £? ¢ . /? < $$?/ # . # "¢ @@ # < < /$A ¢ ? " !. # < & A @ & @ & " .@ " ³@ " A /A. < < A?/. " ¡ ¯.?A " $ < # @ < # . A ¢ ? /. " ¡ ! " ®? #"" A ¢# $ < @$/ < # @? "# ¡ $ ± < # < # $ # £?A " A ²# < # . A;. # $ # !@.$ # A ' = .?/ " ?$A$ # ! < $$? " A # @? " < A ¢ . < # A$./ <# " .@?F < /$/. " B. # " / . ? A$/ . " ¡ $£ ? # $. < A & . $A " @ " $$ 9 @ '9 G C9 G 9 G ± ' "¢ ;? 9 " ;. # $ # !$ <" . < # . @ < " A " @ & @ & " .@ " ´ . 9 ? ;. # $ # ! ¢ $ < ? < £? ¢ ? $ " A/. # < $ ? < .?A ¢ ?$. " ? < # . A. ' E/A < 9 . < ? < < < A@.$ < ? # ®/ < # A@$;. # $ # !$A$ " !. # < & $/. " B # ?F 9 . ? $ . ? $ # < $$ .$ < ? "" < £? ¢ ®A$;. # $ # !$/. " B # ?F 9 @ ¢ $ < > ® > A # £? ¢ ?$. " ? < # . 9 £? " £?$. # < " /. " ¡ < # < 9 ?? ¬ ¤ © ¥ ¤ ¬ ¤ © § « ¤§ ª § ¬ª ' E@ # .?$ ¡ ® " A ² #<# . $? # ; < A?;. # $ # ! ' -. # <Π
? /. " ¡ A ': ; . # $ # !$?Π
$ < ?.@ < # .V : I
Π
×Sol
Π
→
P(Sol
Π
)
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A$. " ? < # .$A
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$ < A$ " @ " $$APX
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.r ∈ {ρ, δ}
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'∀Π ∈ NPO
9∀I ∈ I
Π
9∀s, t ∈ Sol
Π
(I)
9d(s, t) = ks − tk
1
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i
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? /. "¡ ANPO
<h
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h
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³∀h
9∀I ∈ I
Π
9∀s ∈ Sol
Π
(I)
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h
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Π
(I) : d(s, t) 6 h}
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> . < .? < ;. # $ # !V
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> . 9 $. # << . ? < .@ < # .V
; # ² < ³∀I ∈ I
Π
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Π
(I)
9V(I, s) ⊆ V
h
Π
(I, s)
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ANPO
9 $. # <I
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Π
(I)
$ < . /? /. " B " < #"" A "¢# $ < @³|{t ∈ Sol
Π
(I) : d(s, t) 6
h}| 6
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> . $/ " # $$ < " $@.A # < # .$ @$$ # $AA .? " < /. " B. # " A ¢ ? " !. # < & A@ & @ & " .@ " / $ < $@ # > A$$?$ 'GLO
[δ]
7 $@ << $@ < # . 9 . $ ¢# < $$F@ " ?$ # ; < ?F$ < ?@ < ?$A;. # $ # !$h
> . $ ' .?$ A ² # $$.$A ¢ .A " . <# . A ¢ ./ <# ? " .@ " ! <# ¯/.?A$;. # $ # !$h
> . $ ± ' : /. " ¡Π
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< ? < #h
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s
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Π
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s)
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[δ]
$ ¢#" ! < #<" £? "# < A$$./ < # " .@?F < $ # . $ # < 9 /.? " ;. # $ # !£? # / < A ¢ < "# @ << ! < # 9 A < # ? ./ < # ? " .@ " < /$/. " B. # " ' 7 $ " @ " $$GLO
[δ]
¯ #" $ < A /.?GLO
[ρ]
± 9 #" ? < A # $ < # !?A?F@ & .$$³ ª ¤ ¥ ¦§ ©« ¤ «° « §¦ ««¬ ¦§ ¯@£? ¢ . /.? # < £? "#² A((
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[δ]
?F # $ < @$A$/. " ¡ $/.? " $£? "" $. $ # << .?;? ./ < # ? " .@ " < /$/. " B. # " ' E ¢ $ < A$@ < $/ # < £? " $? < ?$A . < / " @GLO
[ρ]
A$ 9 @@ << $ < # @ < # . $ < $? $ < ® " A A?@ < # . A = % 55 ®//. < @.$ < < ®/ < # A? . @./. < < ! " . " A$./ < # " .@?FA?/. " ¡ ! " @.$ # A '
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7 $ " / . / .$ # < # . £? # $ ? # < . ? $. < .$£ ? A # ; $/ . " ¡ $ # @. ? $. < / < # A " @ " $$GLO
[δ]
/.?A$;. # $ # !$G > . $¯ # ' '9 < < A. ?$. " ? < # .s
9 " $$. " ? < # .$ ;. # $ # $@.$ # $ < < @ "" $ $? " < < As
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9 ) * 3 >B
9 33 + 3 >B
< ) * 3 )+ >B
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<Y
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X
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' ) 333*3 >B
9 )*3 >B
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$. <" $; # < $. " $!/ & $$. < ®A! F # ? . /B
' E.$ # A .$ # < < ?@. "" @ < # .C
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1
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2
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0
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0
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< " £?∀k < l ∈ {1, . . . , p}
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$ < . $?/ # ? < /B
' ¤ ¬ª § ¤ ¬ª¬«¥ $G(V, E)
¤ ¬¬ ° ¥ ¤ ª ¤ ¬ª|V | = n
¦ ¬ ¤ ¦ ¤ ¬ª¥«∆
ª ¤ ¬ ©«° § ©© % ¤ ª ª¥ ¤ © ª ) 333*3 ¦ 33 +3ª ¤ ¬ ¦ § ¨ «¥¥ ¤ ° § ©« ¥ ¤ «¥¥ ¤¦§ ¬ ¦§ § ª ¦ « § « © ¬ ¦ ¥ ¤ ) 333*3 ¨B
¦ 3 3 +3 ¨B
ª §U
ª ¦ ¬ ¬ª © ¤ © § ¬«¬ ¦G
« ¤ ª|U | > |V |/(∆+1)
ª¥¥ ¤ ª ¤ ¬ª ©« § ¬ ¦ ¬«¬ ¦U
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" $ < # @. ?¯ D ± £? $ #U
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< £? /. ? 33 +3ω(G) = n
9 " $ // . < $@ " $$ # £? < A # < # " /.? " $A?F/. " ¡ $@. # @ # A < /.? < .? < !/ &G
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P
u∈U
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9U \{y}
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´y
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9 ? < < A # <(( ∀u ∈ U
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£? ## / "# £?|U | 6 ∆|V |/(∆ + 1)
' E@ # < # " /?;A?/. # < < A? " ' 7 $ " /./.$ # < # . @ # > A$$.?$ 9 .?$@.$ # A .$? # £? < A$./ < # " .@?F/.? " $ ;. # $ # !$G > . $ ' = .? " $/. " ¡ $ < ?A # $ 9 @ " ; # A® "¢ < ?AA$$. " ? < # .$F # " $.? ## $/.? # @?$ # . ' $B$$/ $ < $$. $/?$ $/.$$ # $/? # $£? .?$ . < .$£ ? " $ < # .$A ¢ // . F # < # .$ $ ? " < < $$. < << # < $/. ? @ < # $ # $ < @$ ' ) +1
∈ GLO[δ]
) * 3 ¨B ∈ GLO[δ]
) 333 * 3 ¨B,
33 + 3 ¨B ∈ GLO[δ]
) *3)+ ¨B ∈ GLO[δ]
-. # <G(V, E)
? !/ & < $. # <U
? $.?$ > $ " A. # < A$. > < $ ' 4¢ $ "U
$ < ? ./ < # ? " .@ " $ # < $? " < $ # " @ & ! < A ¢ @ < < # . A ¢ ? $. <u
AU
®U
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®U
¢ "# ./$ " $. " ? < # . 9 $. # < $ #U
; # ² ³∀u ∈ U
9|h{u}, U i| 6 |h{u}, U i|
<∀u ∈ U
9|h{u}, U i| 6 |h{u}, U i|
' 4 $ " < # . / @ A < $ # / "# £? < ³P
u∈U
|h{u}, U i|+
P
u∈U
|h{u}, U i| 6
P
u∈U
|h{u}, U i|+
P
u∈U
|h{u}, U i|
@£? # A.2|hU , U i|+
2|hU, U i| 6 2|hU , U i|
< @@ # $ < £? # ; " < ®|E| − m(G, U ) 6 m(G, U )
9 § 9m(G, U ) > |E|/2
' 5 ; @β(G) 6 |E|
<ω(G) = 0
¯ " / # $. " ? < # . / . ? ) +1@ .$ # $ < ®/ AU = V
± 9 . . < # <δ > 1/2
' E.$ # A .$ #< <"¢ # $ < @ $? " < < A ¢ ?C
4
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¯ ¤ ¥ ¦§ ©© ª ¦ § ¨ © © ¤ © § ¬«¬ ¦ ± < / " A?F #¡ /. #<m(U, G) 6 B|V |/(B + 1)
/? # $£? "¢ .∆ 6 B
' = # "" ?$ 9ω(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. / "¢ ? < $ " $ < " # $$?A " # / < # < # . < ;? < G < £? " / # $. " ? < # . $ < A. / " $B + 1
$. < $ ' E@ # < # " /?;A? / . #< A " / . / .$ # < # . ' = " /. #< GA? " G 9 ) 333*3 >B
< 33 +3 >B
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" $//. < $@ " $$ # £? < A # < # " $. < # A < # £?$ ' 8 < < A. ² £? 9 @. #" $ < A . < A$ 9 ) 333*3 >∈
GLO
[ρ]
¯ @.$ # A < " $;. # $ # !$ G > . $ ± 9 " /? ; A?/. # < $ < @.@ " ? ' E.$ # A .$? # $ < @I(D, C)
A ) *3)+ < @.$ < ? # $.$ ? # $ < @f (I) = G(V, E)
A ) 333*3 $$.@ # < ® @ & £? $. ? $ > $ "C
j
A " #""C
? $. <v
j
9 < @ < ? <v
j
v
j
0
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j
<C
j
0
@.$/.A < $@./. < . < A$ " < $@.?$³f : I = (D, {C
1
, . . . , C
n
}) 7→ f (I) =
G(V, E)
; @V = {v
1
, . . . , v
n
}
<E = {v
j
v
j
0
: j 6= j
0
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j
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j
0
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' 4 @.$ < ? @ < # . A "¢# $ < @f (I)
/A? < /$?/ " ?$n
2
m
2
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j
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j
0
)
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<f (I)
. <" $ " A$. " ? < # .${0, 1}
n
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i
= 1 ⇔ C
i
∈ C
0
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A ) *3)+ ´s
i
= 1 ⇔ v
i
∈ U
/ . ? "¢ # $ < @f (I)
A ) 333*3 ' 5 # $ # 9 " .@ < # .g
$ < " .@ < # . # A < # < ³g =
A: {0, 1}
n
→ {0, 1}
n
9s 7→ g(s) = s
9 @£? # # A? # < ? A?@ < # . ; # A < $? @ < # ; ' = @.$ < ?@ < # . 9 " " < # .(( s
$ < " A$G ⇔ s
$. " ? < # . A ) *3)+A $C ))
$ < ; # ² ³(( s
$ < " A$G ))
£? # ; ? <(( ∀j < j
0
9s
j
× s
j
0
= 1
# / "# £?v
j
v
j
0
∈ E ))
/
@£? # £? # ;? <(( ∀j < j
0
9s
j
× s
j
0
= 1
# / "# £?C
j
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j
0
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@£? ¢ ®$. < .?$ < £? # ; " < ®(( s
$. " ? < # . A ) *3)+ A$(C, D) ))
'$A?F # $ < @$. < A.@ $A$.? <# .$ # $ $ ' /?$ 9 < .? < $.? > < # .
s
; " ? $ ? " $ # $ < @$I
<f (I)
9 § 9∀s ∈ {0, 1}
n
9m(G, s) = m(C, s) =
P
n
i=1
s
i
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$ < . /B
9 " .$ " A! F # ?AG
$ < . /B
9 ! " < ' 8² 9 $. # <k
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> A # $ < < $A ¢ ?$. " ? <# .s
$. < " $ $$?I
<f (I)
< /./. > $ < / . ? < . ? < ; . # $ # !V
0
@.$ # A / . ? ) 333*3 " ; . # $ # !g(V
0
)
/ . ? ) *3)+ 9 . @.@ " ? ""¡ ! < £?$ # ) 333*3 >B ∈ GLO[δ]
9 " .$ ) *3)+ >B ∈ GLO[δ]
' 4 /. # < $? < " .$/.?@.@ " ? " /?;A?/. # < < A " / . / .$ # < # . ' -#B
¢ $ < / " ?$$? " < ? . # $ " . F@ < A$. < $A @ < $® < .? < $. < 9 $. # < A$ " @$A!/ & $B
> !? "# $ 9 " //. < A ¢ //.F # < # . A # < # " A < .? < ./ < # ?G > " .@ " A$/. " ¡ A ) 333*3 >B
< 33 +3 >B
$ < /. < 9 /.?A$!/ & $@. F$ 9 ®2/(B + 1)
' 8 @ < # ; < 9 . £? " .$£? 9 /.? " @.?; < ?A$. < $ 9 < .? < $. " ? < # . A ¢ ? # $ < @I
A. # < # < !?. # $m/B
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9 @ & £? $. < $/ << < A@. ?; # ?/ " ? $B
< $ ´ . 9 A$ " @A A!/ & $B
> !? "# $ 9 " $ . $m
<n
A ¢ < $ < A$. < $$. < "# $/ " " < # . ³2 × m = 2|E| =
P
n
j=1
d(v
j
) = B × n
9 .d(v
j
)
$ < " A! A?$. <v
j
A$ " !/ & ' 5 # $ # 9 Aβ(I) > m/B
. A A? # <β(I) > n/2
9 < " //. < A # < # " A ¢ //. F # < # . A ¢ ? @.?; < ? # # "U
9 £? # $ << .? .?$A < # "" ?/ " ?$n × B/(B + 1)
/ " A. # @AU
9 "# $? //. < A # < # " A³δ(I, U ) = (n − |U |)/(n − β(I)) > 2/(B + 1)
'GLO
[δ]
: / . / # <P
$ < $ §¦ « § $ # "" ; # ² 9∀X
<∀Y ⊆ X
9P (X) ⇒ P (Y )
' 7 / " ? $ 9 " /./ # < $ < A # < . < # ; # " $ # "" $ < ; # /.?? #²# < A ¢ $ " $X
< ?F/.?? ? < # ² # < ' 4 /. " ¡ ) 333*3 .?@ " ? # A " @ " # £?A < # "" F # ?$. < A$/ . " ¡ $A ² # $$ ? A$/ . / # < $ & A # < # $ ' -. # <P
? / . / # < & A # < # <X
? $ " ':P
> / < # < # . AX
$ < ? $ "S = {V
1
, . . . , V
q
}
A$.?$ > $ " $AX
£? # ; # ² " $ < . # $/./ #< $$? # ; < $³∪
q
i=1
V
i
= X
´∀i, j = 1, . . . , q
9i 6= j ⇒ V
i
∩ V
j
= ∅
´∀i = 1, . . . , q
9P (V
i
)
' -. # <Π
? /. " ¡ ANPO
A. <" $ # $ < @$$. <" A. A ¢ ? $ "X
< ; < ? "" < A ¢ ?; " ? < # .p : X →
N
A$ " < $AX
´Π
$ < ¬¥ ¤ © ¥« ¦§¦§¤ ¬¬ © ¬ ¦ $ §¦ « § $ ¢ # " F # $ < ?/./ # < & A # < #P
< "" £? < .? < # $ < @I
AΠ
; # < ® $.?A? /. "¡ A? < B/³β
Π
(I) = min
(
q
X
i=1
α (V
i
) : S = {V
1
, . . . , V
q
}
$ < ?P
> / < # < # . AV
)
.α
$ < ?.@ < # . A ¢ ; " ? < # . A$$ " $A < B/ # A # @ < # @ . <|
9 /. # A$F # ? 9 /. # A$. B 9''' 4 # > /@ # ! 9 " @. " . < # . 9 " $/. " ¡ $A/ < # < # . @ " # £?$ 9 $.?$ > !/ & $®A! . 9 $.?$ > !/ & $/ " # $ '''9 $. < < .?$A$/. "¡ $A/ < # < # . < & A # < # ' " $ ¢ ! # < /.? < .?$@$/. " ¡ $A/ < !? $ " ¯A ¢ " < $ 9 A$. < $ ± ? . # # ?A$.?$ > $ " $; # ² < ?@ < # /./ #< ' 5 # $ # 9 " /. " ¡ A @. " . < # . / < ! " $$. < $ ? ## ?A$ < " $£?A@ " ? # A " / < # < # . @ " # £?$ " $/ < ! ? # # ?A@ "# £?$ ´ /.? " /. " ¡ A/ < # < # . $.?$ > !/ & $®A!. 9 $.?$ > !/ # A? #< /@£?$.?$ > $A / <#<# . A. #<< AA! . / ? @.$ < < ´ / . ? " / < # < # . $. ? $ > ! / & $/ " # $ 9 " $$. ? $ > ! / & $ # A ? # < $A. # ; < < / " # $ ' 8 ² 9 " # > /@ # ! " ¡ ;A?! < A ¢ " < $A$? . # # ? A . < $ 9 A$. < £? " $.A$;. " ?$A$. < $A$? . < ¢ F@ ¡ A/$ " ;. " ? A " . < ' 4 $ < # " # < 9 " @ " # £? 9 " . < # . ¯ $ ? " $A! $A$ " $! / & $.?$ ? " /. # A$ A ¢ ? $ " A ¢ " < $ ± 9 " / " # < $. << .? < $A$/./ # < $ & A # < # $ ' E & @? A@$/. " ¡ $ #< "¢ . < A . ?$$ < ?A$A$$. /./@A 9 < . < A. " # ?®A # < $ " !. # < & $A $. " ? < # . // .@ & ´ @ $. << . ? $A$/ . " ¡ $ > @ " A " @ & @ & ./ < # . "" ' 4 /. "¡ A@. " . < # . 9 /F/ " 9 A . ?$$// "# @ < # .$ A$ " @.@/ < # . A ¢ / " . # $A? < /$¯A < .?. # $ 9 A ¢ F$ 9''' ± H 9 G H ´ @ " ? # A " / < # < # . $. ? $ > ! / & $/ " # $¯ // " ? $$ # ¦ ° ¦ $ § ¬ ªª ± # < ; # < . < < A$ " @.@ / < # . A@ # @? # < $ # / # $¯@ '9 G D /.?? < .?A ¢& . # . A$ < ?A$ "# $®@/. " ¡ ± ' #§
Π
ª ¦ ¬¥ ¤ © ¥« ¦ § ¦ § ¤ ¬¬ © ¬ ¦ $ § ¦ « § « « ¦ § ¤ ¬α =|
« ¤ ªΠ ∈ GLO[δ]
.?$ "" .$. < £? < .? < ./ < # ? " .@ " /.? " ;. # $ # ! > . " # $? //. < A # < # " AG /.? " $/. " ¡ $A/ < # < # . <& A # < # < £?@//. < $ < << # < $B/ < . < # £? < /.??/./ # < & A # < # £? " @.£? . < # ; # " ' -. #<Π
? /. " ¡ A/ <#<# . < & A #< # <X = {x
1
, . . . , x
n
}
? $ " ®n
" < $ ' .B < ?/ < # < # . @.? / < # < # . A$ " < $AX
$??/ " ?$n
$ " $ 9 . /? < / $ < < .? < $. " ? < # .V
1
, V
2
, . . . , V
q
/? ;@ < ?s ∈ {0, 1}
n
2
£? ¢#" ? <# < / < $ " . ³(( s
i
j
= 1 ))
9 $ # < $? " < $ #((
"¢ " <x
i
$ < A$ " $.?$ > $ "V
j
))
< £? # ; # ² ³∀i = 1, . . . , n
9P
n
j=1
s
i
j
= 1
$ #s
$ < ? / < # < # . <∀j = q + 1, . . . , n
9P
n
i=1
s
i
j
= 0
$ # $ ? " $ " $q
/ # $ # A # @$$. < ? < #"# $ $ ' 4 ; " ?A ¢ ?$. " ? < # . /.?α =|
$ < " . A$.?$ > $ " $. < " / < # < # . < @. @ . / ? < F @ A " @ A # " # <n
A "¢ # $ < @ 9 # " $ ¢ $ ? # < £?Π
$ < ; # A < /. " B. # " < . ³∀X ∈ I
Π
9ω
Π
(X) = |X| = n
' E.$ # A .$ " ;. # $ # !V
£? # @.$ # $ < ®/$$A ¢ ?$. " ? <# . ®?$. " ? <# . ;. # $ # @ & > ! < "¢ @ < < # . A A ?F " < $ ?/ " ? $ < $. # <{V
1
, . . . , V
q
}
? . / < # ? " .@ " " < # ; < ® @;. # $ # ! ' 4¢ ./ < # "# < A " / < # < # .{V
1
, . . . , V
q
}
$ # ! #² £? ¢ ?@? @ < < # . "# $ " A ¢ ? " < ®? ? < $.?$ > $ " ¢ " # . # <" $. " ? <# . @ < ? "" ³ A ¢ ? < $ < $ 9 # " ¢ $ < / $A${V
1
, . . . , V
q
}
A$ # ! " < . £? "¢ . /? # $$ " # # ' -# " $. " ? < # . @. / . <k
$ # ! " < .$ 9 . $?//.$£? ¢#" $ ¢ ! # < A$k
/ # $$.?$ > $ " $V
1
= {v
1
}, . . . , V
k
= {v
k
}
<"¢ . £? £?P
< < & A # < # 9 " $$. < $v
1
, . . . , v
k
$. << .? .?$A$ $ " $A # $ < # @ < $ ´ A ¢ ? < $ < $ ³∀(i 6= j) ∈ {1, . . . , k}
9 $ #¬P ({v
i
, v
j
})
9 " .$∀s ∈ Sol
Π
(X)
9m
Π
(X, s) > k
9 < /@.$ £? < 9β
Π
(X) > k
' = #"" ?$ 9 " $? < $$.?$ > $ " $V
k+1
, . . . , V
q
@. < < @ & @? ?. # $A?F " < $ 9 # " $; # ² <P
q
i=k+1
|V
i
| = n − k > 2(q − k)
9 $ # < $ ? " < $ #q 6 (n + k)/2
' 7 .@ 9 @.@ " ? $ # . 9δ
Π
(X, {V
1
, . . . , V
q
}) = (ω
Π
(X) − q)/(ω
Π
(X) − β
Π
(X)) > (n − ((n + k)/2)(n − k)) = 1/2
' 7 . < .$£?@//. < $ < << # < ' -. # < ?/./ # < & A # < #P
. < # ; # " <X
n
? # "" A ¢ $ " $ < # @ < < @. # $$ < /.? "¢ # @ " ?$ # . ; # ² <P
' E.$ # A .$ " $? # < A ¢# $ < @I
n
= (X
n
, P )
A?/. " ¡ A/ < # < # . < & A # < #Π
'β
Π
(X
n
) = 1
9ω
Π
(X
n
) = |X
n
|
< ? ./ < # ? " .@ " /.? " $;. # $ # !$ > . $/ < # < # .X
n
A$$.?$ > $ " $A < # "" $A? F 9 / " ?$/? < > < ? $.?$ > $ " A < # "" < . # $ < /.?; " ?b|X
n
|/2c
' E $? " < < 9 / "¢ // < ; < A " $. " ? < # . //.@ & 9 ¢ $ < @/A < /$$ # . > A # /? # $£? ¢ # " ! " " //. < . < ? /.? " /. " ¡ A@. " . < # . ## ?A$ C ¯. < ® H A /? # $ 9 ¯ ±± ' 6 # $ # "" # $$$ ? < . ? < ? ! ? A # "" ? $ // . F # < # .$ ® "¢ # A A ¢ ? " B$? /?/ " ?$ ² 9 ;@A$;. # $ # !$? /?/ " ?$!A$ ': # $ < @
I(C, S)
A ?/ . " ¡ A@. ?; < ? A ¢ $ " $ ## ?¯ *3+3 ± $ < " A. A ¢ ? $ "C = {c
1
, c
2
, . . . , c
m
}
A ¢ " < $®@.?; # < A ¢ ? #""S =
{S
1
, S
2
, . . . , S
n
} ⊆ 2
C
A$.?$ > $ " $A ¢ " < $AC
A ¢ ? # .C
' 4 ? < $ < A < .?;? $ "˜
S ⊆ S
A@ A # " # < ## ?£? # @. ?;C
' $ $ < # < # @ # ?F # $ < @$B
> . $ 9 @ ¢ $ < > ® > A # ?F # $ < @$I(C, S)
A. << .?$ " $$.?$ > $ " $S
i
A " #""S
; #² <|S
i
| 6 B
' .?$ . < .$@ << ; # < A *3+3 / *3+3 >B
' *3+3 ¨B ∈ GLO[δ]
. ? $ "" .$. < £? < . ? < . / < # ? " .@ " / . ? " ;. # $ # !G > . " # $? // . < A # < # " A1/(B + 1)
' 4 ; " ?A ¢ ?$. " ? < # .˜
S
$ < A. /m
SC
(I, ˜
S) = | ˜
S|
' " $ ¢ ! # < A ¢ ? /. " ¡ /. " B. > # " < . /? # $ £? " / # $. " ? < # . 9 @.$ # $ < < ®$ " @ < # . < . ? <S
9 $ < A; " ?n
' 4 ;. # $ # !G > . A $ # ! # @ # @.;. # $ # A ¢ ?$. " ? < # .˜
S
< .? < $ " @ < # .˜
S
0
A. < ?/ " ?$? $.?$ > $ " A # ¡ A˜
S
´ " $./ < # " .@?F/.?@;. # $ # !$. < $ # / " < A$$. " ? < # .$ ## " $ ' E.$ # A .$?@.?; < ?˜
S = {S
1
, S
2
, . . . , S
p
}
A < #""p
´S
˜
$ < ?@.?; < ? # # " $ # "" ; #² " $A?F@.A # < # .$$? # ; < $³¯ # ±∀i = 1, . . . , m
9∃j ∈ {1, . . . , p}
< " £?c
j
∈ S
j
< ¯ ## ±∀j = 1, . . . , p
9∃i
j
∈ {1, . . . , m}
< " £?c
i
j
∈ S
j
<∀j
0
6= j ∈ {1, . . . , p}
9c
i
j
∈ S
/
j
0
' 4 @.A # < # . ¯ # ± < A? #<" "# $ #"# < A " $. " ? < # .˜
S
9 " @.A # < # . ¯ ## ± $ # # "# < ' = ¯ ## ± 9 . $ # < £? ¢ . /? < @.$ < ? # ? $ "˜
C = {c
i
1
, . . . , c
i
p
}
Ap
" < $A # $ <# @ < $¯? " < /$.?$ > $ "S
j
A " @. ?; < ? ± £? # ; # ² < ³∀c
i
0
∈ ˜
C
9∃!j ∈ {1, . . . , p}
< " £?c
i
0
∈ S
j
' -#"¢ # $ < @ # # < # " $ << "" £? < .? < " <c
i
∈ C
// < A$?. # $A?F$.?$ > $ " $S
j
9 " .$ @ ¢ $ < / <# @? " # ; # A$ " < $c
0
1
, . . . , c
0
p
0
´ @?F > @ # /.?; < 9 /@.$ < ?@ <# . 9 // < # ® ? $@.A$.?$ > $ " A˜
S
9 #" $// # $$ < A.@A$ " $$.?$ > $ " $S
p+1
, . . . , S
n
9 @ £? # / < A ¢ < " # "¢ # @ " ? $ # .˜
C ⊆ ∪
n
j=p+1
S
j
' 4 $$.?$ > $ " $S
j
AS\ ˜
S
< < A < #"" . /B
9 . A A? # < $?p
" " < # . ³p 6 B(n − p) ⇔ p 6 Bn/(B + 1)
9 @£? # .?$ ¡ ?//. < A/.@³δ(I, ˜
S) =
(ω(I) − p)/(ω(I) − β(I)) > (n − (Bn/(B + 1)))/n = (n/(B + 1))/n = 1/(B + 1)
'-# ? @ < # " < ®@.?; #