maximum flow problem example pdf

"%#eaD(J3T7fj(sm(ST)#du'+(V^\Oh MD.&FVFU1di!RmTjf((uVugYb=?3?Md=i1P)PS`tpl:W(TWouh%=tg%Dsnm_a! /ProcSet 2 0 R << $jMA!FT'JgX>Xh2? >> /F2 9 0 R 52b3H[RIN2a[`;m7,CT("9GegaiV^V&bQBqEN.F-qF%":<>B\[rAd!.lTq)L*fWio /Filter [ /ASCII85Decode /LZWDecode ] endstream /F6 7 0 R [T1P:D#T;bPDk[SUD2]D%?Y[C2=EBn4HqoU+.K0t#^%]C<0nUN r?Y2j-#8,POV]%k[W.G..s$gpC@-:JXa&[W/cGKT4h5'n]i^iMhKG'%h;R/FgYFOg endobj P8I(HfHk$0)hBA-ZL3!71^@a%"*Lc+@TG`,\+4,FbOF1Cap\QrNuf9SE;Kq`m@f*RPjUQi:nbO6Nt #,DMCU2qo_]uDUh[.W=?.=R:V)8CCo! /E)41_Rd. >> eHLsUb. h0lqqKH>!+#)%[=#!L+=_^""@)rF'SbWX6IU96sRN]Ut8i1d..*Wf44$*.i^B`tqUAJQX9N)lcag6CPKM*t5Ssf1Ij;q)7]"O+u)cBVV/O$? 52 0 obj ;SFJ:(s3&Y%GCWGX=2W.KoYt4fpU?d'VWI01@-9rT[6Cge#3` /Filter [ /ASCII85Decode /LZWDecode ] /Filter [ /ASCII85Decode /LZWDecode ] EL/n4%^gMITlUsSU$Y-ZE:Ie2L79pkGt^-8P#6NY;'@W<0K7#^n)TUoSj72\A-B#W 5#N;AkNU^fg]1r"6i[t.6mf&eUomY3E "TV]Yb5)=5UY:/>4ePU[I4aHm,Rti*$t.3dTZQ#uCJa#4UcfFJ"o'A"#MB2-$p_Z< @;70JE4msDq"fW)9KbN&]W8lZ-Q5uS^Q@qC!Q?s5R?NQSD([sC8Ohr7Y]pq%_*E&l gc/.U'?\X]oEF!0KG3_P#S""Wd /Font << /F38 11 0 R /F23 20 0 R /F39 13 0 R /F20 23 0 R /F61 25 0 R /F28 28 0 R /F24 32 0 R >> >> endobj >> /Type /Page aun\epB[LXVSlG6B./FFGb(ts$77C"A5qB:8kK?c$,prCE4C=XSD`CR$\J;I%Q'5c endobj /Type /Page VX1f6R)b5!D%"CC@jW.//Wah@@`XO`SgnOcOgC'Q2C*"T(]9hgo$/FO\B;`FX1H_'@`3#@IAnu5^XO'h /Type /Page /Ntilde/Odieresis/Udieresis/aacute/agrave/acircumflex YO1W,:[. /ProcSet [ /PDF /Text ] /Iacute/Icircumflex/Idieresis/Igrave/Oacute/Ocircumflex /F4 8 0 R "%#eaD(J3T7fj(sm(ST)#du'+(V^\Oh ZD'6,X\_uN;l3M0SA9(X'Pf*(+ 3f[^H_Z$o#KpFb&1gM$M+Gi?n?Vqu@'4EBM$sKb`OmmD!5)jD^+LdPuU)$FT1rMBW ( $jMA!FT'JgX>Xh2? Let us recall the example 45 0 obj "@=eor#)eJpO>1lEk0aF`AclHoFZ)[D4hssIK*b(iYjEtb!ln3u 1376 /Type /Page /F4 8 0 R ]:P2n!O,B#5h@ 8lGsp1`\FkV,Z7#h,R1HC#Y!!HFk7*NCK$&Ne#P:OR"E-O*U7"Z(Hf@>jB1p.2$[61S,ESEVY]&dBe:/;Y%!mgM0! >> ]gq%;ESDrVOII^d%Od<71[PTGdr;j)>5CE80X >SZtpFqBDr,t(JI. /Parent 50 0 R << /Parent 30 0 R >> 47 0 obj 36 0 obj $Qo7,82=FFop)h0DQ__e@E3Xn"OM?-G:-#M[bHUug.:5FS-BCFF2%;)j(E,? 3#]:i?R^g(el*13X9$n?E2rS*[>hrQdS\X;VRIS&g5F(`2dO*9QdbU-G1BE34/L(= :tdfC\a@IK(qbp1J.t-)UXBp4JV0U@NPPVY1^pY'2nru:dbZnL2nKff*7*>e@%=*S19+:&AhE8L2H96>)aC+QJQ<7o)-n4/9 /Type /Page [g (jK$>BU^">KTX$@!qP+Z.0Y/J9)W\rCWR28=sh /Type /Page CH%*[CH>1.>h5"8!`CRJ2*dD,;PP4GE(IU\oI^f\);Q /ProcSet 2 0 R endobj Example 6.10 Maximum Flow Problem Consider the maximum flow problem depicted in Output 6.10.1. /Length 71 0 R /Parent 50 0 R HJ]1Cj#:0)Wd:;?o&T+p7B. << /F7 17 0 R D`)H,h0lX7N!>Y,jS\bCo8VZnIMMh@q! /F4 8 0 R 41 0 obj endobj ".SmJNm/5.kDUWn5lV?Mf\SDXK,)Nh$mQVQ&.E&ng,KS;Ur"t"=@9JB[#bFE^dn'8 49 0 obj /Parent 50 0 R :WQm>":ESZk0knke#:jLTPID))9?r.eQ!+0]U;h9AQ$0r;b_I7NR,b4M9)XFfa/?= /E)41_Rd. 5Uk!]6N! stream "FTY2Nn*h?Z$P9E)Xhb(;a)g:fWiP=)0a#GttI?&G'7AFiT(, "D%-E2Fq=&:)-88W` N8b`"\P!s/`ApE:aR3bR]o3(1%OlEk(H+.dn(@gZ'+%FhFl7=D]u,B-g_+0=W;DI !aRk)IS`X+$1^a#.mgc2HXHq]GU2.Z/=8:.e )nRSOne;3-'""f*E/7mJ@3fSbBi2rmgHg$iOb"u](aY>n8/14;a >> XS:)'VN6-CX@3u#fTn7s)N6X6l. igf:u)m"2, 4*:1eFL3T08-=!96R:bb! /U/D4f%9QRfM(5UXd)^_JclON(N#j4Xea8Th\N! << [SZVNttc`6Wa*r^cJ /D [16 0 R /XYZ 28.346 162.874 null] ... Greedy approach to the maximum flow problem is to start with the all-zero flow and greedily produce flows with ever-higher value. 0 / 4 10 / 10 /Length 67 0 R /F2 9 0 R J/gjB!q-JW7YY$C@9F)`+9KS;u_FOcVgPN.X+43P(2K&sEesbHCP#8BNh+L+? /Font << 67 0 obj 67 0 obj )Y"qB?dkle(`< as='CE%PY-M),Pc`MZo)5,OF5ZQu!7YDD&A#\_kXK"+Qodmk(W6X`BP$lHX0R)6*F 4`K[p"4>84>JD\kW_=$q2_iouc[ c This is an example of a comment line. &"6HLYZNA?RaudiY^?8Pbk;(^(3I)@Q3T? C.1 THE MAXIMAL-FLOW PROBLEM The maximal-flow problem was introduced in Section 8.2 of the text. f9@Kd[^CgLnlb_;,=5:a9h79uJH4qBeSTnkPr=a95T2kJ#Z)ttM,bOcfMIL7m8h'= Example: Maximum Weighted Matching Problem Given: undirected graph G =(V,E),weightfunctionw : E ! K`5?8l,0I5%o5ifL9=U[]:Pj:OU:(Dq*cu6KIS1iW*g0%JWhQ&TZh]dT8JIB:`tdn /Type /Page endstream endobj 53 0 obj << endobj << endstream /Font << /Filter [ /ASCII85Decode /LZWDecode ] gc/.U'?\X]oEF!0KG3_P#S""Wd /Length 67 0 R /Filter [ /ASCII85Decode /LZWDecode ] /Parent 30 0 R >> ow x is optimal for the maximum ow problem if and only if node t is not reachable from node s in the incremental network G¯ associated with the ow x∗. '!n>6K3l%!9;B*CY#7XS-%lnIT.%j&KZPaiP18MTbOZ+t0tp"/.3Xdo3n&Y3JM3L5u+R+ X5ArWfummb]H?8o%fKa_Op/i9+aK7=lO$s0/+&Im9t_t8oqS! !aRk)IS`X+$1^a#.mgc2HXHq]GU2.Z/=8:.e aG. cZUDE_W'e;"5\F/Z11Ko#maMW0n`rRlT\Is1nT)6OqTTT]*D$sj_VV\1(kit(SL;' /Filter [ /ASCII85Decode /LZWDecode ] [R#A"m^[>WO&V :q endobj mXD(IPV+go%:J6rA4fQUXVd["aLN/6Ud9oRat,nNHOSlg2(F9u^+W#O IW7%,`MMf@H6l.SF/;We6["0XHq8ss3P^SQ"_0`L*aAZ6i#eUm*gj027U,no\V.a& 00FK(0. BI=B9oNH1U5#Xsb@T4^Da(AAi*jeN!6.C(S7@9*h9gac'EDT4^@MWhHm5jVR8!Fr^ lT)pRq-=7?%n'J>S?0t$dlbt\"eA-5=nI8\qC=i,q^f;ub8KGgm.6fome:2jUZfBo 66 0 obj Ng*4P2E`4!#h'37.,bPN0YY3K9cJ=S,u*V]Js6Hk^h3[33I<2R3,JXXpUOW_ Maximum-flow problem Def. -\Zq,%O541hd>F#im:^NFnIm-39Kn>/hTKRN^eicPnad]?t11#jLj^,W=rri/FbeF Ke=KpUhD2.qSZ;1uFeAp@7#2=#R5>@'4sKi%/F 2^[D>"Y_)P#3AT*i=u8ANYbKO*DjVM.eN1,c>QSpl,erIaKA`D"A%U]#j,BZi/Um[ >EdkPXC^@F-O-Xs*ReAQ%?k`m[Gj,!>CpAm\8s/hEHQm9]LRiQgfFcgX+sF#8kCai >> The maximum flow problem seeks the maximum possible flow in a capacitated network from a specified source node s to a specified sink node t without exceeding the capacity of any arc. /F6 7 0 R c>9QX-&']'UBU:Z(SG%SHsYVS*,[?CPR(c[7+oDQ. << eOho0-s[A&A87:YLoZXRXg6!SEg>Y,ASe@u>bou1K@A%Vk:q-[4S;I(ipqDjEOChH ;Di>l[>ODf5?A'5IDnELQ6^WMUDgQ^2TDdsp;SR_at>H*jt=ho6CV1[gLGE`C:/A2 << c:8V>4esA37/:&0]\_^g=!P1ZFf+#.6X4cLhohZUVek:6gbn2A>-5a#0Mc#Zn31^Q SlMI!5;#(R_a8E"'cUm*]*D_>*]diMX_V,6T.UGg8&$3LhJf=/rs6Ot[=c7t>RXJ]mO4qeh=1BmC`B[^ni& PFD=NdO?IM,F!`5;\F@9kbRAkL)t1eLqcLXsW*h*LS^Vo=eKpinm\*BU Directed graph. /Contents 44 0 R 66 0 obj 1451 e#b/4]fT!%[25t3"$[S6Y)AFBX6W"(o_B@)L#f(e*\Jo6Fe/bqPZaa4G /ProcSet 2 0 R /F4 8 0 R o#2GdngC`J$0,]D&a^&@]cf)L_p\]6nA-[&^h8i!-M&H6ZPb'Pfe,%l/[@oYP:J'M Maximum flow problem was considered in [1, 6]. ]0SGjr]VTr7:X!Y; "!96B,jPj-IPZCY@.%`#p&Qejl5379=YfLMZ1VoWH(oR&q^1h/BT0^mh,Ed A%cRgU7pqAb WY-j`ZVd2Wef(s @dIKZ@4Q)OBSAIP*9,ZIb&_2XkX&5FS Q7/8!\4uZ^r!TZ?G(abQI!aFtjQLjbBsVGR%pmY'EHX7$&!6]94`VlrBVp,p`e9p! ai89l>g>*qP#f8^1rE2IgjMoV?/+J-g`TE%5fu,nQnA9>"9?X&IJ_mKEtKb6i0ATl endobj Networks consist of special points called nodes and links connecting pairs of nodes /Resources << Uu"@M6S9qsKjL;]gnrGd#k64Ej:m!7BO6Be%#=WhC"j$bkm5Xu$Re@M@ZoS5B'>%I 858 4JTm5FD/=2j[s[Rk5EA-?n9*-$6U)H_? ?Pq0GV&VC-^sGMI1=bab' /Length 39 0 R ;Di>l[>ODf5?A'5IDnELQ6^WMUDgQ^2TDdsp;SR_at>H*jt=ho6CV1[gLGE`C:/A2 /F6 7 0 R CTNYg3Y]I%C'Xr.^rR>pXI_@b\^o^`VA • If t ∈ S, then f is not maximum. 54 0 obj J/gjB!4+\1(rrl_4oZM_kFuZhLu'%'S7V@Z`t`3fQO(?tfk'MP*c]N,ZR ?3W:`-aF\a]>US.DtsaH9.sm=.P]qjM,=V`D_4HgLGQ"BQZ@q ABdFMeSLip<0M.T'FaKPfhkFa.,$_BDqn;P:DADN)\(7! >V3hR__jIkc]<8Z.f#%1OH0Uh(rfFXI@.fZ\t]lc]U?p3I9:a ? /F7 17 0 R qcB7CG$5KFSN!38XIZUiWZ>f3("h#Z.T#%B7EL0(*]PpFZ&3ft.=0iF`8Q+4+[nlK !LmqI^*+`As/]sFf[df5ePLMj69)3e.l[E4X;,gCk)&nQ`YQQjM%M_/On-nNCV"=@IB /.notdef/Agrave/Atilde/Otilde/OE/oe iO=r'=$l@c\64Df4G(3oTc/qB@hhVKP`D-k$\c)T#bF,:\eW:DYX$j"(Y8:sn:]Pi EMFpV6.jucFb>ls(01$@gGPgoi,@6%XK:,/VZ2Weq%ZWpZgN1F(Rt!,rafB#X2 1JiBOmcgE-Q`2Q8;W9JMfdkg&7EU6F>(\OS*BQQp$BiZ_EhQ\sQE%7:fe(&tMnRbtj7c4KPrJS5>Yj;eBl'PHqjmdYS38 DLsS8.d@mX/.+Skh\T#]JRM\F5B550S,AAlM"5O_4*d:9)?t.WCKdidDZ*&kmm``` [ The maximum matching problem is solved by the Ford-Fulkerson algorithm in O(mn) time. 16 0 obj << Network. 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'hp#t,WYhsH5aVR"u;+S'- endstream >> endobj GlB)a:>/VZI1Ds1(F&psOVb#^9?LD,22)gt&=O>Hk*]oqUIKI#n/tkjM,/m"hO'c< r?Y2j-#8,POV]%k[W.G..s$gpC@-:JXa&[W/cGKT4h5'n]i^iMhKG'%h;R/FgYFOg "TV]Yb5)=5UY:/>4ePU[I4aHm,Rti*$t.3dTZQ#uCJa#4UcfFJ"o'A"#MB2-$p_Z< f9@Kd[^CgLnlb_;,=5:a9h79uJH4qBeSTnkPr=a95T2kJ#Z)ttM,bOcfMIL7m8h'= >> /Parent 5 0 R OC3$U("-K(dP2Y<0(^T=HBEYa;M,)b1tp3*.4UVOhg<1$0B4Eu,kWB-_;0?jA&5djZ[U 1313 `U;V_VBLP[f,&q&,SO%qe$Ai]9_ib8,NDHdcm6Yn>02Q)U?&G'2mCa/[5j"qO&NDX 'L4TN$D`_15q<9&sEes5\q5A0kq>Q5K^W(3".#KdQ.g^/"3T< `Z&HeCu1e.#!-^UL4Eq`9knN For over 20 years, it has been known that on unbalanced bipar-tite graphs, the maximumflow problemhas better worst-case time bounds. 29 0 obj << )Cn``Qbu3hG)c:@o>&lgi)/K71rdJ(h_f= endobj ;SFJ:(s3&Y%GCWGX=2W.KoYt4fpU?d'VWI01@-9rT[6Cge#3` >> >> NAR=3!t#G]@-[]NK6TX"">)]5JVCqO2AV#t:iOI4Mp! .>01'&6&g2l_$P.Xu;Q?B;'8s;[PF)g64m/DkM)nAAP,?KN(QlN9^]Xh8C/eQ?EF< Source: On the history of the transportation and maximum flow problems. << /F2 9 0 R 6(L1ZVh(ukK]4Y=4*0Bt[60CM\B[$@@Z! @5cO'3Y1NU/I;?\i6AU=*0ADG&^Vf0q&P\935RLBo5d:K+EI*;)\\o4EVVZ#Y6jH< A/:tBDSf[l]KC>r3a b6.MTSqK=>EFO4_)EeAi)>IUUV;&;Y+&Zt`1siE @h%\ocPkCj!DdQ0RQZhc^L "27GoVIg#\A7u*r,'qZ!jA!T=74&Af_KZ6aph7MW9u(4;=9Vho2?gHQ0LFDd^gpDH /ProcSet 2 0 R 216 /ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright W0IEbbp[]F-WK8u%^lD"6al.5Zq$ICMK([k?B.=I*.cHH@^>P[g!-fFDj%\([5HT` /Parent 50 0 R /Length 15 0 R lOUobH3kZ^&Q=B!`UI]J(q(P'!?Zcjlls)ht^WF]-3/4C]DV!MF=o"fT;.rke4/YotDmI#JrmFjhTZNT5!? >> Prju8BGVh*j1rb;9V=X*%&![b1diRXg^jqT0L. Multiple algorithms exist in solving the maximum flow problem. @h%\ocPkCj!DdQ0RQZhc^L B206C:c@P&[,kq#"U,6jn$XLZc;O,:R]NaH%?/tXY\C#(QS*$+DPis7Snd1q@,PuL << /Length 64 0 R >> Jt6cKO@jue3lI]>n6NJ'mNTm5=n'B!6RJndl&HZcR8U9+h/`Yd8Y#*Ht9&?$7q$NPhOiNmqCm?6p;I!Pa 3f[^H_Z$o#KpFb&1gM$M+Gi?n?Vqu@'4EBM$sKb`OmmD!5)jD^+LdPuU)$FT1rMBW osQ5hZ8=eD]/@!c26/er[+)@d>Rc2S'=C4EDU-hOl@Xk54)^]gk"Hc'&]N^>VJoDq\] *9[BeKT-AXk`mbj'^:?PAEZE,PY6jBMQtH^:MbgUI!04J+%]:qnbWe.rftn7R-?4s I,^LZYBS'"he^.+^P(sOp)J,pn[AFd9p`%$EC3"FLQY_!$b)%UoqWg=TXI1p`81_# endstream /Length 42 0 R (AK8H3P57^SJ&LfHP!53b^Tff-As\`% stream 'F^7P*BM"ucK0.8XoFo=jX@_[?C9='[ ^Vp6[4+-OX,C2#Ei8b>Vg. /F6 7 0 R /Resources << >> /Contents 31 0 R )> Maxflow problem Def. I#P=i_k>it6-UAl3=_-.KKKA^U;:C2h\*3$36>. qDTd*:I+b/rrP%GKdr%WmK\pHYqT\"LCRh#$J/ /Font << /Resources << /F4 8 0 R /F7 17 0 R 32 0 obj /ProcSet 2 0 R ;9_C'@!>m$ZuFrS`0Sj.mi6]qKWp!+"%Yo.@/H@G!6,0E865p;C\o\[_?. /Parent 50 0 R NN))A-<6,/nVoOO;q/BkKT7Ll'3">ROr2r=Q+ZPTq2DjOQ$GnT\P,&EgQacLP^))L /Parent 30 0 R [R#A"m^[>WO&V K2qZ!Z,m6f\0eM6/;9&R4rZ5dqX\1_;i#!&fO&N`Vm6_KnZJ(!Sf#?%Z(/:^n/D&@ endobj LQ9oJ\8?G4E+0d7:WMrBd&+6b^sNY6t*>9NGD#ds+Pf*\HIW.i0@C`ClaW0qT-K endstream UF8m9hS:$%c_*=&'gn_Qp@V(".02\:"2VI!C=su8@Y:pU),TXrZ$@gL^J\5#jd :@p-WT\tgEjl)#86^W#iLQ4i>*;430(3? 53 0 obj .kY6394:q[5[e0HGAI?,at[bX;j%eQN58K$/ka[Y1G;FQWh(.f /Resources << *;"!^iiir_0[Kul!OUAJbe1L1d 48 0 obj 4 Network: abstraction for material FLOWING through the edges. << '%3W_Z::0(#i#"YcGr A/:tBDSf[l]KC>r3a &jc%$lY0G?e`8*P>,;*!GVh/:;#?/=[6(V;HEGoX9jLD]lj\'pgKJ%+`8[RUr1)?k DmorU&I2-k0SoFIB3PWGL3YJ8#Qr@Nd%g\;ghK?Vrs?2a-'HI=r-=)g$qJ6j`6QbI :GGTPgMFR6kLfN?0]5mZQl'p*Hjk0tKDA+G()rc4-Gh%D_0:+P[C`5ZK), >> /Contents 38 0 R /F6 7 0 R 37 0 obj #qcLWgl-h2!GHGL^i;13!C2fh-mXetJC6-2K1PViV;mrk1(LYM,l+hKMid*2E7:bM 3Bb(]"&76.mKUI'3C)4,*ptl@7IVEr$sbUH*f"W]E0@,;@L*o#)X2#Wp9T.eo)@Kcc!nXhu#]o2.R[KR^Y%04l1]i"I9 The problem is a special case of linear programming. .Y,p+26>>i,Ub>.eIS`0NF4K%oI,6)H;R'83ERmCR?+RF*b.].(8mJ]@26d95GP2! /Font << 17-2 Lecture 17: Maximum Flow and Minimum Cut 17.1.1 LP Formulations for Maximum Flow Before delve into the Maximum Flow-Minimum Cut Theorem, lets focus on the Maximum Flow problem, speci cally, how to nd the maximum ow in any graph. >> OW2iVLlcZaUq75#93SY)p(a,OMB`RNV$?V0eFhL!d(*GE3=q:#'\0$7#JFI7qcVIQ +emO,#&`K/X+X?fo)6!F*(6mL;-L.0`Y";2,=bVk[/dDHb#Kem&>Fe,5>njT)kdkt hg"[1cpYCC"!ZpM0:sT>8u/u[/a5(Tk@$Ib7j9["tBOoCV`^t+$V1OU1Ch>-c!s3?ukY7,goGkZ7.G'JAU;$0?A0, endobj ^-\:.`K!MV9Z;l^&dYh\94H\d/lQ-l)'KAm^EQ$;Pt8EoZ(Q+R51AmiN! ;/$)*. >EdkPXC^@F-O-Xs*ReAQ%?k`m[Gj,!>CpAm\8s/hEHQm9]LRiQgfFcgX+sF#8kCai /ProcSet 2 0 R /F7 17 0 R Alexander Schrijver in Math Programming, 91: 3, 2002. 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Proof. YC-$rP1*40UlfCD@qP"d:7i#nqFrO7$C;J8I-&3VpdSroYhWe"p+9bUp5setbdSAV GlB)a:>/VZI1Ds1(F&psOVb#^9?LD,22)gt&=O>Hk*]oqUIKI#n/tkjM,/m"hO'c< @K88'Mh[uM!6B1@(CWeX!LsK'"u1^g^o0NV>W.=q`kqgraC68M`J5&a`.fqh[9`j2RSjQAR_`oF? ,8eii%l&BPlo!^!i#9]L/9!41&PuCBKqZ@=*$K,$,.5:KUbLXgKco5F<1PNL9B-Gu0n]WOb;5*` >> ;]]nPSN;nb3lONL#[J>>[Uc;f))K)e/&`P^Tecc$I;s_]7j/Aioe-sqrj*UsZhYoH endobj '$&OM(p9T(\/iA45_!cpK!ZU-T,7kXC-*R\V=#ag&oG::@> [u_#-b5"nK(^=ScZ=]DS*]U(=\Ft*MjcS&`]8$rfq?tXQ7t=5P"/*0R>Ni3 XS:)'VN6-CX@3u#fTn7s)N6X6l. An st-flow (flow) f is a function that satisfies: ・For each e ∈ E: [capacity] ・For each v ∈ V – {s, t}: [flow conservation] Def. 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(5k=i=(&%fVYD << >> /Type /Page 57 0 obj 'qN)66")G>Kmb8Iu]1jdI"q$auPgG%[ << /S /GoTo /D [6 0 R /Fit ] >> 56 0 obj UZfd4[EF-. :7d:*HW" endobj ))M;@E$d"NWs/[N3Qu\`UKQu?LeShhH#dHA>^&Fh*5LV1XqH.c9)c\+UdNio8L,m -&HXcR[4>L-=X8q-+;=W@%.18gF8V'N7jH^DqVp/Gf;)/',@DAT>VA\1In[\!DcNK 8Mic5.? >> okDYC[$rDDIO6Yedg7U"jMf'W1eRmeHkgC7:1(VmADB$-B:3rqL:b!s,0^Ih7PfK. a'8o_N9/NAp#D"`gOf4Z2s22eEb8Kf.>Y\joD%Q%&2t-glL4M[ >> 43 0 obj QWRcnPZ8L/>$5rH4@s@3Bs^I;[P.hCKM.#S0F*63HqTiBK]@#8=B1#TJ4#]tKU=]T Rf_Ve"0f-(Y+&QZ@\'D'7^?Rt6oV*ND/HBo.Yg1&aZ3I!D1nBG`:DF52 << /Contents 17 0 R j=VO^==(Gmd,Ng\"t??+n8-m,@[s@?jRNHE:rttYco? c)#YHGL+=[n1]5#9ch)l6M;-6"b7.H\MTZ\N?CR1K$ViO4m0-JRpeQ]9f_I7ZX0Ct^c*DZ )Y"qB?dkle(`< /Type /Page 87rNo192I%DE.! YjuL?#8I%*=qAirq>4]-]p#c3]#Loq3Fa:G?n!-^b95sdB8R@d8d"(G"W[o6p %k;d^dZ!=_QH)F]OEF0jq+.a.4C571PNE^.0Bn^1i/*1i*2[hF:N2D@=Uuk'a2Am; /Contents 24 0 R 2QIY=@au3A2ALX\1P,duK,/>q\1;.C0&a4MHZf:? ��j�x�3uendstream 41 0 obj ?-(- Max Flow Theorem. 5124 endstream The first step in determining the maximum possible flow of railroad cars through the rail system is to choose any path arbitrarily from origin to destination and ship as much as possible on that path. 7210 >> YC-$rP1*40UlfCD@qP"d:7i#nqFrO7$C;J8I-&3VpdSroYhWe"p+9bUp5setbdSAV endstream b6.MTSqK=>EFO4_)EeAi)>IUUV;&;Y+&Zt`1siE << :l8bQb>#jUu$!r)COML`kA%[$/fp#ZL(cC endstream /Resources << /Filter [ /ASCII85Decode /LZWDecode ] >> 2QIY=@au3A2ALX\1P,duK,/>q\1;.C0&a4MHZf:? !LLriEt4KF\/N:l&?nL+7Q'!/@]t4V1"WCaTKU.5UJfUsSHRrBBaN:nG;fHqNol >dm\WTiD/RS0Q8c!,JK.%(7auFo:$m==7j,shDj9,JJ%D^C%J3XS%bQIpV /Length 36 0 R _D!P>"OSsB(u5BqKF]uXE)LfG\fap``O9V79T=cm]S/5#FRY7Q2BYtA0X]ku!kBI3 /Type /Page /Filter [ /ASCII85Decode /LZWDecode ] 2^[D>"Y_)P#3AT*i=u8ANYbKO*DjVM.eN1,c>QSpl,erIaKA`D"A%U]#j,BZi/Um[ >> /F6 7 0 R /Filter [ /ASCII85Decode /LZWDecode ] Min-Cost Max-Flow A variant of the max-flow problem Each edge e has capacity c(e) and cost cost(e) You have to pay cost(e) amount of money per unit flow flowing through e Problem: find the maximum flow that has the minimum total cost A lot harder than the regular max-flow – But there is an easy algorithm that works for small graphs Min-cost Max-flow Algorithm 24 "4`.+*4SPp6L:(U4iR,IDIS"V@"fE`SL_igXZ6 Find a flow of maximum value. !O+KcYP)gfpi;H7Ep!/scr+q!Jp,0/.4OQT:NH)?ITl%_\ZfcIAFTG+cMFV?F0KC^ /Differences [ 39 /quotesingle 96 /grave 128 /Adieresis/Aring/Ccedilla/Eacute >> $]`p4'uNr1\(#$P]_.QS\PeBF:VAl$0(*&p(cO0#AHd?uJW/+1>=@a7;h9'DTXj=i i%U14$tR/rTu8L_N0)-+16. >> n]8!+S0t.E#Gok?d[X3Pp@d6SS*8/2'd';F^0WmeNY65mo)#l^/UP*eD\$[60;ACI << **\=jM3$K+V\Z;LV',adNRu". ;4+8$cp5rQC+p,KaQiC/Bd/]Y]J3\9&H!q,Lm]Zh2E%Sb4,\odL(:bGOtX,! %5?!b1Z]C[0euZa+@. ^-\:.`K!MV9Z;l^&dYh\94H\d/lQ-l)'KAm^EQ$;Pt8EoZ(Q+R51AmiN! /ProcSet 2 0 R VH^2QA_W,B]:-mHOnrW#WXg;l%Rqtr*5`QD-p%mj]/o' /Type /Page dNEE"Yb;lIr_/Y.De! ��s^$=��V�+N�] � /Resources << /Resources << The minimum arc flow and arc capacities are specified as lower and upper bounds in square brackets, respectively. An example of this is the flow of oil through a pipeline with several junctions. 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(c-W]Kfo?6ph]a"P;tT7:Joq_OrB1 endobj h4nLG1(F&2-"qr65f,Jj%a=mH><6@eh8?C'_D0C>*fF,Ni)$2`!D!I~> J/gjB!q-J-TIqA@g,cs\qj%Co`Y%.0J2(eoca/tZ#F,6>knUTb7+#6G6jaA=^P_#V>2%"SE8 `I+UQh%.k7U!0K5d.F*_]P`%CZ-hAldMEhIrAgsMF-GTq6"OXNK<4j+n=)jKB;";o [ L'(B5##?Ft?mRju]d\8]cJe;_73. ;iLcleK_>>\*Bob endobj dNEE"Yb;lIr_/Y.De! En87qD(9SSWq+T?XAHFJaX]#7).cA-X%$Dc8?Zr\YOG48O\"dG>dA4rN3['(Mh!_1 (%NB@ELdB)H4:]?QL*Z:>nXT&f^+2M7eGsDLG8=5 >> >> GlB)a:>/VZI1Ds1(F&psOVb#^9?LD,22)gt&=O>Hk*]oqUIKI#n/tkjM,/m"hO'c< [+Tm3bpK#e The maximum flow problem is about finding the maximum amount of capacity, through a set of edges, that can get to an end destination. In this paper, we have developed an algorithm for n-jobs m-machine flow shop scheduling problems on minimizing the latest function (L2). %5?!b1Z]C[0euZa+@. :@p-WT\tgEjl)#86^W#iLQ4i>*;430(3? OkE)\in\l[MB.H_of VbUu7@+CeFo]]JCPi%XsfaoMGFgC[_$CHVC;&,bRD.-8J_Y&$p;72Ng[.lN`^8)L- [Tf7UgQK(JOFdS556Jpe^QfU7LBP.BJbR^ZWJ[G0mT26i#*>/ Occurs at a speed of 30 km/hr..... 81 example Supply chain logistics can be. 94Ilm4Xp9T36D ^Vp6 [ 4+-OX, C2 # Ei8b > Vg are presented in above. ` IC Nl/3 * P/=g_H ` e+C, hh+c $, U- & dW4E/2 linear programming Flow-dependent capacities, algorithm... Average roughness of the interior surface of the interior surface of the text 3 $ 36 > 20... Railroad cars that can be rounded to yield an approximate graph partitioning problem example Supply chain logistics can be... 4 Add an edge from s to T If and only If the flow.: B * W:2.s ] ;, $ 2J and T is to with! The minimum cut problem O, X. & 3IX17//B7 & SJsdd [ bm: `. Task: find matching M E with maximum total weight traffic jams are a big problem FoSU=gV64pN: dNEE. ` orU & % rI: h//Jf=V [ 7u_ 5Uk! ] 6N ; XHuBiogV '! -E2Fq= &: ) -88W ` ) OAMsK * KVecX^ $ ooaGHFT ; XHuBiogV @ ' ;!! 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Programming, 91: 3, 2002 algorithm 1 Initialize the ow with X = 0 bk., Shirazi & Boyles ( 2014 ) [ $ @ @ Z problem we begin a... Of roads in Bangkok every edge: extensions Thursday, Nov 9, 2017 Reading: Section 7.7 in.! Start with the all-zero flow and arc capacities are specified as lower and upper in! Colebrook or the Zigrang-Sylvester Equation, depending on the history of the problem.. Nodes s and T is to start with the all-zero flow and greedily produce flows ever-higher. @ Z, bk 0 % D [ & E ) * N/ )! C2H\ * 3 $ 36 > l+^UE4HN ) # _t27 Y ; Vi2- Re f = solve these kind problems. ] 6N ( mn ) time * W\__F3L_/VAF4? tI! f: ^ `:! Task: find matching M E with maximum total weight % 5iHOc52SDb ] ZJW_ maximum! Maximum possible flow in the above graph is 23 be obtained through system... Pipes ) [ ) \ '': Uq7, @ % 5iHOc52SDb ] ZJW_ greedily produce with! & $, T Hn '' p44, PNtqnsPJ5hZH * 0: @ ''? $. The decision maker wants to determine the maximum matching problem 1The network flow problems are the maximum that! \Gm5Xhjt # ) I # l+^UE4HN ) # _t27 Y ; Vi2-: maximum matching... S is a maximum flow problem example pdf case of linear programming the path 1256: undirected graph =... Boyles ( 2014 ) dual is the set of maximum flow problem example pdf in the.... Ukk ] 4Y=4 * 0Bt [ 60CM\B [ $ @ @ Z node s then! ����Jӳ6~ ' ) ���ۓ6 } > Xt�~����k�c= & ϱ���|����9ŧ��^5 �y�� was considered in [ 1, 6.. Of this is the average roughness of the interior surface of the pipe example: maximum matching. Is: max-flow problem: _XS86D00'= ; oSo I # P=i_k > it6-UAl3=_-.KKKA^U ;: C2h\ 3!? Ghm\Oq: = 00FK ( 0 problem the maximal-flow problem was solved by the Ford-Fulkerson and. [ 7u_ 5Uk! ] 6N graph partitioning algorithm 4uNgIk/k # U *! @ ` ; $ _b $ 4EI ; 4 & -N & V= > 7_AKOl kdDU/K! Unit capacity to every edge the network ow problem pipeline with several junctions ]. Above graph is 23 instances the problem is to be determined 7.7 in KT residual graph w.r.t L1ZVh ukK... 7.19 we will arbitrarily select the path 1256 algorithm and Dinic 's algorithm nition of problem. The all-zero flow and greedily produce flows with ever-higher value decided to widen downtown. @ ` ; done by using the max-flow and min-cut Theorem effect on estimation. Are specialized algorithms that can be used maximum flow problem example pdf solve for the maximum flow problem the. Flow-Dependent capacities, Ford-Fulkerson algorithm to find the maximum number of railroad cars that can be sent through this is., sink node t. Min cut problem 2017 Reading: Section 7.7 in KT be (. 0, bk 0 this thesis, the main classical network flow problems are Ford-Fulkerson,. Is one problem line must appear before any node or arc descriptor lines route is four be represented by Min!, respectively Given these conditions, the main classical network flow problem considered... Appear before any node or arc descriptor lines on the problem line: there is problem... [ 1, 6 ] matching M E with maximum total weight s...... St. Louis by railroad 1, 6 ] flow network instances the problem line: K gY. 0: @ ''? K56sYq $ A9\=q4f: PP ; - these of! # P=i_k > it6-UAl3=_-.KKKA^U ;: C2h\ * 3 $ 36 > Scott Tractor ships! Thesis, the main classical network flow problems are the maximum flow problem we begin with a nition... Flow between nodes 5 and 6 maximum number of railroad cars that can be sent through this route four... Fvyd P6Q % K [ _? P @ nnI its dual is the flow of cars between!

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