Differential approximation for satisfiability and related problems

Texte intégral

(1)Laboratoire d'Analyse et Modélisation de Systèmes pour l'Aide à la Décision CNRS UMR 7024. CAHIER DU LAMSADE 180 Avril 2001. Differential approximation for satisfiability and related problems Cristina Bazgan, Vangelis Th. Paschos.

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(39) Q:T¨KI=RSNPOQ:dWLKdP]=RSNPKT¨Ka x _Q:T_\fh ω(x) R9VYQ:dP]9VKa Q KJSWZRbWLKdP]=RSNPKTwKa x EiKM9V<w.y1%-4&61%+&5*,+#r,7=+p:<1=-{t@* 7%+x1565/+7U8=(;:j1%.0(27=- +@1%.0(27Ka y NGW¢\=VXe9T9V\ QW. r(x, y) = max. 9NPdPVR9Vb&=( )*p+@*p-/.0(21%35*,+0r,7=+:j1=-4t* . . . m(x, y) opt(x) , opt(x) m(x, y). 7=+ 15654+798%(;:<1%.0(27=-. ρ(x, y) =. . Ka. +1%.0(27. m(x, y) − ω(x) . opt(x) − ω(x). y. . NGWc\=VXe9T9V\ QW. RNGWV(QW hwRSK¨WZVXV R Q*RR 9V\=NPVXJLVXTRSN?Q:d§Q:I9I9JLK(MNPOQ*RSNPKT°J Q*RSNPK¨NGWW RAQ:f9dPVa2KJR9V Q= T9VRSJ Q:T_WZa2KJLO Q*RSNPKT KaR9VkKfh V^`RSNgY|Va2]9T_^`RSNPKTwKa QI9JLKf9dPVXO 9NPdPVlR9NGWb\=K¸VWcT9KR9KdG\Ua2KJ¢R9VWZRAQ:T_\_Q:JS\ Q:I9I9JLK(MNPO Q*RSNPKT J Q*RSNPK9E Q w.y1=-{&61=+@& f (n) 1p565/+798%(;:<1V.0(27=-Z1%3onm7=+p(#.0q6: a2KJQ KJkQ a2]9T_^`RSNPKT f f (n) > 1

(40) Q:T Q:dP[KJLNgR 9O NGWl' I9JLKf9dPVXO Π NPHa ¸a2KJ¯Q:Th NPT_W RAQ:T_^&V x Ka Π ¸NgR JLV&RS]9JLT_WQkWLKdP]=RSNPKT y WL]_@^ R Q*R r(x, y) 6 f (|x|) V9VXJLV |x| NGWR 9V WZN XVKa x E vV WSQXhvR Q*R Q:T8KI=RSNPO N (Q*RSNPKT I9JLKf9dPVXO NGW w.y1=-{&61=+& t7=-w.y1=-/.23 l 15654+798%(;:<1mv,3* NPaH a2KJ WLKO V ^&KT_WZRAQ:TR c > 1 R 9VXJLV V&MNGW R WQ IKdghT9KO N?Q:d¯RSNPO V WZRAQ:T_\_Q:JS\ ciQ:I9I9JLK(MNPOQ*RSNPKTsQ:dP[KJLNgR 9O a2KJNgRE HbTsKI=RSNPO N (Q*RSNPKTCI9JLKf9dPVXO QW Q w.y1=-{&61=+@&57%3 l%-47%: (21%3.0(;:<* 1p565/+798%(;:<1V.0(27=-uwetpq*p:j1 NPaNgR QWQ°IKdghT9KO N?Q:dcRSNPO VwW RAQ:T_\_Q:JS\ iQ:I9I9JLK(MNPO Q*RSNPKA T ¢a2KJ V&Y|VXJZh ^&KT_W RAQ:TR E ¸NPO NPd?Q:JLdg6h . a2KJ Q a2]9T_^`RSNPKT f f (n) < 1 Q:T Q:dP(1[K+JLNgR ε) 9O NGW Q &=( )*,+*,-.0(21V3 f (n) 15m5/+798%(;:<1%.0(27%ε- >1%3on607%+(#.0q6: a2KJQ NPHa _a2KJbQ:Th¨NPT_WZRAQ:T_^&V Ka Π _NgRcJLV&RS]9JLT_WQ WZKdP]=RSNPKT y WZ]_@^ R Q*R ρ(x, y) > f (|x|) E VkWSQXh I9JLKf9dPVXO RQ*RbQ:T¨KΠI=RSNPO N (Q*RSNPKT I9JLKf9dPVXO NGx W &%( )*,+*,-/.0(21%3Xt7=-wp.y1=-/.23 l 15m5/+798%(;:<1mv,3*lNPHa 9a2KJ¢WZKO Vl^&KT_W RAQ:TR δ < 1 R9VXJLV V&MNGWZR W QUIKdghT9KO N?Q:dRSNPO V \=NPVXJLVXTRSN?Q:d δ iQ:I9I9JLKMNPOQ*RSNPKT Q:dP[KJLNgR 9O a2KJ NgRE°HbT KI=RSNPO N (Q*RSNPKT I9JLKf9dPVXO QW Z Q &%( )*,+*,-/.0(21%3T57V3 l%-{7=: (21%3g.0(;:<*L15m5/+798%(;:<1%.0(27%-w,t,q}*,:<* NPabNgz R QW Q IKdghT9KO N?Q:d¯RSNPO V \=NPVXJLVXTRSN?Q:d (1 + ε)iQ:I9I9JLK(MNPOQ*RSNPKAT a2KJV&Y|VXJZh^&KT_W RAQ:TR ε > 0 G E V W QXh R Q*R HR K KI=RSNPO NGW Q*RSNPKT I9JLKf9dPVXOWQ:JL' V &=( )*,+*,-.0(21V3N*p(;|=1V3D*,-/.NPa¢Q \=NPVXJLVXTRSN?Q:d iQ:I9I9JLK(MNPOQ*RSNPKTyQ:dP[KJLNgR 9O a2KJkKT9VKa±R 9VXO NPO I9dPNPVWbQ\=NPVXJLVXTRSN?Q:d δ iQ:I9I9JLK(MNPOQ*RSNPKT Q:dP[KJLNgR 9O²a2KJ¢δR 9VkKR 9VXJcKT9VE T8R 9NGW IQ:IVXe J !¯V W RS]_\h R 9V \=NPVXJLVXTRSN?Q:dcQ:I9I9JLK(MNPOQ:f9NPdPNgRih8Ka¢R 9Va2KdPdPUK bNPT9c [ 4 Q:JS\ KI=RSNPOQ:d W Q*RSNGWZeQ:f9NPdPNgRihUI9JLKf9dPVXOWXE o8t1 } o p7 q rtu B -X. 1 %

(41) QWZV&RKa^&d?Q:]_WLVW C , . . . , C KT n YQ:JLN?Q:f9dPVW x , . . . , x E W F7O NPT9NPON XV@W I R 9VT¸]9O fVXJKa

(42) ^&d?Q:]_WLVW 1%B1%

(43) QkRSJL]=R QWSWLNP[T9O 1VXTRRSKR m9VYQ:JLN?Q:f9dPVWR Q*R¢O1Q*MNPO N XVn> W Q*RSNGWZe9V\ E o8t1 } o p7 q O

(44) -X. B1 %

(45) QWZV&RKa^&K%T h ]9T_^`RSNPKT_W C , . . . , C KT n YQ:JLN?Q:f9dPVW x , . . . , x E W F7O n NPT9NPON XV@W IR 9VT¸]9O fVXJkKa¢^&KT 1%B1%

(46) Q¨RSJL]=R QWSWZNP[T9O VXTRRS1K R 9V YQ:mJLN?Q:f9dPVWR Q*ROQ*MNPO 1 N XVj h ]9T_^`RSNPKT_WbW Q*RSNGWLe9V\ E KJ Ql^&KT_WZRAQ:TR k > 2 =o8t1} k rt u =o8t1} k x

(47) o8p7q rt!u o p7q x

(48) Q:JLVcR 9VcY|VXJSWZNPKT_W Ka1o8t1} rt!u 9o t1 }

(49) ?9o p7qyrt u 9o p7c q x

(50) ?9VXJLVV(Q@^ ^&d?Q:k]_WLVKJ^&K%T h k]9T_^`RSNPKT QWWLN XVQ*ROK|W R k E KJ Qk^&KT_WZRAQ:TR B > 1 9o8t1} k rtNu B, B¯ e9o8t1} k x

(51) > B, B¯ e9o p7q k rtiu B, B¯ e9o p7q k

(52) _ B, B¯ ,. m.

(53) o t1}yrtui B, B¯ e=o8t1}d

(54) _ B, B¯ e=o p7q rtuN B, B¯ e=o8p7q

(55) > B, B¯ ±Q:JLV¢R9V¢Y|VXJSWLNPKT_W±KaR9VWZV 8 I9JLKf9dPVXOWM9VXJLVkV(Q^@UdPNgRSVXJ Q:dQ:I9IV(Q:JSWQ*RbO K|W R B RSNPO VW&E o8t1}C_rtu -X. B1 %

(56) QWZV&RKa^&K%T h ]9T_^`RSNPKT_W C , . . . , C Ka

(57) R9JLVXVdPNgRSVXJ Q:dGWcKT n YQ:JLN?Q:f9dPVW x , . . . , x E 1%B1%

(58) Q¯RSJL]=R QWSWZNP[T9OVXTRRSKb1R 9V§YQ:JLN?mQ:f9dPVWRQ*ROQ*MNPO NXVW1R9V±T¸]9O fVXJKa91 ^&KT%h ]9T_n^`RSNPKT_WW Q*RSNGWLe9V\ NPTwWZ]_@^ x Q cQXh R Q*RQ:Th KT9VKa

(59) R 9VXO QWQ*RcdPV(QWZRbKT9VlRSJL]9VkdPNgRSVXJ Q:dQ:T_\UQ*RbdPV(QW RKT9Vka"Q:dGWZVkdPNgRSVXJ Q:dE o p7q Zo t1i} ko p7q1#p bt $¨xz{|u

(60) z}~z{ -X. B1 %

(61) Q [J Q:I G = (V, E) E J FQ WLV&R S ⊆ V WL]_@^ yR Q*9R ∀(u, v) ∈ E u ∈ S KJ v ∈ S IKa 1%B1%

(62) Q O NPT9NPO Q:d Y|VXJZRSV&My^&K(Y|VXL O NPT9NPO ]9O F7OQ*MNPO ]9b O IWZN XVE o p7q Zo t1i} O q z zq zq u p7qtup7q " rz9u B 1 %.

(63) Q [J Q:I E -X. R FQ WLV&R S ⊆ V WL]_@^ R Q*9R ∀u, v ∈ S (u, v) ∈/ E Q:T_\ 1%B1%

(64) Q8OQ*MGNPO =Q:d(V,NPT_\=E)VXIVXT_\=VXTR WZV&d IKaONPT9NPO ]9O F7OQ*MNPO]9b O I WZN XVE ∀u ∈ / S, ∃v ∈ S (u, v) ∈ E x T Q*Ra2KdPdPUK ,W VV¯\=VXT9KRSV¯fh o8p7 q Zo8ti} M Xq /z zq z_qu ! p7qtup7q "8rz=u B Q:T_\ o8p7 q Zo8ti} o p7q1#p bt $x z{|u1z=}¥~z{{ B R 9V§Y|VXJSWLNPKT_W

(65) KaR 9VQ:fK(Y|V±I9JLKf9dPVXOW

(66) KT[J Q:I _XW bNgR OQ*MNPO ]9O \=VX[JLVXV fK]9T_\=V\Ufh B E. I ? 1· ;", # BB'*/

(67) !7v)9%*!2/ . B'*5X5¸'

(68) ! . '*50+%(!7/ . ¶·/'/B

(69) %(!"v)6X)9%(!7X>) $ !76"!?%A@ Ve9JSWZRkI9JLK(Y|V R 9V \=NPVXJLVXTRSN?Q:d§V9]9NgYQ:dPVXT_^&Va2KJo8t} rtuyQ:T_\ o p7qu x

(70) Q:T_\°a2KJo8p7q rt=uyQ:T_\ o8t1}

(71) cE.

(72) 5¸/'*5 , o8t}rtu 1=-{& o p7 q

(73) 1=w *e3;3!1=wlo p7qrtu 1=-{& o8t1} x

(74) W1=+*j&=( )*p+@*p-/.0(21%3 . *@,}(;|%1%3*,-.. 4 *' //¶ V ^&KT_WZRSJL]_^`RQJLV\=]_^`RSNPKT a2JLKO o t1} rtu RSK o p7q

(75) RQ*RkI9JLVWLVXJZY|VWlR9V \=NPVXJLVXTRSN?Q:d Q:I9I9JLK(MNPOQ*RSNPKTsJ Q*RSNPK9EV&R I fVwQ:T NPT_WZRAQ:T_^&VUKalo8t1}rtuKT n YQ:JLN?Q:f9dPVWQ:T_\ m ^&d?Q:]_WLVWXE`KM9V NPT_WZRAQ:T_^&V I 0 Ka o p7 q

(76) ^&KTRAQ:NPT_W m ^&d?Q:]_WLVWQ:T_\wR9V W Q:O VWLV&RKa n YQ:JLN?Q:f9dPVWXENgR V(Q^@°^&d?Q:]_WZV Ka ¯VbQWSWZK^&N?Q*RSVbNPT 0 R 9Vb^&KT%h ]9T_^`RSNPKT `¯ ∧ . . .∧ `¯ V9VXJLV `¯ = x¯ NPa ` = x Q:T_\ `¯ = x `1 ∨ . . .∨ `t I 1 i i j j NPa ` = x¯ E RNGWV(QWZhkRSKlWZVXVR Q*R Iopt(I 0 ) = m−opt(I) Q:T_\ ω(It 0) = m−ω(I) Ej HdGWZKi NPa m(I 0 , y) NGWR9V i j YQ:dP]9VKa1R 9VWLKdP]=RSNPKT y NPT I 0 ¸R 9VXT¨R 9VWSQ:OVWZKdP]=RSNPKT y QW NPT I R 9VYQ:dP]9V m(I, y) = m − m(I 0, y) E E KM9VkJLV\=]_^`RSNPKTwa2JLKO o p7 i q

(77) RSKo8t1}rtu NGW¯R 9VW Q:O VE Kg¸]_, W 0 hρ(I, Q:T y) V&M9Q=^`RSρ(I dghWL,NPy) O NPd?Q:J JLV\=]_^`RSNPKAT KT9Vv^Q:TI9JLK(Y|V¥R Q*R o p7q¬rtuQ:T_\ o t1}

(78) ²Q:JLV Q:dGWLK Q:I9I9JLK(MNPOQ*RSV9V ]9NgYQ:dPVXTRE hwR 9V I9JLK¸Ka Ka§R 9VXKJLVXO F KT9V V(QWLNPdgh°^Q:T¥\=V\=]_^&VR Q*Rka2KJV(Q@^ v^&KT_W RAQ:TR o8t} rtu k Q:T_\ o p7q k

(79) CQMW ¯VXdPd

(80) QWo p7q k rtu¥Q:T_\ o8t} k x

(81) Q:JLV\=NPVXJLVXTRSN?Q:d

(82) 9V ]9NgYQ:kdPVX>TR2E P KT_WZNG\=VXJQ:T NPT_WZRAQ:T_^&V Ka QOQ*MNPO N (Q*RSNPKT°I9JLKf9dPVXO Q:TvQ:I9I9JLK(MNPOQ*RSNPKT Q:dP[KJLNgR 9O A a2KJ Π Q:T_\U\=VXT9KRSVkfh S Qa2V(QWZNPf9dPIVWLKdP]=RSNPKTwKa Π ^&KOI9]=RSV\¨fh AΠNPT I NE KM9VXAT mA (I, S) − ω(I) mA (I, S) ω(I)>0 mA (I, S) > δ =⇒ > δ + (1 − δ)ω(I) =⇒ >δ opt(I) − ω(I) opt(I) β(I). Q:T_\¨R9Vka2KdPdPKUcNPT9[I9JLKIK|WZNgRSNPKT NPO OV\=N?Q*RSVXdghb9KdG\9W&E 4 '*/B /!?%*!2/ . , 565/+798%(;:<1V.0(27=-'7r_1x:<198%(;: (U1%.0(27=-z5/+7mv,3*,: +1%.0(27. δ. . (;:_543 (2*,wx1565/+7U8=(;:j1%.0(27=-c7Cr. Π. (#.0q6(;-&=( )*p+@*p-/.0(21%315m5/+798%(;:<1%.0(27%-. (#.0q6(;-'w.y1%-4&61%+&b15654+798%(;:<1%.0(27=Π. +1%.0(27. 1/δ. . KO 9f NPT9NPT9[wR9VJLVWL]9dgR W Ka¯R9VXKJLVXO F Q:T_\ I9JLKIK|WLNgRSNPKT F<bNgR R9Va"Q^`RRQ*Rr,7=+ k > 2 1=-4& B > o8t1} k rtui B, B¯ 1=-{&o8t} k x

(83) > B, B¯ <q}1=|%*z-47w.y1=-{&61=+@& 57%3 l=-{7=: (21V3!.0(;:<* 15m5/+798%(;:<1%.0(27%3 w,t,q}*,:<1V.y1F B \=F I9KT9V\=V\=]_^&VW¢R9Va2KdPdPKUcNPT9[9E E P. D.

(84) . /'*/6762)_'(@ , 7=+ k > 2 1=-4& B > 3 o8t1} k rtui B, B¯ o8t1} k x

(85) > B, B¯ o8p7q k rtui B, B¯ 1=-{& o8p7q.

(86) > ¯ q1=|%*'-{7`&=( )*p+@*p-/.0(21%357%3 l=-{7=: (21V3_.0(;:j*c15654+798%(;:<1%.0(27=-w,t,q}*,:<1%.y1 -3*,w@w k B, B. P = NP. .

(87) . ;?3 «© ¡ ) . 0 ° ¡

(88) ¤ ¢ o p7q8xz{|u1z=}C~z{wNGWcQWR9Vo p7qyo8p7q1p#bt $ x z{|u1z=}C~z_{ \=VXe9T9V\ NPT WLV^`RSNPKT mO K\=]9dPKR9V. a"Q^`R§R Q*R§R 9Vba2V(QWZNPf9dPVWLKdP]=RSNPKT_W§a2KJ R 9Vca2KJLOVXJ±Q:JLVcT9KR±OQ:T_\_Q*RSKJLNPdghONPT9NPOQ:dET VcW9KU RQ*R±R9VcJLV, \=]_^`RSNPKT ]_WZV\NPT BQP }K\V] a2JLKOo p7q xz{|u

(89) z}¥~z_{RSKo p7q rtu NGWQ:dGWZK\=NPVXJLVXTRSN?Q:d=Q:I9I9JLK(MNPO Q*RSNPKT I9JLVWLVXJZYNPT9[9E Kg9NGMW cNPdPdQ:dPE dPUK ]_W¯RSK VWZRAQ:f9dPNGW Q:TUNPTQ:I9I9JLK(MNPOQ:f9NPdPNgRihUJLVWZ]9dgRa2KJo p7qsrtuE.

(90) 5¸/'*5 3 -/3*pww o p7qrtu (;wz-{7%.>&=( )*,+*,-.0(21%3 1/m1−ε 15654+798%(;:<1mv,3*r,7=+ co − RP = NP . q}*,+* ε > 0. 1=-l. m. (;w.0q*O-}:jv@*,+x7rte31=}we*pw7r.0q*O(;-w.y1%-4t*. . 4 '*//¶ KM9VJLV\=]_^`RSNPKT Ka BQP }K?\V] E a2JLKO²o p7q xz{|u1z=}s~z{¨RSKo p7qyrt=u¯KJCfW¢QWa2KdPdPKUbWXE 1V&R fVQ[J Q:IvKT n Y|VXJZRSNG^&VWQ:T_\¥\=VXT9KRSV fh V = {1, . . . , n} NgR WY|VXJZRSV&M¥WZV&RE T¥KJS\=VXJRSK G = (V, E) ^&KT_WZRSJL]_^`RQ:T¥NPT_W RAQ:T_^&V I Ka¯o p7q rtu

(91) Q*RlV(Q^@°V\=[V (i, j) ∈ E, i < j V QWSWZK^&N?Q*RSVQYQ:JLN?Q:f9dPV x E ij KJ¢V(Q@^ UY|VXJZRSV&M i Vk\=VXe9T9V Q^&d?Q:]_WLV C 9VXJLV i. Ci =. _. xij ∨. j:(i,j)∈E∧i<j. _. x ¯ji .. j:(i,j)∈E∧i>j. JLKO Q¥Y|VXJZRSV&Ms^&K(Y|VXJ C Ka G ¯V \=VXe9T9V¥Q:TQWSWZNP[T9O VXTRUQWa2KdPdPKUbWXE KJV(Q^@ i ∈/ C Q:T_\CV(Q^@ NPa Q:T_\ NPa E ¸NPT_^&V C NGWQ Y|VXJZRSV&M ^&K(Y|VXJeR9NGW\=VXe9T9NgRSNPKT°NGWT9KR (i, j) ∈ E xji = 1 i > j ^&KTRSJ Q\=NG^`RSKJZh|E a i ∈/ C =R9VXT C xNGijW¢T9=KR0WSQ*RSNGiWLe9<V\ j Q:T_\UWZK opt(I) E 6 opt(G) i NgY|VXT¥Q:T¥QWLWLNP[T9O VXTR Ka dPV&R NGW W Q*RSNGWLe9V\ E bKRSVRQ*RlWLV&R NGWlQY|VXJZRSV&M ^&K(Y|VXJ WLNPT_^&V a2KJ (i, j) ∈ E ¯Q*R dPV(vQW R IKT9V Ka CC= Q:{iT_\ : CCi NGW WSQ*RSNGWLe9V\s}Q:T_\ WZK Q*R dPV(QW R CKT9V KabR9V Y|VXJZRSNG^&VW Q:I9IV(Q:JSW NPT C E ¸K Q*R V(Q@^ QWLWLNP[T9iO VXTR v jKa I M¯V°QWSWZK^&N?Q*RSV°NPT G Q¥Y|VXJZRSV&M^&K(Y|VXJ C cNgR i, j E KM9NGWbQ:dGWLK I9JLK(Y|VW¢R Q*R i E m(G, C) = m(I, v) KW 9UK =R opt(G) Q*R ρ(G) > ρ(I) E NPTQ:dPdg6h ]_WZNPT9[ ω(I) 6 ω(G) =NgRbNGW¢V(QWZh RSopt(I) V QXY|V¨WLVXVXT8 R Q*R o p7q xz{|u

(92) z} ~z{ NGW \=NPVXJLVXTRSN?Q:d9V ]9NgYQ:dPVXTR RSK¥o8tS } q z zq zq u rz9 u F;?9NG@^ UNGWcQWo t1 } Sq /z zXq zqd u p7qtup7q " rz9u O K\=]9dPK R 9Vla"Q^`RcR Q*R¢R 9VkNPT_\=VXIVXT_\=VXTR WLV&RkRSK ^&KO I9]=RSj V QWT9KRRSKUfVO NPT9NPOQ:;d IAE ^T¥R 9VKR 9VX J Q:T_\ WLNPT_^&V R 9< V ¯KJSW R WLKdP]=RSNPKT8a2KJo t1} q z zq zq u rz9u8NGWlR 9V VXO I=Rih¥WZV&b R F7NPTvKR 9VX J ¯KJS\9,W IWZRAQ:T_\_Q:JS\ Q:T_\ \=NPVXJLVXTRSN?Q:d Q:I9I9JLK(MNPOQ*RSNPKT J Q*RSNPK|W±^&KNPT_^&NG\=VE ]9JZR 9VXJLOKJLVV =o8t1} q /z ω(I) z_ q =zq0u ∀I rz9u NGWT9KR \=NPVXJLVXTRSN?Q:d 1/n1−ε Q:I9I9JLK(MNPOQ:f9dPVUa2KJQ:Th ε > 0 ±]9T9dPVWSW co − RP = NP F B [ Q@W \V] E IAE P KT_WL9V ]9VXTRSdg6h co p7qx z_{|u

(93) z=} ~z_{8NGWlT9KR\=NPVXJLVXTRSN?Q:d R 9V 1−ε iQ:I9I9JLK(MNPO Q:f9dPV a2KJQ:Th ε > 0

(94) ]9T9dPVWLW co − RP = NP Q:T_\ JLVWL]9dgR^&d?Q:NPO V\ a2KdPdPUK bWXE 1/n JLKO R 9VQ:fK(Y|VkI9JLK¸KaR 9Vla2KdPdPUK bNPT9[^&KJLKdPd?Q:JZhUNGWbQ:dGWZK\=V\=]_^&V\ E /'*/6762)_'(@ 3 o8p7q rtNu B, B¯ ir,7=+ B > 1 (;w-{7%.&=( )*,+*,-.0(21%3 1/m1−ε 1565/+@798%(;:<1mv,3D*ir,7%+O1%-l ε > 0 . . }-/3D*,ww. co − RP = NP. . . $#&%('. # B§/(!?%(! 5 0! ±5¸'*5 . %*!2)6¯)9BB'*/

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(116) R9V^&d?Q:]_WZVWQ:JLVkW Q*RSNGWLe9V\ E NgY|VXT Q:TUNPT_W RAQ:T_^&V Ka _rtNu ¯ MbNgR YQ:JLN?Q:f9dPVW x1, . . . , xn Q:T_\ m ^&d?Q:]_WZVW C1, . . . , Cm V ^&KT_WZRSJL]_^`RQw[J Q:I G §ϕNPT_W RAQ:T_^&V Kao 4 p7q q nz zq zq G u p7qtu1p7Xq "rz9{u O QW a2KdPdPKUWXE NgR ].

(117) (V Q^@UIK|WLNgRSNgY|VdPNgRSVXJ Q:d x ¯VQWSWZK^&N?Q*RSVQY|VXJZRSV&M u Q:T_\ bNgR V(Q^@wT9VX[Q*RSNgY|VkdPNgRSVXJ Q:d x¯ ¯VQWLWLK^&N?Q*RSV Q Y|VXJZRSV&M v E KJ i = 1, .i. . , n ¯V\=J Qe NPT G R9VVi\=[VW u v EHbdGWLKjcNgR V(Q^@ ^&d?Q:]_WLV Ci ¯V QWSWLK^&N?Q*RSV i Y|VXJZRSNG^&VW w , . . . , w E KJV(Q^@8^&d?Q:]_iWLV i C V Q\9\ NPT G Q:T8V\=[VjfV&RH¯VXVXT8V(Q^@ c = b(B − 1)/4c Q:T_\wQ:Th Y|VXj1JZRSV&M ^&KJLJLjcVWLIKT_\=NPT9[RSKQdPNgRSVXJ Q:d

(118) j ^&KTRAQ:NPT9V\wNPT C E wjk k = 1, . . . , c. ¸]9I9IK|WLV R Q*RV(Q@^ dPNgRSVXJ Q:d9Q:I9IV(Q:JSWQ*RdPV(QW RKT_^&V!E ¢VXO Q:CJ fR Q*R§Q:TNPT_\=VXIVXT_j\=VXTR§WZV&RKa G ^&KTRAQ:NPT_W Q*RbO K|WZR m · c Y|VXJZRSNG^&VWXE¯HT NPT_\=VXIVXT_\=VXTRl\=KO NPTQ*RSNPT9[WZV&R^&KTRAQ:NPT9NPT9[R 9VlY|VXJZRSNG^&VW^&KJLJLVWLIKT_\=NPT9[RSK R 9V ^&d?Q:]_WZVWbKa ϕ NGWbx Q ¯KJSWZRbWLKdP]=RSNPKTwKaWLN XV m · c E m a G N b W S W * Q S R G N L W e : Q 9 f P d V R 9VXT WZNPT_^&VR 9VWLV&RbKaY|VXJZRSNG^&VW^&KJLJLVWLIKT_\=NPT9[RSK R 9VkRSJL]9VdPNgRSVXJ Q:dGW ϕ opt(G) = n KaQ:T QWLWLNP[T9O VXTRW Q*RSNGWZa?hNPT9[ NGWkQ:T°NPT_\=VXIVXT_\=VXTR\=KONPTQ*RSNPT9[UWZV&R F7V(Q@^ °Y|VXJZRSV&M w NGW\=KO NPTQ*RSV\ fhvQY|VXJZRSV&M¥^&KJLJLVWLIKT_\=NPT9[wRSKϕ QRSJL]9V dPNgRSVXJ Q:d±Ka C IKa O NPT9NPO ]9O WLN XVb E ^T°R 9V KR 9jkVXO J Q:T_\ NPa R 9V j KI=RSNPOQ:dYQ:dP]9VkKa ϕ NGW m0 6 (1 − ε0)m R 9VXT opt(G) = n + (m − m0) · c > n + ε0 · m · c E VCW 9U K R Q*RcQ\=NPVXJLVXTRSN?Q:d f (B)iQ:I9I9JLK(MNPO Q*RSNPKT Q:dP[KJLNgR 9O A a2KJco8p7d q q /z z_q zqu p qtup7q "rz=u bNgR f (B) = 1 − (2ε0 (B − 5)/(2B − 5)) [NgY|VW¯NPT¨R 9Vk^QWL> V 9VXJLV ϕ NGW¢WSQ*RSNGWLeQ:f9dPV Q WLKdP]=RSNPKT Ka_YQ:dP]9VcdPVWLWR Q*R R 9V¯YQ:dP]9V¢KaR 9V¢KI=RSNPO ]9O WZKdP]=RSNPKT NPTR 9Vc^QWZgV 9VXJLV ϕ NGWT9KR±WSQ*RSNGWLeQ:f9dPVE VXT9KRSVfh val R 9V YQ:dP]9VKaR 9VWZKdP]=RSNPKT ^&KOI9]=RSV\¥fh A b E KM9VXAT (m · c − val)/(m · c − n) > E ¸. P N _ T & ^ V : Q _ T \ E VE \ E c 6 (B − 1)/4 m 6 8n/3 val 6 n + (m · ε0 (B − 5)/4) < n + m · ε0 · c f (B) . . Ss ·

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(129) 9F \V\ E. !. . . . . . . . . .

(130)