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(1)Integral Symmetric 2-Commodity Flows Aubin Jarry. To cite this version: Aubin Jarry. Integral Symmetric 2-Commodity Flows. RR-4622, INRIA. 2002. �inria-00071963�. HAL Id: inria-00071963 https://hal.inria.fr/inria-00071963 Submitted on 23 May 2006. 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..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Integral Symmetric 2-Commodity Flows Aubin Jarry. N° 4622 Novembre 2002. ISSN 0249-6399. ISRN INRIA/RR--4622--FR+ENG. THÈME 1. apport de recherche.

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(78) >׈  ‚ ‘ ¡¾hœOˆ_‚ ⠈. i.

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(80) 

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(83) >׈ ‰LXD¾hœQ‚>d†@?@ˆÉ€ O9 ‚ ‘ ¡¾hœOˆ_‚ ⠈ < =@C\BP Ü > T \BW'ŒŽTCBW'ŒX[`FŒLŠT'W'P@ChfLXoN[\M\MX[W^ŒŽ` s‰TX‰<>=@CMX[WCBA ¬°WX[A š › T XDž › XQ¬Él[NQ§ŽP@CDEh™.  . .  :    L .  .  . .  . . .

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(112) T'=@CHLCM\MX[`@^D\MX[A‹ADX8^ŒŽT Um™ÎkXOª¹CBlmCBWOfT'= ŒLŠŒŽ`fTCBW'ADCM^ŒÝNQTC © XªŒLH`@X[Tk]rNQW'TkXQ¬T'=@C½@`rNQ§LX[§ŽP T'ŒX[`™k<>=@CB`ŒŽ`¦L'P n@L_CM\BT'ŒX[` +™ +FªÜC&\MX[A‹] P TC‰T'=@C © XOª L ¬±X[WÂT'=@C‰L_CM\MX[`@^i\MX[A‹ADX(^ŒŽT˜U´Ë°¥§ŽsmX[W'ŒŽT'= B A +i˰LCM\MX[`@^iLTCB]Ô'ÔՙÜÒlmCB`(T'PrNQ§Ž§ŽU¡Ë°L'P n@LCM\BT'ŒX[` +™ c(Ô ª¹C\MX[A‹] P TCT'=@C © XOª L¬°X[WŠT'=@C½@WL'TŠ\MX[A‹ADX8^ŒŽT U¦Ë°¥§ŽsmX[W'ŒŽT'= AcvË®T'= ŒŽW^FL'TCB]Ô'Ôՙж(P n@L_CM\BT'ŒX[` +™ 2&ŒLHN—WX[P `@^ÇP ]iXQ¬ÉT'=@C^Œ CBWCB`(T]rNQW'TL XQ¬X[P WHNQ§ŽsmX[W'ŒŽT'= Ai™. . L     . . . .  !    ".  .  . . < =@C] P W']XhLCoXQ¬T'= ŒLkL'P n@LCM\BT'ŒX[`¦ŒLTXF] WXOlmCoT'=rNQTHshŒŽlmCB`¤N © Xªƒ¾ÖXQ¬CBlmCB`¦lhNQ§ŽP@Chrª¹C > \ONQ`DL] §ŽŒŽTоFŒŽ`FT ªÜX © XOª L—NQ`@^ Ë®T'= ŒLADCONQ`@LT'=rNQT è  @€ ¾ÔLXkT'=rNQT8ESÆ-‚¿¾ è [ˆdNQ`@^ E Æ-‚¿¾ è [ˆ_™µ˜`SX[T'=@CBW>ª¹X[W^ LO8ªÜC\MX[A‹] P TCN © Xª FL'P@\=ßT'=rNQT‚¿¾ [ˆÜÆ ESƔ‚¿¾ è [ˆ Ë®n(U m@ªÜC%ADCONQ`iT'=@C%¬®P `@\BT'ŒX[`ØLP@\=iT'=rNQT C F i‡& m‚@ˆÉ€ {Ôՙ. ;. . . %;. . ( k -€ƒ) ‚ +-„% †& ‡‰ô ˆ (W&× ¾¦’f‡ w ד–|  •x &  zx ‚¿¾/i. t' ró y x. . . H¾. !#"!$%. . . . . [ˆ¹Æ5E SƔ‚¿¾ è [ ˆ . . &. (. '. . Ji. ; ’m‡”“–• > „ & Cv i‡ d¾ ®; ‚ ˆ ¼¾É‚@ˆ ( = 'LU @ )‚> k n ) n †@? k n ) n ˆŠ Ø‡ )   D¾ ; ‚> k n ) n †@? k n ) n ˆ     ¾ ®; ‚> k n )  n †@? k n )  n ˆ –¾ 3; ‚> k n )  n †@? k n )  n ˆ è  > ? k n ) n  ( =   ' > €  > k n )  n "    

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(114) >׈H )#"" ¾ 3; ‚?†

(115) >ˆ ¼¾ ®; ‚>d†@?@ˆqi > ? ã I e `) h Cv i‡

(116) ‚ ˆ€ œ.

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(120) ".  & . $. /. tFu !t ö Q.

(121) E#+. 

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(123) CBW XQ¬¹lmCBW'T'Œ\MCML ?*  ܂>ˆ L'P@\_=¤T'=rNQT‰¾ ‚>d†@?@ˆX[W¾ ‚?r†

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(125) CBWÂXQ¬×CM^smCML S Ç‡”L'P@\_= T'=rNQTk¾ ®‚ @ˆŠŒL X8^ ^i^ CM\BWCON[LCMLOrLX‹T'=@C%`(P A‰nCBWHXQ¬L'TCB]@L ŒLÂnX[P `@^ CM^in(UßT'=@C\ONQW^ŒŽ`rNQ§ŽŒŽT UiXQ¬ S Ç‡o†_¾É‚  ˆÜŒL X(^ ^ m™Š<>=fP@L T'=@C‰\MX[A‹] §CÕR8ŒŽT˜USŒL ŒŽ` é ‚_ë ‡‹ëæˆ_™. ;. ]. -. ;. ) . ( (  &.  (. =. k n ) n. _. y óó w x ®ó . @!. š. 

(126)  . ;. ܾ.  ( 

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(128) CBshŒŽ` ` ŒŽ` sÇXQ¬ŠCON[\=*§X8X[]êË3¬°WX[A > k n )  n TX > k n )  n ÔՏ

(129) T'=@C © Xª C (PrNQT'ŒX[`@LNQWCon WX[ÈmCB` ŒŽ`> k n )  n ™É¥>TŠCON[\=SL'TCB]DŒŽ`@L'Œ^ CÂT'=@C §X(X[]fT'=@C © XOª C fPrNQT'ŒX[`@L¹NQWC ÈmCB] T¬±X[WŠCON[\_=>ßCB`@\MX[P `¨ TCBWCM^Á™Š¥>T¹T'=@C%CB`@^ÇXQ¬ÉCON[\=i§X8X[]8T'=@C © XOª-C (PrNQT'ŒX[`@L NQWCWCML'TX[WCM^S¬±X[W > k n )  n ™<>=@CBWCÕ¬±X[WCh NQT&T'=@CSCB`@^¡XQ¬T'=@CßNQ§ŽsmX[W'ŒŽT'= A ¾ ; ŒL—N © Xª¬°WX[A švTX؞‰XQ¬ lhNQ§ŽP@C4EQ£r™*ZX[WCMXlmCBWOCBlmCBW'U CM^smC‰ŒL ] WX8\MCMLLCM^¦NQTADXhL'TkX[`@\MChrLXS¾ÖNQ`@^¾ Á ; ^Œ ÁCBWnfUØNQTHADXhL'T‰EoX[`CON[\=*CM^smCh™Â<>=fP@L ‚¿¾/i[ ˆÜÆ5E SÆ-‚¿¾ è [ ˆ_™ L p  L   b     \ "  qL    ! . . . . . . . .   . .  . . i. µ `vT'= ŒLÜL'P n@L_CM\BT'ŒX[`fª¹CH^ CML\BW'ŒŽnCHNQ`ÇNQ§ŽsmX[W'ŒŽT'= A T'=rNQTн@`@^ L>N © XªÃ¾ØL'P@\=vT'=rNQTÜT'=@CH\BP T \BW'ŒŽTCBW'ŒX[`،LÂT'W'P@C¬±X[W‚'‚ ‘ ¡¾dˆ†O‚ šœ{†'ž'œQ†£[œ{ˆ†O‚°ž'œQ†9š{œ†£hœ{ˆ'ˆ_™. ( k -€ƒ ‚ +-„% †y{‡‰wMxDˆ wOdx ‘iïL.’f‡. . t' ró y x. .

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(132)  až'œ  ‹š{œ!= I  k£[œ)#@%"=  I 

(133) H& SƔ‚ ‘ è   iˆ 

(134)   š ›  ož ›    k£ › %;ÁƔ‚ ‘ è Ji  ˆ?  »š[› ž9›= !I H£(›.  (ÇÆ³‘  ) . “–•. š{œ.  ožŸœ!. ". ". =. ". . . ".

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(138) ˆŒL×\ONQ§Ž§CM^oT'=@CWCMLŒ^PrNQ§ \ONQ]rN[\BŒŽT U5X[`@\MC ´=rN[Lon

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(140) ˆ_™. ] 7_ ] 7_. i. ŒŽ`´T'=@CvL_NQADCDAFNQ` `@CBWOÂ‚ ‘ è  © OX ª    LX Á\ONQ`inCk¬°X[P `@Á ^ ™. %;. i. ] 3_ ] 7_.  {ˆkŒLT'=@CFWCML'Œ^PrNQ§¹\ONQ]rN[\BŒŽT˜U´X[`@\MCvWCBADXOlmCM^¡T'=@C. ¥Â\M\MX[W^ŒŽ` sFTX—§CBA‹AFNØEh@¾¦ŒL¹¬°X[P `@^Çn(UÇ¥§ŽsmX[W'ŒŽT'= A#Eh™. -. <>= ŒLÇNQ§ŽsmX[W'ŒŽT'= A \MX[A‹] P TCML + © XOª LH\ONQ§Ž§LߥH§ŽsmX[W'ŒŽT'= A EX[`@C¦T'ŒŽADC¤NQ`@^Ã\MX[A‹] P TCMLßT ªÜX \ONQ]rN[\BŒŽT˜US¬®P `@\BT'ŒX[`@LOrLX—T'= ŒLHNQ§ŽsmX[W'ŒŽT'= A­T_NQÈmCMLHJ â>ãMäæå¸çÇè«é ‚_ë ‡‹ë戹L'TCB]@L™. ( ( ß&¾. y óó w x ®ó.    &.      U  )  I. =. D(. L.   U )  %        )      ‚ ‘†O‚ šh›O†'ž_›O†£(›Mˆ†O‚ š{œQ†'žŸœQ†£hœˆ'ˆ     $   1 =   )*)  D U U  D   Š‚'‚ ‘ i¾dˆ†O‚ š{œQ†'žŸœQ†£hœ{ˆ†O‚°ž'œQ†9šœQ†£hœˆ'ˆ (. ". tFu !t ö Q.

(141) E#2. 

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(143) i i %; 3 i i j ŒÊ¬ë è %;˜ë ‚ d† @ˆ€ %;±‚ ׆ rˆ i

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(146) \MX[`@L'Œ^ CBW—‚>׆@?rˆ  ¦‡ LP@\=¦T'=rNQT 

(147)  ‚ ?r

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(152) >. . > @?. . ?

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(154) >ˆ\ONQ`—`@X[TÉn

(155) CÜ`@X[`—`fP §Ž§rNQTT'=@C>L_NQADC¹T'ŒŽADCh[ªÜC¹=rNMlmC%‚ ‘ Së  è  ˜ëæˆ ‚    ë  è  ëæˆ è ˆ_™ÜU"N´L'U8A‹ADCBT'W'Œ\¦NQW'shP ADCB`(TO ªÜC=rNMlmC¤N[LvªÜCB§Ž§—‚ ‘ ‚     è   ˆ_™ ϳŒŽT'=ßT'=@CMLCknX[P `@^ LOªÜC%\ONQ`ß`@XOª”\MX[`@LŒ^ CBWNo\BP T â 6³„ƒªÂŒŽT'=8¬±X[W>ŒŽ`@L'T_NQ`@\MChrš œ   â NQ`@^ižŸœ   ⠙ÏÖCªÂŒŽ§Ž§] WXlmCT'=rNQT‰‚ ‘ ¡¾dˆ_‚ ⠈ «£hœ[™ÏÖC%È8`@XOª”T'=rNQT‰‚ ‘ ³ë  è  ëæˆ_‚ ⠈ ‚     è   ˆ_‚ ⠈_@LXǂ ‘ "ë  è  ¿ëæˆ_‚ ⠈ «£ œ è #‚    ®ˆ_‚ ⠈_™. ; %;. i. ; %;. ŒÊ¬‰ž ›   â [X Wؚ ›   ‚ ‘ ¡¾dˆ_‚ ⠈ «£hœˆ_™. [. i. i.

(156) . %;°‚ â. ⠏ÂT'=@CB`. :[. ;. ˆ¤Æ. . ;. i. %; i %;. ^i. %;  ‚ â. %; ëæˆ_‚ ⠈ à £œ. ˆ‹NQ`@^ƒ‚ ‘ aë  è. . ;. . i. ;. ;. LX. @. X[T'=@CBW'ªÂŒLChš[›  â NQ`@^ž_›   â LX  ‚ ⠈܀  [‚ ⠈ è (£ ›NQ`@^5‚ ‘ ë  è  ëæˆ_‚ ⠈ £hœ ´£f›O™µ˜`ßT'= ŒL \ON[LC%ª¹C=rNMlmC&NQ§LXØë  è  ë ‚ ⠈€ ë  è  ë  ‚ ⠈ è EQ£(›HNQ`@^i¾É‚ ⠈€ ¾ {‚ ⠈ è £f›LX—¾É‚ ⠈ŠÆ ë  è  ë h‚ ⠈ è £(›>ªÂ= Œ\_=njŽA‹] §ŽŒCML ¾É‚ ⠈ŠÆë  è  ë ‚ ⠈ Ø£(›™ <>=fP@Lo‚ ‘ ´¾dˆ_‚ ⠈ «£hœ. i. i. ;. %;. i. %;. ;. i. <>=@CBWCÕ¬°X[WChÂT'=@C\BP Tv\BW'ŒŽTCBW'ŒX[`̍LDT'W'P@Cج±X[W؂'‚ ‘ ¾dˆ†O‚ šœ{†'ž'œQ†£hœ{ˆ'ˆ_™¹UÚN´L'U8A‹ADCBT'W'Œ\ NQW'shP ADCB`(TO@ŒŽT ŒLHNQ§LX—T'W'P@C¬±X[W‚'‚ ‘ ´¾dˆ†O‚°ž œ †9š œ †£ œ ˆ'ˆ_™. . . . i.  L  /  U  -   ". .

(157) . . . . Qi. L. ¥ §ŽT'=@X[P sh=ÇT'=@C\BP T \BW'ŒŽTCBW'ŒX[`njL>T'W'P@Ck¬±X[Wk‚'‚ ‘ ¤¾dˆ†O‚ š{œ{†'žŸœQ†£hœ{ˆ†O‚°ž'œQ†9šœQ†£hœˆ'ˆ_ŒŽTÂAFNMUv`@X[T  nC—L'PÙF\BŒCB`fT‰TXßshPrNQW_NQ`fTCMCDT'=@C‹CÕRŒL'TCB`@\MCDXQ¬ÂN © XOª *¬®WX[A#š{œoTXߞŸœ&XQ¬ÜlhNQ§ŽP@C‹£hœ—NQ`@^´N © Xª  d¬®WX[A žŸœ‰TXišœ‰XQ¬ÜlhNQ§ŽP@C—£[œoL'P@\_=¦T'=rNQT‹‚¿¾ è  è  ®ˆHƃ‘ÁN[LL'=@XOªÂ`*ŒŽ`¤ºŒŽshP WCDЙ <>=rNQTŒLªÂ=fUi¾ŒLÂ`@X[T ]rNQW'T XQ¬ÉT'=@C½@`rNQ§×LX[§ŽP T'ŒX[`™ ÎHXªÜCBlmCBWO×¾´ŒLkP@LCÕ¬®P §TXß\MX[A‹] P TC—T'=@Co½@`rNQ§ © Xª L¾hœoNQ`@^¤¾(Àœ%¬°X[WkT'=@C—LCM\MX[`@^*\MX[A—¨ ADX8^ŒŽT Um™. ;. QRQ. úDSUTVWW. ;.

(158) EOe.  . 1 1. 2. S1. 1. S2. 1. 1. 1. 1. 1. 1. T1. 1. 1 1. 2. T2. i. . . ºŒŽshP WC‰ÐU9ÉT'=@C\BP T\BW'ŒŽTCBW'ŒX[`،LÂT'W'P@C¬±X[W‚'‚ ‘ Ö¾dˆ†O‚ š œ †'ž œ † [ˆ†O‚°ž œ †9š œ † [ˆ'ˆ. . ( k -€ƒ ‚ + „w ï †‡‰| óˆ &Éô³‘Ç’mwO‡”x ïL.“–•. t' ró y x. . š{œ†'ž'œ „ ×£hœ Ç• d¾’f‡Ã“¼•& $ %U @  )# D(   ž_›% &šh›= !Ik£(› & @  /U    (ÇÆ”‚ ‘ i´¾dˆ 

(159)   š{œ! ‰žŸœ= I£hœ @  /U    ( ;ÁƔ‚ ‘ iÖ¾  ˆ 

(160)  ۚœ  ‰ž'œ= D !Ik£hœ =)       /@  /U  (Ú¾hœ: D šœ! ‰ž'œ!= !Ik£hœ)U   ‚_ë  è  ; ëmi [ˆÜÆ5E[¾hœÆ-‚_ë  è 8; ë è [ˆ& Rd¾À  œ € ‚_ë  è  ; ëi´¾hœˆ & %@ ¾(Àœ U @  )   ( 

(161)   ž'œ! —š{œ!= Ik£[œH& k¾hœH’f‡Ã“¼• d¾(ÀœH’f‡ “–•& Šï Öz  + w ï | ó& ô wMx ïL. R‘i’m‡”“–• @$  )   @ U = %&  š › †'ž › †9š œ †'ž œ   „ D"D£(›O†£hœF 5•&  ܾ³Æ ‘ @$  D(2   šh›  Fž_›O=  o£(›!)U    U  U    5 )4     D ߂'‚ ‘ iþdˆ†O‚ š{œ†'ž'œQ†£hœ{ˆ†O‚°žŸœQ†9š{œQ†£hœˆ'ˆ& U  U  @

(162)  (&  (*)D¾hœ 

(163)   š{œ! ‰žŸœ =  k£hœ "&¾(Àœ:

(164)  žŸœ  ‹š{œ!= D !I k£hœ)   D¹‚¿¾hœ è ¾(ÀœOˆÜƳ‘ D" )   DHU  OU        ) I  D F‚'‚ ‘ i³¾hœ<i"¾(Àœˆ†O‚ šh›O†'ž_›M†£(›Mˆ'ˆ& U )# (&  (*)@ 2@$    / @"  $   1      E â>ãMäæå¸çßè"é ‚_ë ‡‹ëæˆH) ) &

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