Approximate Constrained Bipartite Edge Coloring

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(1)Approximate Constrained Bipartite Edge Coloring Ioannis Caragiannis, Afonso Ferreira, Christos Kaklamanis, Stéphane Pérennes, Pino Persiano, Hervé Rivano. To cite this version: Ioannis Caragiannis, Afonso Ferreira, Christos Kaklamanis, Stéphane Pérennes, Pino Persiano, et al.. Approximate Constrained Bipartite Edge Coloring. Discrete Applied Mathematics, Elsevier, 2004, 143 (1-3), pp.54–61. �hal-00371101�. HAL Id: hal-00371101 https://hal.archives-ouvertes.fr/hal-00371101 Submitted on 26 Mar 2009. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(70) \. ^ /38:( '-#/3^P]-]HI-3+3 K -]/38-i+3-HI.#/&LR*0a$^'/O$*#-]*0 /&Y$*RLE/6$#*0 $*LT/3-;5+6$*":4."/3%#44THTL/E< HP7:+3 K "L-;4>PDL/38E/3+6$LT/6e¯fG-Nq+O'/i7+3-'-;T/ $+&-HI./3L51*0o/38-YL:L/3%$*"aLE/6$*M#-Y*0 /3Q/38:-Z025"L"*PDQ7+& K "-;4e. PD8%8¾E('/O 0$ K L7$+3/&L/3-+6$78 ¼ E(HI-)+$p'/O$*-*0 ½ b'¿]#À\#Á^d]PDL/&8Â½ Â\ËÂ

(71) mg-;+&/3%#-#BTPDL/&8¿ ½ !L"$#&% TB À½ ' !L"'(#*) TB $*HvPDL/&84!$SI4Q.4 HI-;5+3-;-VEBcLvPD8(68'541-1*0Z/38--HI-^LªÁ $+3-$*"L+&-$HT< "L-;t$*""L<#"+&-H[eaN8(7M$*+3/&($"i#"+&L1%+3-)7+3-E-;T/3-H K <$!'-;/r*0W#M'/3+O$*I/O,+ #T/6$LQ7$L+6N0c/&8-025+34¡.b -0 /d\PD8-;+31 - -32Á>$*M4 H /(N$Q"L5+;eª7M$*L+Y.b -55 /d HI-;5/3-N/&8$*/X/38:--HI6 - -,%D#"+&-H1PDL/38#5"L57 + /Ne B*P]-D#5'/&+3./c$,E-;+3-[*0a'/O$*#- 0 Ê Lmg-;Y/38:-NL:L/3%$*"g'/O$*#-]*0 X PD8(68 B[02+$I<GLT/3-)-;+98 TBc#5'%'/O^*0Z$ K L7$+3/&L/3-+6$78u¼;:1$*Hv$E-;/Q*0 #M'/3+O$*I/O< +=:eTs+]$*T<Î/&-;-;+>8 IB5/38- K L7M$*+3/&L/&-D+6$78¼;:,½9b@?^BA#Áb'¼;:d&d 8$D/38:-Ê-;/6N0img-;+&/3(-C ?p$D H AÈHI-#qM-H!$. ?ª½<EGF ; GF 2G¿ IH JF K ; '(FL2À T $*H. Ap½. J(F ; '(F2GÀ IHEGKF ; $FL2¿IM z.

(72) . . s+!+O$*78 ¼ BD/38:-G'-;/!*0-HI-Áb'¼ ;d(1HI-q-HU$102"L"PX#eDs+$I<u-HTb GFC" ' 3 d 2uÁQb'¼YdrPDL/38 GF72U¿$*H ' 92 ÀBÁb'¼ ;dr#T/6$L,/EPN!-HT-»cb EGFC"J d $*Hu@b E KF J K dOefG-!;$*""N/&8-'--HI-1±j2k #¯h`lg±,-HI5-#e\"(EB

(73) "L-;/V K -1/&8-14!$SI4.4 HI-;5+3-;-,0i¼¾$H1"L-;/ b GFdWb+3-'7-#/&Lmg-;"< b '&Fd&d K -Y/38-YHT-;+&-;-,*0\$^mg-)+3/3-)S$F 2µ¿ b+&-'7-#/&Lmg-;" < '(F 2ÀQdiLG¼êN8:-^-HI5-QE-;/,ÁQbE¼ d]$*"%'#T/6$LXV @b $FCd]7- *0]@b E$F E KF d02+ ½¾? !!6Â BT$*H1V b '&F(di#57L-c0]b J(F J F K d

(74) 02+ ½w? !LOÂ

(75) eTN8-'-HI5-$+3-;$"L"L-HªÃ)±3¬*®6®R-HI-#eNÊX+6$789; ¼ :025+ 8 (Q K /O$*L:-H02+34 ¼ B K < $HHTL 817-0c/&8-,-HI5-,b EGF2 E KF d

(76) 02+ ½¾?5 ! L6Â B:$*H 81#57L-*0\/38:-,-HI5- b J&FC J F K d6BI025+ o½w? !LOÂ

(77) eXF-)S$417"L-D02+X/38:-Ê/3+3.M#/30i+O$*78G¼ 02+354£¼ (DHI-)7(#/&-Hs

(78) L.:+3-?e. . . . . . . . . u1. v. x. y. u. v. x. y. u3. v3. x3. y3. u. v. x. y. u5. v5. x5. y5. 1. 2. 4. 4. .

(79) . . 2. . 1. . . . . 1. 2. 2. 4. w. 1. z1. 2. z. 4. w. y’. x’. y’. x’. y’3. x’3. y’. x’. 1. 1. 2. 2. w3. z3. w. z. 2. 4. 4. 4. 4. ÉtËé÷8é8ì*Á$»Ë Ð Ô Ï Á×Gæ½?Á$»*Ë Ð Ô Ï Á¦× æ8Ô8Ê$¿"½oÁ$»åqÄ Ð

(80) Ð »q¼ Ï ÄSÊ$¿ÉÊ$Ë Ò¾$Æt½

(81) ÉtåqÄSÒ"Ò ÄS¿Ét½%À"Ê$»q½:È#Ä Ð. éCìÁ$»¦»q¿Ë8»q¼Ä/ÂÞÚ{× ßÔ Ð » Ð » ÏÐ »q¼

(82) »ÊD½

(83) »q¿=Ô/¼ Ï ÆÔ/ÉÊJ»q¿ËD»q¼ÈBÁ$ÉtÆÇ»Ö½?Á$» ¿Ä8½!½

(84) »q¿^»q¿Ë8»q¼ Ô Ð » ½?Á$»å Ð Ä8¼

(85) ¼»q¿Ë8»q¼È#»çÔ±¿¿^½

(86) ÄË Ð Ô Ï ÁG× ÉÊÄ Ð ¿» Ð ½

(87) ÄË8»q½Ë Ð Ô Ï ÁG× é x’5. y’5. ". s+R$*T<8 IB[/&8-'-)/,*0#E/3+O$*LT/6 +=:Y(YHI-#q:-H$02""LPX#e¦sM5+,-$58-HTb GFC" ' di#5'/&+6$L-H!/3 K -R#"+&-H1PDL/38J#"+7/1/38-RE-;/D*0o#M'/3+O$*I/O +QBI/&8'-;/D0iE/3+6$LT/6 +=:+&- 5g.L+&-/&8-,-HI5-b EGFCJ di*0\+6$78¼;:/& K -^"L5+3-H1PD/38 #"7 + /]e. . . . &

(88) ¬g±Xlg¶:¬*®)²L° ²*jX²(I°Ej2kIj±=8

(89) ° Fj±j,j3Å*²@®° ®ZlQÃ6¬h`¬g±²(kQ¬X°DFjj&³;kTj3®r¬D¼ c²L°DF1VË 8 Ã6¬h`¬g±3®?F:²(ÃGF Qlg²(T°Clg²(¯®R°DFj,®j6°D¬Ã6¬g¯®°C±&lg²(I°(® +² lg³1¬gIh ² Y° Fj±j j3Å5²@®°(®l!Ã6¬h`¬g±²(k¬^°D F jj&³;kIj3®¬Y¼;:!c²L°DF!VË 8vÃ6¬h¸¬g±O®"?F:²(ÃGFDQlg²(I° lg²(¯®Q°DFj,®j6° ¬RÃ6¬g¯®° ±3lg²(I°(®3+=: C. $#&%. L #-1¼%$!E. K +O$*78v0Y¼ :B

(90) $*T<"L-;t$*"i-HI5-1"L5+3!*0Y¼;:/&+3mI%$*"L"< <I-;"(H:$!"L-;t$*"i-HI5-1"L5+30Y¼ K <G#5"L+&L/38--HI5-b@GFC' #dr*0Y¼PDL/&8/&8#"+Y.E-H K </38:-Q-HI-b EGF J dXLv¼;:*e "L-$*+3"<gB K </&8-1HT-#q/3L50D/&8-1E-;/Y*0. . * . . . ©.

(91) #M'/3+O$*I/O +=:*Bc`0X/38-#5"L+&L0,/&8-!-HI5-^*0R¼;:J4J$*LT/6$LY/38-J'-;/Q0Y#5I¤ '/&+6$*I/O +=:B/38:-Q#5"L+&L*0o/38-Y-HI5-D*0D¼>4!$*I/O$*/&8-QE-;/D*0]E/3+6$LT/6 + $DPN-;""Ce ,3E.41-X/38$/DPN-X8$;mg-Y$^"L-)g$*"a-HI5-Y#"+&LR*0]¼ªPD/38V¯ËD8F"L5+6iPD8(6814!$LI¤ /6$Li/38-r'-;/]*0 #E/3+O$*LT/67+QeTN8-;¦B¯/&8-X-HI-*0i¼;:X;$* K -r#5"L+&-HPD/38VtË 8 #"+O$5o025"L"LPXegs+D$I<R-HI5-^b@$F ' d=2Áb'¼Yd\#"+3-HPD/38$"L5+C/NBP]-,"L5+ /38-X-HI5-@b EGFC J #d\$Hb E KF J K d

(92) *0iÁb'; ¼ :*d

(93) PDL/&8 /NetN8(iLmg-$Y7$*+&/3%$*"-HI5-Y#"+E¤ LX*0¦¼ :iPD8(68Y4!$*I/O$*[/38-]E-;/0a#5'/&+6$*I/O +=:cLYPD8%8R/38-]#+&g3

(94) -HI-$+3.#5"L5+3-H[ e -;; / GF 2¿b2+3-E7-#/3mg-;"L< '(F 2ÀQdD$H"L-;/ @b GFdrb+3-'7-#/&Lmg-;"< @b '(Fd&d K -^/38:-QHI-;5+3-;-R*= 0 GF]b+&-'7-#/&Lmg-;" < '(F2dN¼Re[N8:-Q#+&g3D-HI5- K -)/'P]-;-; E$F

(95) $ H E KF K K b+&-'7-#/&Lmg-;"< -;/EPN-;-; J(FX$*MH J F K dR$+3-!P E/3+6$L-H < @b $F2db+&-'7-#/&Lmg-;"< b '&Fd3dr#5"L5+6#e¦N8I.B PN-;$.E-/&8-V[Ë 8 b GFCdYb2+3-E7-#/3mg-;"L<V[Ë 8 @b '(Fd&d #"+O5/.M'-H K <v-HI5-Q$H)¢6$#-;T//3 $FRb+&-'7-/3Lmg-)"L< '(F2d^/&"L5+/&8-#+3t3 -HI5- K -;/EP]-;-; EGF

(96) $*H E KF b+&-'7-/3Lmg-)"L < J(F\$*MH J F K d6eN8%417"-;/3-o/&8-Q"L5+3 *0D; ¼ :,PDL/&8VMË 8#5"L+O# e . . . . . . . . . . ÆPYB02+$I<uI/&-;-;+ 8 TBX#E(HI-)+1/38-4."/3(414QTHI/E<-;/EPN5+3¥1:½ b G ?RB A DÁQb 1:#d&dr#E/3+&.#/3-H$02""LPX#e -;/60Dmg-;+&/3%#-I?$*HA¨$*+3-Q/&83$41-,PDL/&8!/38t'-^0i+O$*78G; ¼ :e"%':B. . ½< ! L

(97). :3. $*H. u½< 6 !L : M N8-R'-;/,Áb1 :*di%DHI-#q:-H!$ ÁQb : d½Áb'¼ : d. H4Tb FCE$d; ? H4Tb CF J K d ;? H4Tb@J )

(98) F d ;? H4Tb@E K

(99) (F d; ?. uVË. uVË. uVË uVË. 8[;?uÂ . 8[;?uÂ

(100) . 8a;?UÂ

(101) 8[;?UÂ . "L"]/38-!-HT-QLuÁQb1:*dY8$;mg-!./;$*7$5#L/E<gB\$*H$*-HI-;$u;$+3+&<"<v$* LT/3-;5+6$"$*415.T/c*0 HP02+i-$584141THTL/E<geN8-;+&-X$*+&-,V*Ë48Q#441THIL/&L-eN8'5.+6#-Y02+,/&8- (ng/38#441THIL/E<(D"LT;$/3-H$*/ F BPD8:L"L-Y/&8-Q#5+3+&-'7MHILJ'L:¥ (,"LT)$*/3-H$ / F eX-;S$417"-02+Y/&8-1E/3+3.M#/3G*0D:-;/EPN+& ¥ 02+&4¡+O$*78 ¼ %HI-)7(#/&-Hs

(102) L.:+3-Q?5e. . I /&.L/&Lmg-;"<gB;$I/&-;+O$*"HP*0 /&8-rV*Ë 8Q#54141THI/3- #+&+3-E7Hi/3R$Qb7$+3/&($*"2d "L-;t$*"¦#"+3Q*0\/38:--HI-N0N¼;:»$*!-HI5- K -)/'P]-;-; ? $*H AÈ;$*+&+3<TLR-Y./. ~.

(103) cL 1:^#5+3+&-'7MH/3J$*-HI-#"+3-HFPDL/&8#5"L5+ iL. *0AHM*P9025+,#54141THI/E< ¼ :e ;. LM#-R'4-,*0i/&8-Y-HI5-0i/38:-5+6$78¼;:$+3-R7+3-"L5+3-HaBI.:+4."L/&(#544THI/'< HP 7:+3 K "L-;4¡8$5Q'541-1$5HHI/3L5$*"XE/3+6$LT/6e 0X$-HI-1%7+&-#5"L+&-HPD/38 #"+ ; ¼ :BL/D%E/3+6$L-HF/3J;$*+&+3<!$.:L/r$*415.T/D*A 0 HPª025+Y#54141THI/'< K cL 1:e:N8-E- HM*P #5'/&+6$*I/OD$*+&-+&-;7+&-'-;T/3-H <$Q'-)/D*0o#E/3+O$*LT/6 :*e :B /3Q4."/3(44THI/E< HM*PuPD/38#5'/&+6$*I/O PN-Y)$*1+&-HI.#-Z$*!LE/6$#-,0 K .'/38-,02""LPDL:Q E-;+3m5$*/&L5 e. . &

(104) ¬g±QlgQ¶:¬*®)²L°C²*j²(I°Ej2kIj± 8 D°DFj±j1j&Å5²@®° ®lÃ6¬h`¬g±²(k¬°DFj1j&³;kIj3®¬ ¼;: c²L°D! F V[Ë 8Ã6¬h`¬g±3®2F:²(ÃGF Qlg²(I° lg²(¯®Q°DFj^®j6°X¬Ã6¬g¯®°C±&lg²(I°(® +=:!² lg:³¬gIh !² ° F j±jYj3Å5²@®°(®Ylg!²(I°Ej2kg±&lh¬ v¬ ;lh #MjWÂbCVË 8d )¬g±XÃ6¬6Q¬³g²L°C²j&®X? !LOVË 8G²(1° Fj j6° c¬g±1: ?: F ²(ÃGF Qlg²(I° lg²(¯®R° Fj,®j6°W¬RÃ6¬g¯®° ±3lg²(I°(® : C. /&8-:-;SI/Z'-#/&L5P]-1E8P8P /3$*77:+3SI4!$*/&-Q/38-+3+&-'75HIL:LT/3-;5+6$*" #M'/3+O$*-H4.:"L/3%#44QTHIL/E< HM*P¨7+3 K "L-;4B$H[Bi.'/38:-+3-HI.#/&Lµ$ K mg-gB PN-Y K /6$*$^7+&m$ K "L<gTHG'"./3102+X/38:-L/&($"[LM'/6$#-Y*0 e. L

(105) *

(106)

(107) T 8G (2 g L c .

(108) *8 &. 5-;-;+O$*" B#LT/3-)+6$"4."L/&(#544THI/'<AHPbPD/385./

(109) #M'/3+O$*I/Od( ÆrÇ\n¯#47"L-;/&=y©ACe$@P]-;mg-;+;BL/\(i'/&+6$*8T/'025+3PD$+6H/&,02+34."%$*/3-]/38-r#5'/&+6$*-HQ4.:"L/3%#44TH¯¤ L/E< HPu7+& K "-;4 $5D$ ;n?,LT/3-)-;+i"L-$+N7+&+O$*4$H1E"Lmg-rL/6c"L-$+N7+&5+6$*4¤ 41L:+3-;"%$*S:$*/3 K <v'-;/&/3$5'%HI-/&8-I/&-;+O$*"L/E<#5'/&+6$LT/;e /&8(^PW$<tBcPN K /O$*$*7/&L4!$"02+6$/3L5$*"

(110) E"./3 e. E(HI-)+$*g$L/38-!+O$*78U¼ $*Hv"L-;/ K -!/38-!T.4 K -;+Y0R#"+O^.'-H/&8 '-;/*0#M'/3+O$*I/O +Qe "L-$+3"<gB4J$*S5VE6G($µ"PN-;+ K .Huu/&8-41L4Q.4 I.:4 K -;+,0X#5"L5+6Ê

(111) . J#L-;T/D02+^-)SI/&-;HI!/38-7$*+&/3%$*"i-HI5-#nT#5"L5+3L:F0Y¼Re fGK -;^PDL/&8^-)/'P]+& ¥ & $*HY/38-W'-;/

(112) * 0#E/3+O$*LT/6 : BE"mI/&8#+&+3-E7HIX"LL:-$*+ 7+&5+6$*4 aÇ e 0:/38-i4!$SI4Q./ 4 HP% '4!$"L"L-)+/&8$* : Â4!$ S 5V'O tBI/38:(c41-$c/38$/N/38:-,LT/3-;5-;+i"L-$*+i7+&5+6$*48$5N HPuPD/381m5$*".Â4!$ S 5V'O tB¯4-$*b K < -;414!$XÉ¯d\/38$/N/&8-;+3-X-;ST(E/6cQ"L5+3^0i¼ & PDL/&84!$* S 5VE6 #"+OPD8%84J$*LT/6$L,/&8-!'-)/Q*0r#5'/&+6$*I/O + efG & #T/3I.:-1PDL/&8:-;/EPN+&¥ & &

(113) !\$*Hv/38-#5+3+3-'7HTL '-;/O0i#5'/&+6$LT/6#BT.T/3L"aPN-DqH!'541 - v'.M81/&8$*/D/&8-YE"L.:/3L5!* 0 aÇ 5Lmg- &. $F02+6$5#/3$",bCE/3+6$L-Hd,4."L/&(#544THI/'. < HP*0Xm5$*".-! Â Ne "L-$+3"<g B ª%^$ K K "LPN-)+ .:H02+c/&8-D41L4Q.4uT.4 -;+ *0¦#5"L+Oc' . !#-;T/[02+i#5"L5+3L:/&8-D-HI5- *0D¼ e. b. b. b. b. b. b. b. &. ÆPYB*P]-iPDL"L"g.M'-N/&8-02+O$#/&L5$*":'5"L./&LQ0/38:-N"L-$+ 7+35+6$4 aÇ : ^+OHI-;+c/& K /O$*$1E"./302+Y/&8-#+&+3-E7HI!LT/3-;5-;+X"LL:-$*+X7+35+O$*4 aÇ PD8:(68. |. &.

(114) (^7+&m$ K "L<µ#"Lt'-1/&/38-!7:/3L4J$*"N5-ge

(115) f-!PDL""N.E-!/38-!+6$HI541L;-H /3-686 5¯.:-,7+357g'-H K < $8$;m5$*µ=@?)©*ACe. . . +&.HTL. . K -Y/38- HM*PU K /6$*-H K <1E"LmTL:aÇ -;/ e:s"LP ;$* K -RHI-#417t'-HI/& & HM*PX

(116) ! :§[-Q02+r-$68G#441THIL/E<ge $68 (Fi;$ K -^02.+&/38-)+ K +35¥g-; .7I/&Q.FcE-;/6 F B F

(117) BeeLeB F 0]Â mg-)+3/3-)S¯n¯HI(¢3LT/N7M$*/38MW02+&4 F/& Fib2Cey-teLB K /38-D-HI- -)/'P]-;-;9?u$*H AULQ-$68Q'-)/c*0¦mg-;+3/&-;S¯n¯HI(¢3I/ 7$/38025+34!

(118) $,7-)+'02-#/ 4!$*/O8:LTdD-$68G;$+3+3<TL:$*$*4.T/ 4F Y*0 HP 02+Q4141THTL/E< 3B E.68/&8$*/ F G½ (F½ ?eNfG-G;$*""/&8-7+&T#-HI.+&-*0HI-417gEL HP Ql° GÃ F:²(k ®°C±²y¶g¶a²(k^b '#-,/%NEL41"($*+\L!E7L+&L/i/3Q/&8-,7$/38!E/3+377^/&-8:D 5g.-X7+357t'-H L=@?)©*A2d6e. . .

(119) . . &

(120) . .

(121) . l°CÃ : F ²(k®°C±²y¶g¶a²(k!Ã6lg. . !. %#jZ³g¬gjR²(¶:¬h ¬Q²(lh ° ² j C. $#&%. 7$/668L:X'/3+&L77:LX;$ K -]7-;+'025+341-Hr$[02"L"PX#e M'(HT-;+

(122) $E"./3^/& aÇC:D$*H/&8-X$&'T#%$*/&-H M H *PX025+N4141THTL/E< LQ:-;/EPN+&¥ 1 :e -;/ ½¾?e -;/7- K -/&8-D-HI5-,;$+3+&<I,/38:-X'4!$"L"L-'/

(123) Int;-;+&Q$*4.T/ F \*0 M H *Pv02+]#54141THI/'< . * . 3ecs

(124) LHv$G'-;/ F *0ZÂmg-;+&/3-;SgnTHI%L¢35LT/^7$/38^02+354 FD/3 FX#T/6$L-HI5- /38$/X;$*+&+3<1:Int;-;+3$41.:I/ON*08HP 025+X#414THIL/E< &BILM#"L.HTL- e,3ET#($/3$*415.T/ F ] PDL/&8 F $*H!'. K /3+O$#/X$41.T/ F i 02+&4 /&8- HPu02+X#54141THI/'< K )$*+3+&L-H <Q-$68-HI-XLF e-)7-$*/i/38%7+&T#-&o02+ ½ ÉI6xT! LB.T/3" HP . . +3-;4J$*LM#eN8%PDL""

(125) HI-#5417g'-R/38- HM*P &FcLT/3'-;/O*0WÂmg-)+3/3-)S¯n¯HI(¢3LT/D7$/38 F K -)/'P]-;-; F

(126) $H F -$58;$+3+&<I$*415.T/ F D0AHPª025+X#414THIL/E< &e. . . . . f-cq+O'/

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