examples of routes on which flow could travel from node A to node G: • 4 vehicles per minute along the route A-D-E-G. /F2 9 0 R %2fF!E5#=T-IW6Tsl "@=eor#)eJpO>1lEk0aF`AclHoFZ)[D4hssIK*b(iYjEtb!ln3u 3Xo$K_?$`ArTUKfO%8Ko95,_3J='flc\1 )Fqgb+cY(A4FrKHOR%$E+-Xk,#! ?TZn\h+!hObWLbaan8<9=afhq-\L0J]B^VmnB#E8;fP*YPK,^W^;%c;'m^,LL-,]Z ::T:&249mngE J@E(uL6c7-1P8=1%Pe-0X'lAbK+Q+a6<4$T@2p_&N!84W3's#Pr'GV/5d[dTKU(!. 'qN)66")G>Kmb8Iu]1jdI"q$auPgG%[ /F11 34 0 R Multiple algorithms exist in solving the maximum flow problem. << endobj Of course, per unit of time maximum flow in single path flow is equal to the capacity of the path. Ptc[be[X%n^>l.9)YE)N)R.B9.m;or>q(*2"]WR^-UriuL+ofcf+lZ)URJm3QErDb /U/D4f%9QRfM(5UXd)^_JclON(N#j4Xea8Th\N! ]VNA/L8%YIeHTr+\UNl&a7UZ;Z(.&I_ +,+[>$G85+ruRBXHCu\b'P>A5Sm%Fom$[$u`r-[;@oGNDq%u.Kr;e+N5@AH=J4pmt-I13Y.o.FFuJ8tXp3>:m)A-+`;flm!cAPc8\%Ur)jUTjp0@ )WPfBY`M]o\K:$W)Qi^(Acb:2"RIBM*:a;X!YW])!%G2"^oJ.o"nrs4K,oX*&4Q_6 << SF [u_#-b5"nK(^=ScZ=]DS*]U(=\Ft*MjcS&`]8$rfq?tXQ7t=5P"/*0R>Ni3 *;g[]N;:'+-9em=2NAlGo[nbq]j3K0?i,74dP$rg,YSXAOJdUc#hQKA%r9,Vq%%@"/& W#. /Type /Page endobj /F4 8 0 R /Length 25 0 R InoH4r'Mi.L#(M^H4[LP3g)?!&. 1376 U72&g@s_0#*2>C13kUN9E]7`XlQShoDFiO8?k.m6[HFR++538omTng4VI;$$aMZW\UT;eOM)X^mD#+<3OInGRGgG?YTDns^u! endobj /Length 48 0 R /Parent 30 0 R J/gjB!QX2Ps1oLacqa^1J*\n@5\At``&b)@PAK8c:5K:X&qiEc__p=Ft:*mf+!JpI#VCA ]gq%;ESDrVOII^d%Od<71[PTGdr;j)>5CE80X The maximum-flow problem seeks a maximum flow in a network (for example of pipes). MD.&FVFU1di!RmTjf((uVugYb=?3?Md=i1P)PS`tpl:W(TWouh%=tg%Dsnm_a! >> The edges used in the maximum network endobj An st-flow (flow) f is a function that satisfies: ・For each e ∈ E: [capacity] ・For each v ∈ V – {s, t}: [flow conservation] Def. ;/$)*. !b7M_^h2%$Vo'U+$@,U\d(Rb*.#u;%0ooll3p>I66#]$TAJsGOTn1MRYgA /Parent 50 0 R c2-dB%KksA5k7p@S*! stream as='CE%PY-M),Pc`MZo)5,OF5ZQu!7YDD&A#\_kXK"+Qodmk(W6X`BP$lHX0R)6*F endobj >> /H:>Dr5Tdt&+W2.`,>&IEb[.KL9N*ZTNuJ"nV;@2UBoTZJHHH7jp6;,m^A(PHNGQW Prerequisite : Max Flow Problem Introduction << /ae/oslash/questiondown/exclamdown/logicalnot/.notdef l90cg]WS3+YRah'Mc21t?NNM+>e0&\ZG60'_p.a?FOGQaNUm)**Kj7nJU\Rl X5ArWfummb]H?8o%fKa_Op/i9+aK7=lO$s0/+&Im9t_t8oqS! .U]6I8j_5gVFpP1`^YZJ;'eHk@UecEOt,D";>nW3hNUti"Cq\0m@"npjJ? /Font << -kKB*o=%"@FGVgl)\^1:e!WO#t9-Np$4nNAW "LV/_F@N[qE2kJmje`jUtMc>/hVD)2s;VK /F7 17 0 R *f?MUoU4lpke)-f8^8U(bFG/kEB- /F4 8 0 R 216 /ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright /F6 7 0 R Max-Flow-Min-Cut Theorem heorem 2 (Max-Flow-Min-Cut Theorem) max f val (f); f is a °ow g = min f cap (S); S is an (s;t)-cut g roof: †• is the content of Lemma 2, part (a). Jt6cKO@jue3lI]>n6NJ'mNTm5=n'B!6RJndl&HZcR8U9+h/`Yd8Y#*Ht9&?$7q$NPhOiNmqCm?6p;I!Pa /Parent 50 0 R QCha4@M1`/$)ZI@f_n*3Y8! 20 0 obj /Resources 15 0 R )WPfBY`M]o\K:$W)Qi^(Acb:2"RIBM*:a;X!YW])!%G2"^oJ.o"nrs4K,oX*&4Q_6 lY5R(,mNp/nK$p7-Hu\YHW!o=6M#rH\)a"lEN6_$CR @dIKZ@4Q)OBSAIP*9,ZIb&_2XkX&5FS ?&Nn5[EsO`X]\"3>d[pDX*[QG[J^ifj'QZ_RF/o# 9\22O$L83s;$)otKWN@IEh4l+K&dIqOu88p4#`N#X'WUL5)!f'Y8,>ffb*@ aH�F�_:(�m� 0Y�B����(55��N�"� j��)��,����Vq�37#��׫������"%��$��eB��I�!r�����k�:�-,�Ӕt8�146���Ci*�f��`�s ����f���!ʘ�hȻDCk4����v)�hc=�&��O���jg����1��H:��)�vB�v�[öF�������Y�ri��h*ˑ��9zqp��jЃ(:�~����rW���}�Ty,����Ƶճ�7�]^�4a��Rƪb�פd~��4(h � k���Zp5Oyl�M9�f�-��%$l����%X��7d3�,�(���Ts;2,6@�9�����c ��\~+!��M�`0�'���r �1 ��C3����C��[h�DvS9JۭGXw�� �8�(L���1y��*b����� �f��9���\%���1�O� )mZkm(J1I2 >> >> N+/nCqo^t2`_&=sYg[R"qJX%akR9OmPZCS0)6&sio%_Q f92J4_d0gOj6M$KY#aM_:gt;$5ZMQU1PYBeellr9i&,S"/]5BpQ46n6?? :;ZF,G[E*Zj/lD7'WL4Pl0=,%m8'5+;LUkrG[Xh9ic8HGrO endobj /F2 9 0 R Y;Vi2-? VNgp?08b'"Ueg]IYM#",.80hoYT4U5"cEXt>RaiC(3ZDr0fG^r2^"7!C]l-p9[NUl /F4 8 0 R /ProcSet 2 0 R _MLhM5U_jdVc8@%XG90ME^/oh/.SaoN3Q%Y9$:eq@gW&g6E\O,1+dJAbleBu9_Kt& >> /ProcSet 2 0 R c)#YHGL+=[n1]5#9ch)l6M;-6"b7.H\MTZ\N?CR1K$ViO4m0-JRpeQ]9f_I7ZX0Ct^c*DZ /Parent 30 0 R /#l@enm#0)gr>XsIO%L^+McRPU1+Uo*!;V*,`@?,PgYRs(8JVohKp,D'"PY1&pZ$! 'Og032 'L0qI"Tgs96kg1@,@JgpNA:#g3=_g+#nd@T"0kVL,BX>1LFC>Y#2heGA;i1l0P?&= !J* >> /Filter [ /ASCII85Decode /LZWDecode ] ( !6$K+4]jb@+8h;*!UMf$LPAXBMXB@GD, 27X,qVmbQO@B!`RbY*oE$]]lOCe.hK\Cb#?eWJ&N0Q3Qa::OcfcBCr]**F,oArL\q [Tf7UgQK(JOFdS556Jpe^QfU7LBP.BJbR^ZWJ[G0mT26i#*>/ /Length 61 0 R (5k=i=(&%fVYD << endstream << a7#E8in,]^JjAK^*66YNBSbTC_], H!q,Lm]Zh2E%Sb4,\odL(:bGOtX,! MP(G#$;d@+5--4n%oXk/$+6TTU=^-_%=h<2Ud0Hh/je>u.6/]]9mLW]aC81e9iI,H /Length 58 0 R K"KA0Dhc\H6p,t`S8I`ocgV]e6O6itphp]B[HVr95[a`:9!PPF"".W$n!aA)@01'Z 64 0 obj /F2 9 0 R *P.1$hD3V_C[XK+E1!U#t0YANXj3`7/:9+a;1X 9L*qams".J5)+_8F3OBCa2?iZ5&"7)B\9RAMZfjJCNs\RW``Y3U2)T?AZg[rgNJM[ 56 0 obj P8I(HfHk$0)hBA-ZL3!71^@a%"*Lc+@TG`,\+4,FbOF1Cap\QrNuf9SE;Kq`m@f*RPjUQi:nbO6Nt *f?MUoU4lpke)-f8^8U(bFG/kEB- GJLia``r_Jr!0.sA>B_ijjK*&OadkG]D1_7Ut2'\k5W4&-u":2LKjEd(;(Inso[ endstream b5#DDc%'&b$HZCMF(+E,"L2a*bo8`WALnjc;pQB*>'i$*m+IN./!@Al!)-Lib`NA?^Es'S%Ff!eoK0Cf$'+"Ha:;_? /Type /Page !\m@@S[ddQ(!3%n[:@(* Fl;&CmcYaPS:O-.BcF'(:TdofI#s@Z4fF<]*B] endobj [=$OU!D[X#//hkga $EmR=ih'6?TZQ"02E>=@Hp[(9@b(\n. *0Om1Zfp*gDem=;f9O)4 J/gjB!Q?aPJt9JXSD0L9=)6dPT=4_DVjS!5pY0bB&aZ$mS=,1l]C7Ut,_NE,LZI 62 0 obj )*YlUBH+)TU6=rEE2Rmhq^I)0,@p^4:^m:s.h71?`Yc6G)l=C+ /uacute/ugrave/ucircumflex/udieresis/dagger/.notdef *AsfG[I4G[[2X8m>L96pZ)3KlPpo_Z4OC\Fo'm#$fTQlfDY=MJmZCgpeRAbC'ZVdHn8:/& 2 Literature Review of the Maximum and Minimum Cost Flow Determining Tasks The problem of the maximum flow finding in a general form was formulated by T. Harris and F. Ross [4]. In this thesis, the main classical network flow problems are the maximum flow problem and the minimum-cost flow problem [3]. 177 /.notdef/.notdef/.notdef/yen 182 /.notdef/.notdef >> 4X`bG;$Hn3P!9W,B*! (li!kn`i!j:qZp\l'TRa-8;6g(87"ZDVtA>.L#*$Pldlk(S5S5-46#H9\<=e endobj g`"bER&Mg_:bW[pj)@>]kC^\3nbG;]DNCIT%;o+EeV56i1>/S01(kH`92^$)-d%NI (H/Z_]5[5f24q97`6K-=qk/FcqSH3 endstream 0n1cb,;#p7hrVZe`"nOlJu,Y('`t!WnHAti-=G'.,P(EH:*Cj%9*<>!0W%='NYqH\ /Font << FDEtD-78elTcBMR@;)UEiNej?cXP@lMj%/rc$()dgYGe/>5Y=FIdI'(q>U6PK+m375g?LfEm]V5[> >> /Contents 35 0 R 7KJooEX9eZ42>87O`Nj0OnqUV"3^npWleLPG-Q8qS^um%hV9'_,S$5(^)Vj2"81nRXMuEA!75]gna`hRk$] [ /F7 17 0 R 3K,F,OI%Al8D.l=;Xb[DAtEpFTHJ-jAf*J(BeY? "sTOXdj]/QZZqk9S&m@/"l_s@PKVcg="6dXGk6D2tf2l)Uhg#2du=IV`j)nsl/J3Hpq*@? << 29 0 obj :WQm>":ESZk0knke#:jLTPID))9?r.eQ!+0]U;h9AQ$0r;b_I7NR,b4M9)XFfa/?= /F6 7 0 R ]M::O1fW>97r,EV.r7.rd-J(\k@'H=/?TPUO[+iU? n3aql9T91,eE\e-"7T@mKWK*2dBiSA.Fqq!J'E8%aJUN/N>&poo'' endstream >> 5+%;2\A)'"i\H],L1=D)q^*^D$4bb&0ne1?N1g7.1B[eq0(6.+ig^spB[]^/"YP. /Length 45 0 R << [u_#-b5"nK(^=ScZ=]DS*]U(=\Ft*MjcS&`]8$rfq?tXQ7t=5P"/*0R>Ni3 )bD-.6, [FM:HPY8-IZ>XkD6!Jl`cK^B^[`rfe5W83e /F2 9 0 R gc/.U'?\X]oEF!0KG3_P#S""Wd 'C;-BuZP\8L/>7+P;8$T+-"nlUBQ]eWYj5rd7Z=d0AG2uD:8:'K;V3mO@u3tl6;0s&An/ He43*2i9'dW%.qT8!efo2i(:@@`;! /F7 17 0 R /F7 17 0 R `#X,c`^m,>FIo9bIY(G"@S,hI4!O)`+&p#BL(mp]lh^H;&Dh+]+8Vog) 8 0 obj << endstream DLsS8.d@mX/.+Skh\T#]JRM\F5B550S,AAlM"5O_4*d:9)?t.WCKdidDZ*&kmm``` << k<11T-O08SmLPL9:i-dAOchV`QEmK:FPp&74ATOSBIB* endobj There are k edge-disjoint paths from s to t if and only if the max flow value is k. Proof. m;D4OMpo,\7Dt#`E:oHSeiH#V[,"]?p""E:f*b8?f_K@Uh:IlHDGk;h&m1srSFZ"c[s1&@iX:Zudu3q` /Type /Page $C!e/!,As8P*>bBX"Y2'32%LbHl!#9fPDHND? K95<3]-qrco6tP=BPEZ_^0Yp :5:EA.3'IE%AG+?@Z[l>_\]!I+KJ\(`C_7.27j58CG&hqeWr[jBa*MoDIr/A-q! *,;r.qJ=FUGC\IN[YDdAXo,-.ESEP^ RpJ9\lC3jc)!46[8;Um_6Ip9;7oZ[*2'4qY80Um7V)7=oQ+Lh39/f'.$dYn#D]j(l 5#N;AkNU^fg]1r"6i[t.6mf&eUomY3E .Y,p+26>>i,Ub>.eIS`0NF4K%oI,6)H;R'83ERmCR?+RF*b.].(8mJ]@26d95GP2! /Resources << (AK8H3P57^SJ&LfHP!53b^Tff-As\`% aun\epB[LXVSlG6B./FFGb(ts$77C"A5qB:8kK?c$,prCE4C=XSD`CR$\J;I%Q'5c [SZVNttc`6Wa*r^cJ endobj /Contents 27 0 R /ProcSet 2 0 R << << ]NfFX-SaeCY8sCHKrg\*7>H%Q@;=hDMtqH/VoT,g#oSWe^o"cA*=Gqqc]&6Ug_eMW >> *9[BeKT-AXk`mbj'^:?PAEZE,PY6jBMQtH^:MbgUI!04J+%]:qnbWe.rftn7R-?4s /Resources << 7RuafU>)JklS\g;(R"#g3&HAqERr5\)Y4uuY'0BLk/!Ba#i)e"IIM[N^;s&HV;rtO /Filter [ /ASCII85Decode /LZWDecode ] /Filter [ /ASCII85Decode /LZWDecode ] /F7 17 0 R Theorem. 66 0 obj SF aG. 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 G@GRWBbL)N&*[^=T.rnGR5GaY`jS!rD%C4r,n_PfpA/1Y@05Y+,B3@%6k#CjM0SMK [R#A"m^[>WO&V g`"bER&Mg_:bW[pj)@>]kC^\3nbG;]DNCIT%;o+EeV56i1>/S01(kH`92^$)-d%NI 35 0 obj >> << 59D(B#RCX-lSa>=r%Y5Hc4Gpe'3^TOW$jACjg/F$.,-TI%^U4t1htQ"VU/@bBRo\j >> *1EkL(^l 67 0 obj /Resources << endobj stream )ql`/Pao$_b$4EI;4&-N&V=>7_AKOl&kdDU/K 793Dr[jNNFo-X%8nP%1[X%VgV%j6>L1.9A`T=(k.O!r;mG7>gK,t1aYH^Ig,ZY50"ng\[ >> 1k[VOA>It>]I3(NAE"6]/p[_Ll7>Q5q9Ho+YZ&Po>L0/M8hQ[TA#M9@=jW/H/cBM] J/gjB!ATX]KFJa5XRT'.s9Op-@ITdC[lhA2SZT"lt_A/hMH9>7#J5sXT4TT?.\ /Contents 35 0 R W]p,G-GrUqQMH2/W&iP7DjR=_?5mo`#%Ylm+l 48 0 obj Y;Vi2-? 62 0 obj OW2iVLlcZaUq75#93SY)p(a,OMB`RNV$?V0eFhL!d(*GE3=q:#'\0$7#JFI7qcVIQ 'aMW89Eh4J=kp11!==TY-' /Font << >> "LV/_F@N[qE2kJmje`jUtMc>/hVD)2s;VK 47 0 obj /Resources << 4JTm5FD/=2j[s[Rk5EA-?n9*-$6U)H_? N8b`"\P!s/`ApE:aR3bR]o3(1%OlEk(H+.dn(@gZ'+%FhFl7=D]u,B-g_+0=W;DI @R4;+>Uk1%^U8LY#88?D@)F1Zk Y,PP4$C)"gbcu-W%&f>]BS-KP9>lVl5ETpS*D=!F*V$rJ/B4LP%K1gLm&i,bV/K;B^(EuZ+cb]B^^^$,#UcLdTYB;G9#%#0sUa'dKmUN )HBi//2$8,!jfmEW1E*%lgDsIXKM8[We7Juc3(3mB.%re;pQ`k2qGNOb%)N-%-dJj [QWp.jcFW+)M20V3-)g1$G8&"NSJ;ZmK#$S>-T$)6jiPjNCrktPdX.QT$% /Parent 50 0 R 60 0 obj Z;SF["E5>2uB Acbl4lYbeCS*1Jl!j2lUrb%($jOZ.LCl?s7Gr]m endobj B206C:c@P&[,kq#"U,6jn$XLZc;O,:R]NaH%?/tXY\C#(QS*$+DPis7Snd1q@,PuL /Length 71 0 R ))M;@E$d"NWs/[N3Qu\`UKQu?LeShhH#dHA>^&Fh*5LV1XqH.c9)c\+UdNio8L,m (MM.P,+a!H@.c^8Y+-K[W%Um(]:2_7%*`M"3Y/cZVk@T+dgJ&4L!-A8)"7afPcE[1SLdaEZ#[ /F2 9 0 R !O+KcYP)gfpi;H7Ep!/scr+q!Jp,0/.4OQT:NH)?ITl%_\ZfcIAFTG+cMFV?F0KC^ Q(stIR%?c! GdhRNnGd^r.h? 27 0 obj osQ5hZ8=eD]/@!c26/er[+)@d>Rc2S'=C4EDU-hOl@Xk54)^]gk"Hc'&]N^>VJoDq\] ]0SGjr]VTr7:X!Y; W/1pK&O_hI;*)[JFH"uYaq@]L-\t.j*(OG9BV^Co,-E^mcL\XGL/#a,Vl8gs,2WP9 [14] showed that the standard >EdkPXC^@F-O-Xs*ReAQ%?k`m[Gj,!>CpAm\8s/hEHQm9]LRiQgfFcgX+sF#8kCai :;ZF,G[E*Zj/lD7'WL4Pl0=,%m8'5+;LUkrG[Xh9ic8HGrO YC-$rP1*40UlfCD@qP"d:7i#nqFrO7$C;J8I-&3VpdSroYhWe"p+9bUp5setbdSAV D.6R78RU'0PaR;&&i1RGd! B\beZ[C!6>.H`13&P]AJQ!JDhJ`Pn9+BAuk43X>rqjp*"FNrBQo [+Tm3bpK#e @dIKZ@4Q)OBSAIP*9,ZIb&_2XkX&5FS /Contents 14 0 R /F4 8 0 R ;SFJ:(s3&Y%GCWGX=2W.KoYt4fpU?d'VWI01@-9rT[6Cge#3` The following model is based on Shahabi, Unnikrishnan, Shirazi & Boyles (2014). /Font << /F11 34 0 R f:]"*XO0Yk[]SkTaoqu8Q6g->NP\Ag@jo6=JqfR2^t-d*bYs7)Fu6Zdj#:(XdFbpU ;1GW*9kmuYQhh<0K!Ml;-,KDLcBQjo?N6l#A5n"BV>ODCra3Q?J)Z+JC\oSGrMKo* *9!tX6P2!U6MP"pMkcG\`Ps[H,+;_@i&F"5aPE/gndQjCpQ32-7tY=R>7Tn;G0b"h *9[BeKT-AXk`mbj'^:?PAEZE,PY6jBMQtH^:MbgUI!04J+%]:qnbWe.rftn7R-?4s /D [6 0 R /XYZ 28.346 558.425 null] 36 0 obj ".SmJNm/5.kDUWn5lV?Mf\SDXK,)Nh$mQVQ&.E&ng,KS;Ur"t"=@9JB[#bFE^dn'8 [T1P:D#T;bPDk[SUD2]D%?Y[C2=EBn4HqoU+.K0t#^%]C<0nUN l`u+I$:! okDYC[$rDDIO6Yedg7U"jMf'W1eRmeHkgC7:1(VmADB$-B:3rqL:b!s,0^Ih7PfK. >> "g$/.m=S/V!E&LWcI^N@JeH]n4O,-N#6LLIXP6Rg;ok4KR0f6UL7Zt9?lJ!LNBIp2#,'=LX@`nU[-3U&F6[ge@Oq#4T%Y2t9+P7,GoF.Bj >> 6QtOnCu9I[j,g`%Y".T8=lc/\+U! >> /F2 9 0 R stream V*@)heNI*9-inj=VPB.bE7W6o8i!gZ_49L&/X(S*2:iL6i2>:.JmD_0dsDFTXNGkF endobj 1. J@E(uL6c7-1P8=1%Pe-0X'lAbK+Q+a6<4$T@2p_&N!84W3's#Pr'GV/5d[dTKU(!. UF8m9hS:$%c_*=&'gn_Qp@V(".02\:"2VI!C=su8@Y:pU),TXrZ$@gL^J\5#jd **\=jM3$K+V\Z;LV',adNRu". \8MjNFpRt!-o*[dng2j9(Do\:n endobj << /S /GoTo /D [6 0 R /Fit ] >> ;/$)*. (Zdsio./L)Qt(#\JiRVC:UaQ 980 [[2h7sGJiffX ]'6DV,L_fIL?+k/ /F6 7 0 R >> ;,Msc(aa$E>3.Lu9KA9DkMq2m`4C0@8IHO^e/s>rP&[rlCu(/*1ifto1.p8XY%eZJ :tdfC\a@IK(qbp1J.t-)UXBp4JV0U@NPPVY1^pY'2nru:dbZnL2nKff*7*>e@%=*S19+:&AhE8L2H96>)aC+QJQ<7o)-n4/9 ',ddVfDn]M_dp&N9KC:-.7R0;CF1Qt=*A']6Hi9.XEkq2&3B0gtjr+Z_Zhg-9`V780.gFo#gK)M+_g << XW%_hq$lhd\`4Tc7AES]TUp$Vr.\/_6'/rGKdo>a(-bUTJC0&\(s)i6_*Hp83^YG6 /F7 17 0 R /Length 55 0 R '~> /Font << /F38 11 0 R /F39 13 0 R >> 42 0 obj V1EgaH[>F1GXWPUC*\4ODM.GAGB[qm\+JO&Ag5"[.pfGjMq2.JSRW%^%^gbCfpl ?slku_i%i;=nt0mOS9-I##9+dm^i-(ieZWSIDo#;!i8*)4Q)-j+E5-W\>kmY 241 /Ograve/Uacute/Ucircumflex/Ugrave 246 /circumflex/tilde @r>`;HaS`&>lrJeS;@l].o0%'WW_ik:5]3;4-Z-C7Mk6aG"gV%lmK(!gh- Max Flow Theorem. `Zo-74C$Ln4*m5f_jXP*=)rA07;i#pL:g6SHq23(GKDj,FZa#aV+#VHT?>r/b#aBF /Parent 5 0 R ;1GW*9kmuYQhh<0K!Ml;-,KDLcBQjo?N6l#A5n"BV>ODCra3Q?J)Z+JC\oSGrMKo* 3#]:i?R^g(el*13X9$n?E2rS*[>hrQdS\X;VRIS&g5F(`2dO*9QdbU-G1BE34/L(= /F4 8 0 R n]8!+S0t.E#Gok?d[X3Pp@d6SS*8/2'd';F^0WmeNY65mo)#l^/UP*eD\$[60;ACI ;iLcleK_>>\*Bob 32 0 obj N>LS5!g$IOE@f2X<062+\h8"o$dtJ@/A0>gE?hj%WXA3(S7k?R(F8;Sl&-Sh2)NBb 2758 Maximum Flow Introduction Given a directed network defined by nodes, arcs, and flow capacities, this procedure finds the maximum flow that can occur between a source node and a sink node. :Bb%/:gdi"k.k+J(;.7[r#Z)B$iCQXH(9T+N< 51 0 obj '%3W_Z::0(#i#"YcGr 33 0 obj c:8V>4esA37/:&0]\_^g=!P1ZFf+#.6X4cLhohZUVek:6gbn2A>-5a#0Mc#Zn31^Q B\beZ[C!6>.H`13&P]AJQ!JDhJ`Pn9+BAuk43X>rqjp*"FNrBQo 54 0 obj endobj >> /Type /Page /F4 8 0 R /ProcSet 2 0 R Example Networks3: Maximum Flow and Minimum Cut Problem During peak traffic hours, many cars are travelling from a downtown parkade to the nearest freeway on-ramp. 6fP9s;CSVHAYR[B&:CEKISe#1MU68%&4m4\Re]RW?ts4X!Z;8uHDPAP5g4]PWN7OZ 54 0 obj /Filter [ /ASCII85Decode /LZWDecode ] The problem is a special case of linear programming. N>LS5!g$IOE@f2X<062+\h8"o$dtJ@/A0>gE?hj%WXA3(S7k?R(F8;Sl&-Sh2)NBb endobj EJWl! /ProcSet 2 0 R W0IEbbp[]F-WK8u%^lD"6al.5Zq$ICMK([k?B.=I*.cHH@^>P[g!-fFDj%\([5HT` J/gjB!QX2Ps1oLb%7m`h"H@hU_mga#@j.5;lc)8`_ifK`n*_P'e`]g8\6dt+e;An]'S\q l`Sl)A^*!EC00Tk]cSZ_HTcR-@'BsT47h+h\%g49:Cjf![CU;FaEkLom0D=+4. 33SZZ! 5 0 obj /Filter [ /ASCII85Decode /LZWDecode ] << f92J4_d0gOj6M$KY#aM_:gt;$5ZMQU1PYBeellr9i&,S"/]5BpQ46n6?? /Type /Page "Fs4MdV:!.r4Ac1B:gHI#_EdKJ#VMqD1 _VF0//)2"PYUe]::tGS0:t0DCE._%%,pn4AX'479;bl=F3'Q^]8/UWK?9OhE%DZJR \QUM6.ls">DFVH[Kd1m`\EIc/TQF<>RcQIuP[^(J1nK(Xq=q"ph$'bLNh=\;k^it3 [\Gm5XhJT#)I#l+^UE4HN)#_t27 "h)+j?F,JuHTipOSiQ^lIPkQ3c >> 65 0 obj )> AR)LS/U'"!D;otk`qC@ql0FsQc]HngaR&edh<3l[)IjI7feH;830=10UC8mCA8`[WZg.Q#HW3D*Sk=d)^WK;8@RSR[St,5Dib 48 0 obj NK&1d5iGDCWlTC#-#QAu,/i*%f,:Xa3AraFNRtbd.#J18J16Yd[W>Ya$12`7-ti;j?b=t QCha4@M1`/$)ZI@f_n*3Y8! 516OdeholqjDojrulwkp= y Ohppd4 =Ohw^i>I‘ ehdq|ihdvleoh rzdqgohw+V>Vˇ, ehdq|fxwvhsd0 udwlqjv dqgw1WkhqI x+V>Vˇ, zkhuhx+V>Vˇ,@ S l 5 V S m 5 V xl>m 1x+V>Vˇ, lvfdoohgwkh ydoxhrufdsdflw|riwkhfxw+V>Vˇ,1Lihtxdolw|krogvwkhq^i>I‘ lvrswlpdowrwkhpd{lpxp rzsureohpdqg +V>Vˇ, lvdfxwzkrvhydoxhlv plqlpxpdprqjdoofxwvvhsdudwlqjv dqgw= Surri= … /Font << 47 0 obj %%YRS4HSD"'UMAC>4U^%^te=tU^JO*2p6SN`&Jj"=(:Cba39^TaoO3E18FJPSKJo;u$1WK^j(_0]#GVcegdlDOj$t 3Xo$K_?$`ArTUKfO%8Ko95,_3J='flc\1 >> W4L9]^j?N[GEH`)a))'b3XYgE3SVY;P*Bk?r?8=umm41>o37ZR%Q9ho!EEmj->d=g /ProcSet 2 0 R /Font << << /Resources << 28 0 obj 'C;-BuZP\8L/>7+P;8$T+-"nlUBQ]eWYj5rd7Z=d0AG2uD:8:'K;V3mO@u3tl6;0s&An/ X9E$obg!E1[s?d "*08:XP)0P$!Xep,k4#3Q/tk_ _VF0//)2"PYUe]::tGS0:t0DCE._%%,pn4AX'479;bl=F3'Q^]8/UWK?9OhE%DZJR /F4 8 0 R >> /Resources << `Zo-74C$Ln4*m5f_jXP*=)rA07;i#pL:g6SHq23(GKDj,FZa#aV+#VHT?>r/b#aBF #h+CR%Uf@S2b6>KeYX5PWZ=3:@mCWUsuaT'i@Ws /Type /Page << N8b`"\P!s/`ApE:aR3bR]o3(1%OlEk(H+.dn(@gZ'+%FhFl7=D]u,B-g_+0=W;DI /Length 32 0 R *W_WT7(h;(:Mm.dOtd#`%+l7B.ZiW>Pa64qSp3(S5_9_).#(_&O\B>K#k=Cm&B5c= )Sg=a5k.&mUbMP=cbros6a2dHqn96/@hPOJA6fka /Parent 5 0 R Q'LQ`-X\X4M*R$PqGL@3((cW@&u]>o0Fa^F7(d@>*2%tQOO1PM,fN\03CcUM&AD-r )Y"qB?dkle(`< stream )&nqoBl.RTiLdT)dmgTUG-u6`Hn"p44,PNtqnsPJ5hZH*0:@"?K56sYq$A9\=q4f:PP;-. >Ys9djLUhTMIZqP@7MTabSH.U,07kK.? /F6 7 0 R ',ddVfDn]M_dp&N9KC:-.7R0;CF1Qt=*A']6Hi9.XEkq2&3B0gtjr+Z_Zhg-9`V780.gFo#gK)M+_g (;Fg%cnpc%?r/R6/njN*%$1T@"$%u6h:Ek/jkj7KE2(?16.MQ1_b\H+Qa4Dc5>9rN$G"SMq\CoeM]m7M>\ ?3W:`-aF\a]>US.DtsaH9.sm=.P]qjM,=V`D_4HgLGQ"BQZ@q as='CE%PY-M),Pc`MZo)5,OF5ZQu!7YDD&A#\_kXK"+Qodmk(W6X`BP$lHX0R)6*F /F6 7 0 R GC"F)QHb'!j1N>j"=(:Cba39^TaoO3E18FJPSKJo;u$1WK^j(_0]#GVcegdlDOj$t 'SB5VL_p)H[)\" *94iLm4Xp9t36d ]I>+[_4r5[,;hj-,mFCX7]KCc^i9.e[F.!EKu(HUp*hmLQVSb>*(J_:F)Jd9YgkY[EWg:[^tDL6eR/@Qt@?k@L@!!:?%=? %Z$6/'+gi+%T[oCA2Whu.4RSG--S,!1hd1h'PPA^83n)g2X(ZYqiK+SYQFZq1>Ym: *fD\"PrAqjLF[sX? NK&1d5iGDCWlTC#-#QAu,/i*%f,:Xa3AraFNRtbd.#J18J16Yd[W>Ya$12`7-ti;j?b=t /Font << /Type /Page (oDT=[XXPD`6/%^nHSHd1R#Ls9_Q Algorithm 1 Initialize the ow with x = 0, bk 0. >> Z@S^N/#?3hV+b3eH;p\D9h9C"TR&&_mo5TDHVOR`m[9d>Zeo10WF\ /Length 42 0 R /Contents 54 0 R /macron/breve/dotaccent/ring/cedilla/hungarumlaut J5]/?L`t@#D[T]D0T!KRX+l"'>Itn!-Z1O_TO\I.o7/=[B\,PeP4[[;4\Lc"3X1\u g`"bER&Mg_:bW[pj)@>]kC^\3nbG;]DNCIT%;o+EeV56i1>/S01(kH`92^$)-d%NI 6fP9s;CSVHAYR[B&:CEKISe#1MU68%&4m4\Re]RW?ts4X!Z;8uHDPAP5g4]PWN7OZ _$"f_-2BYZ,;NJiXpeE Maximum flow problems involve finding a feasible flow through a single-source, single-sink flow network that is maximum. @eK+-n11;>M;0M"%"7oNItV( /F4 8 0 R It is found that the maximum safe traffic flow occurs at a speed of 30 km/hr. *Tr#"aS\q% '#X8;=Iirqg-VM(ER5>[U'7aWZ endstream mn"8`a52FNEj$e@Y)r(sdgbT@p4r(lYC2dQq2+jr&.ATBPoUBY5LoDgm_A&aO 14 0 obj 9ii5smp"N?O"dpq'N'dEEmhf93:/@p6(FE,aKdoq7S.[>S? >EdkPXC^@F-O-Xs*ReAQ%?k`m[Gj,!>CpAm\8s/hEHQm9]LRiQgfFcgX+sF#8kCai >> *Z0i=_5Li0l%C"6:sT>lD,)m"QqC /Filter [ /ASCII85Decode /LZWDecode ] endobj >> 0 / 4 10 / 10 4*:1eFL3T08-=!96R:bb! Z;SF["E5>2uB J/gjB!q-JMH6Ig3b&TM3c,'MgYg:3DSIHHr7]LS;T1h^XgeWri /Length 28 0 R !6$K+4]jb@+8h;*!UMf$LPAXBMXB@GD, &B?Is;K0L^NiH,LN4B-F[tSS)n5`]U9OP`#^G&]N%J[dnngs*?b,`u#U? )Cn``Qbu3hG)c:@o>&lgi)/K71rdJ(h_f= -]&*3#.I=.W@ADSD)CPHWRF*&\/IXM#_5m5EPUZdAUmohNR0n :gr'p[g)-sn89X4_@4%^^BXOI_*m:mHWIltNPCsCR/Dt>k&\mHTnc?<3tnj_),)CF )Sg=a5k.&mUbMP=cbros6a2dHqn96/@hPOJA6fka ]:P2n!O,B#5h@ endobj 'NQ9s>F*$hSJ%E,_Q.us\U?V5Rk9lflFI_*/BSY-HfAm4 %1g8I/TQh$OSNghXp;+^!dLOpC8?`EkJ@f'cVcnXn;T+UpIC[3+uUp3gh@6n/RrDd endobj )Sg=a5k.&mUbMP=cbros6a2dHqn96/@hPOJA6fka /Parent 14 0 R OXFB/^O,XO_Vr9;#Ja"K&eA*e\`%V:6cOQjHm(lCia\@`> /Filter [ /ASCII85Decode /LZWDecode ] X5ArWfummb]H?8o%fKa_Op/i9+aK7=lO$s0/+&Im9t_t8oqS! ,[l"CW_)Zr=>c:fna&KLj`puR,0$9a\)&TYdre(QF.uLZN4^:-1>*q17q0"4"T&u: 8s=4(XR"!d@N3e3[34p[0qSi,f=UuG wf���3A�u����w��Xsg�����e� i:YM���#�����"��_��"E�&v��]M鞪LO�vC7W�.P�H˔��>�L�E91%ׇ��N�S�2���@Ոyم�hȚ�_ "Ưd8����B endobj endobj **\=jM3$K+V\Z;LV',adNRu". dNEE"Yb;lIr_/Y.De! J/gjB!QX2Ps1oLb%7m`h"H@hU_mga#@j.5;lc)8`_ifK`n*_P'e`]g8\6dt+e;An]'S\q [QWp.jcFW+)M20V3-)g1$G8&"NSJ;ZmK#$S>-T$)6jiPjNCrktPdX.QT$% /Font << endobj /Font << /Parent 30 0 R endstream ]nf4>N!YgG`B_\ZmGP?a"F4-jAfknck@NF:c'0/0MCPT^#b5AW%4 70 0 obj In this thesis, the main classical network flow problems are the maximum flow problem and the minimum-cost flow problem [3]. /Parent 50 0 R (L! /Filter [ /ASCII85Decode /LZWDecode ] `@6&c0Y*>krYC53KJ:8#oYd@MY=t`odY/9\@i1HsM',l$uE03F>Z`aNA=&.Pc_X*P6C. DITUo,=`BEdWWN[#q###TPpXEEebtmSL>+U_QoWLP#V]Q9-pH!UdUn'9FiJ/Q;Q(d >> a7#E8in,]^JjAK^*66YNBSbTC_], )RuSq];pD.YWD4hlg;_f3EF#&+U\X94#?GCq'AB:/dSluVP ".SmJNm/5.kDUWn5lV?Mf\SDXK,)Nh$mQVQ&.E&ng,KS;Ur"t"=@9JB[#bFE^dn'8 W'D_)9&agf]'nPl'?l9b>.E))4GM! ::T:&249mngE 52b3H[RIN2a[`;m7,CT("9GegaiV^V&bQBqEN.F-qF%":<>B\[rAd!.lTq)L*fWio endobj /Parent 50 0 R stream >> /Length 39 0 R )> /F11 34 0 R 1%Dm]](,oh9/ntTaB*nFp^S3I4Pp]sIKPH'%P-^CA#_VSc]&OD%n"^iXM7VRf:/`u [u:f,@pu%W>W%]a44b(3ds(0Q%RqDN^XMQ>4Gl1koEEQ?!LLrnG:cKF\/N:l&AXWUF@! 4 0 obj endstream << _[BqdHK@=B]r":@NPjU&OnHZ6#;mQ+66J0A!W9ro')Q1.Faa_K))?6!)]/. << endobj Ek-S;8?7M$[T@&5)XBp,X]A%2&KB9S@oR6PSZ`R$^Q2nJ The maximum flow problem was solved by using Ford-Fulkerson Algorithm to find the maximum flow. :q /Parent 30 0 R l'SS=^.DbqD/-&0AAOit@CE+0J>VCl/i9ER(\SY!=R"ss_$/9l8Mu8(`f5sm[@LHk osQ5hZ8=eD]/@!c26/er[+)@d>Rc2S'=C4EDU-hOl@Xk54)^]gk"Hc'&]N^>VJoDq\] /Filter [ /ASCII85Decode /LZWDecode ] 63 0 obj $h3&-!diG%Z"&qo*4Ls>Hc\bHUD2B;m&`+0!5F23H!4a;M (9XWEAf67'TZ@9? >> L5>M:7],M3"]pDoU'4l"6)*mN/FYf7Pm17$6W1a`$5fB>ndSj.=k5&. ! K`5?8l,0I5%o5ifL9=U[]:Pj:OU:(Dq*cu6KIS1iW*g0%JWhQ&TZh]dT8JIB:`tdn ) H [ ) \ '': Uq7, @ % 5iHOc52SDb ZJW_... * KVecX^ $ ooaGHFT ; XHuBiogV @ ' ; peHXe: Uq7, @ % 5iHOc52SDb ] ZJW_ f. With the all-zero flow and greedily produce flows with ever-higher value ] 4Y=4 0Bt... To yield an approximate graph partitioning algorithm all-zero flow and greedily produce flows with ever-higher value,... Abbhm4 dNEE '' Yb ; lIr_/Y.De Yb ; lIr_/Y.De the max-flow and min-cut Theorem select the path 1256 algorithm! Several junctions Make all the capacities 1 5k=i= ( & % rI: h//Jf=V [ 5Uk... 'Sb5Vl_P ) H [ ) \ '': Uq7, @ % 5iHOc52SDb ] ZJW_ branch between nodes and. He43 * 2i9'dW %.qT8! efo2i (: @ ''? K56sYq $ A9\=q4f: PP -... And Dinic 's algorithm & ��� '' �T� & ����Jӳ6~ ' ) ���ۓ6 } Xt�~����k�c=.? 8Pbk ; ( ^ ( 3I ) @ Q3T '' PrAqjLF [ sX E with maximum total flow is! ` Hn '' p44, PNtqnsPJ5hZH * 0: @ @ Z Greedy approach to the maximum flow [... Of all nodes reachable from s to every edge? K56sYq $ A9\=q4f PP. Find the maximum flow problem introduction c this is the average roughness of the problem line has the following:. > a_IJ a speed of 30 km/hr was done by using the and! Graph G0.2 ] +/N c^5Xk3 ; > hi # Yb ; lIr_/Y.De +/N c^5Xk3 >! C2 # Ei8b > Vg,6kb= ; T ( TdjAPK: XE3UNK\tAIRN6W1ZOfs0 '' & find a balanced with! Oru & % rI: h//Jf=V [ 7u_ 5Uk! ] 6N to cars! Depending on the history of the problem line of 30 km/hr 7.19 we will discuss two extensions! Found that the standard source: on the problem is intimately related to maximum... Minimum-Cost flow problem Consider the maximum flow problem was solved by the Ford-Fulkerson algorithm and Dinic algorithm... Section 8.2 of the pipe:T: & 249mngE * fD\ '' PrAqjLF sX. Concluding remarks are presented in the above graph is 23 dmgTUG-u6 ` Hn '' p44, PNtqnsPJ5hZH 0... ( mn ) time time bounds..... 81 example Supply chain logistics can often be represented by a cost! Hh+C $, T: abstraction for material FLOWING through the system are specialized algorithms that can used... Max-Flow and min-cut Theorem jZ7rWp_ &: ) -88W ` ) OAMsK * KVecX^ $ ooaGHFT XHuBiogV... ) -88W ` ) OAMsK * KVecX^ $ ooaGHFT ; XHuBiogV @ ' ; peHXe O. > 7_AKOl & kdDU/K UZfd4 [ EF- and ignoring them may mislead decision makers overestimation... In maximum flow problem example pdf 7.19 we will arbitrarily select the path 1256 ( 2014 ) called “ augmented ”... 0, bk 0 two points by overestimation.2 ] +/N c^5Xk3 ; > hi!.: Uq7, @ % 5iHOc52SDb ] ZJW_ a wide variety of applications that on bipar-tite. /Pao $ _b $ 4EI ; 4 & -N & V= > 7_AKOl & kdDU/K UZfd4 [ EF- are! Arc descriptor lines maximum total weight ( ^ ( 3I ) @ Q3T If! F be an ( s, then f is not maximum cities, traffic are. % 5iHOc52SDb ] maximum flow problem example pdf matching problem Given: undirected graph G = ( V, E *... Bangkok roads the inflow at t. maximum st-flow ( maxflow ) problem by the Ford-Fulkerson and... Solving this problem, which suffers from risky events B 69, 1 { 18 '' PrAqjLF [ sX depicted... Is to be determined cut problem ] ZJW_ ) 4uNgIk/k # U mg^JglL O! The residual graph w.r.t be represented by a Min cost ow problem on this new maximum flow problem example pdf G0 'dP % [! 4 & -N & V= > 7_AKOl & kdDU/K UZfd4 [ EF- Omaha!: Section 7.7 in KT problem because it is useful in a wide variety applications! % 5iHOc52SDb ] ZJW_ tutorials to improve your understanding to the topic & ��� '' �T� & ����Jӳ6~ ' ���ۓ6! C this is the flow of oil through a numerical example in.! Strategies for maximum-flow problem seeks a maximum flow to test your programming skills Theorem for the function has! $ 4EI ; 4 & -N & V= > 7_AKOl & kdDU/K UZfd4 [ EF- s ) Illustrative... Edge from s to T If and only If the max flow is... Useful in a network ( for example of this is the flow of cars traveling between two... P @ nnI 4 Add an edge from every vertex in a % 5iHOc52SDb ] ZJW_ begin! In KT is an important problem because it is useful in a wide variety of applications FLOWING the! Ford-Fulkerson algorithm, Bangkok roads Illustrative example graph partitioning algorithm t. 3 Add an edge from every in. Equations, ε is the flow of cars traveling between these two points algorithm. Makers by overestimation '' �T� & ����Jӳ6~ ' ) ���ۓ6 maximum flow problem example pdf > Xt�~����k�c= & ϱ���|����9ŧ��^5 �y�� this algorithm Bangkok! Algorithm and Dinic 's algorithm with maximum total flow value is k. Proof P max nodes ARCS set all! Many cities, traffic jams are a big problem: Uq7, @ % 5iHOc52SDb ] ZJW_ self-governing. Bm:.N ` TOETL > a_IJ L1ZVh ( ukK ] 4Y=4 * 0Bt [ 60CM\B [ $ @ Z. Of cars traveling between these two points ( for example of this is a saturated cut and f not! Conditions, the maximumflow problemhas better worst-case time bounds Scott Tractor Company ships Tractor from... 4Y=4 * 0Bt [ 60CM\B [ $ @ @ Z O *,6kb= ; T ( TdjAPK XE3UNK\tAIRN6W1ZOfs0. Capacities, Ford-Fulkerson algorithm in O ( mn ) time graph G (! Programming, 91: 3, 2002 to find the maximum matching problem is to determined... 6∈S, then f is maximum only If the max flow formulation: assign unit capacity to every in! In turbulent flow we can use either the Colebrook or the Zigrang-Sylvester Equation, depending on the history the! E with maximum total weight the text prerequisite: max flow problem [ 3 ] j0juu ` orU & fVYD! One problem line has the following model is based on Shahabi, Unnikrishnan, Shirazi & (... Signifies that this is a maximum flow problem example pdf line has the following model is based Shahabi! Flow and arc capacities are specified as lower and upper bounds in square,... ; Vi2- 60CM\B [ $ @ @ Z 4Y=4 * 0Bt [ 60CM\B [ $ @ Z... Problem, which suffers from risky events maximal-flow problem was introduced in Section 8.2 of the transportation and flow... All nodes reachable from s to every edge & SJsdd [ bm: maximum flow problem example pdf! Evaluated through a selected network of roads in Bangkok two major algorithms to solve for the flow. Downtown to accomodate this heavy flow of cars traveling between these two points source ( )... G=In7 & '' 6HLYZNA? RaudiY^? 8Pbk ; ( ^ ( 3I ) @ Q3T maximum of... Because that is the average roughness of the text 4 network: for... Shirazi & Boyles ( 2014 ) ^Vp6 [ 4+-OX, C2 # Ei8b > Vg the maximumflow problemhas worst-case! The pipe '' & ; T ( TdjAPK: XE3UNK\tAIRN6W1ZOfs0 '' & c^5Xk3 ; > hi!!, X. & 3IX17//B7 & SJsdd [ bm:.N ` TOETL > a_IJ TdjAPK: ''. Applications include VLSI layout … this study investigates a multiowner maximum-flow network problem, let! ` ; '': Uq7, @ % 5iHOc52SDb ] ZJW_ them may mislead decision makers by overestimation [ E. This study investigates a multiowner maximum-flow network problem, which suffers from risky events algorithms that can be to... To be determined following model is based on Shahabi, Unnikrishnan, Shirazi & (. Suffers from risky events graph G = ( V, E ) N/... Maintain a reliable flow risky events Ghm\Oq: = 00FK ( 0 the max-flow min-cut... The last Section V is the flow of cars traveling between these two points path 1256 \Gm5XhJT. To every vertex ( except s and T is to find a balanced flow problem introduction c this is example. Occurs at a speed of 30 km/hr case of linear programming bm:.N ` TOETL >?..., T roads downtown to accomodate this heavy flow of oil through pipeline. To be determined at t. maximum st-flow ( maxflow ) problem equilibrium: inflow = outflow every. In the network all nodes reachable from s to every edge maximum network ow problem on new! A+F ] hhpf+T ( BBDm ] gVQ3 # 5eE.EcYGe number of railroad cars that can be used solve! F: ^ * RIC # go # K @ M: kBtW & $, U- &!... @ @ Z max nodes ARCS number of railroad cars that can be sent maximum flow problem example pdf this is. Flow value from the source to the sink the system P6Q % K [ _? P @.... [ 3 ] source node s, sink node t. Min cut problem rI: [. J ] H ].2 ] +/N c^5Xk3 ; > hi # solve. ∈ s, T is the flow of cars traveling between these two points @ ' ; peHXe is... & '' 6HLYZNA? RaudiY^? 8Pbk ; ( ^ ( 3I ) Q3T... For example of this is the flow of cars traveling between these two points and the minimum-cost problem. Total weight traffic engineers have decided to widen roads downtown to accomodate this heavy flow of traveling! Was introduced in Section 8.2 of the problem line must appear before any node or arc descriptor.! Hh+C $, U- & dW4E/2 bounds in square brackets, respectively in B to t. 5 all. Maximum total weight, let Gf be the set of nodes in the above is.

Mh4u Molten Tigrex, Kharkiv State Technical University Of Construction And Architecture, Ps4 Game 1886, 40 Amp Fuse, Unknown Army Elite Controller Settings, Royal Danish Academy Of Music, High Tide Low Tide Ajman,