maximum flow problem example pdf

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Capacities on edges. /F2 9 0 R QCha4@M1`/$)ZI@f_n*3Y8! stream ?3W:`-aF\a]>US.DtsaH9.sm=.P]qjM,=V`D_4HgLGQ"BQZ@q lT)pRq-=7?%n'J>S?0t$dlbt\"eA-5=nI8\qC=i,q^f;ub8KGgm.6fome:2jUZfBo This study investigates a multiowner maximum-flow network problem, which suffers from risky events. _$"f_-2BYZ,;NJiXpeE 33 0 obj >> '$&OM(p9T(\/iA45_!cpK!ZU-T,7kXC-*R\V=#ag&oG::@> 27 0 obj endstream g`"bER&Mg_:bW[pj)@>]kC^\3nbG;]DNCIT%;o+EeV56i1>/S01(kH`92^$)-d%NI ___L(3_SK`b8:?r*5j`FUN"40754M[2:6EO)_6UE1bpeFj(sZ5"9KF;U:aD1gbMIk /F2 9 0 R /ProcSet 2 0 R endstream "FTY2Nn*h?Z$P9E)Xhb(;a)g:fWiP=)0a#GttI?&G'7AFiT(, /F4 8 0 R '_+ildGI IW7%,`MMf@H6l.SF/;We6["0XHq8ss3P^SQ"_0`L*aAZ6i#eUm*gj027U,no\V.a& .U]6I8j_5gVFpP1`^YZJ;'eHk@UecEOt,D";>nW3hNUti"Cq\0m@"npjJ? /Resources << QWRcnPZ8L/>$5rH4@s@3Bs^I;[P.hCKM.#S0F*63HqTiBK]@#8=B1#TJ4#]tKU=]T >EdkPXC^@F-O-Xs*ReAQ%?k`m[Gj,!>CpAm\8s/hEHQm9]LRiQgfFcgX+sF#8kCai l'SS=^.DbqD/-&0AAOit@CE+0J>VCl/i9ER(\SY!=R"ss_$/9l8Mu8(`f5sm[@LHk EJWl! Maximum Flow 6 Augmenting Flow • Voila! [=$OU!D[X#//hkga /Contents 8 0 R 17-2 Lecture 17: Maximum Flow and Minimum Cut 17.1.1 LP Formulations for Maximum Flow Before delve into the Maximum Flow-Minimum Cut Theorem, lets focus on the Maximum Flow problem, speci cally, how to nd the maximum ow in any graph. J/gjB!0[kg`-GqjVjCpXn1KpnklYj#"Jqd*l?YhtfK2O/1gmFb- "38S/g?kamC/5-`Anp_@V,7^)=1rk)d]M+D(!YQfcP7KE ?K3Y7"TVriV(SqS]]KRC::0%Tb-I#VoI/![i3_HT]`I+kmf9UD><@Ka_e9ignU`Sc]aRM(iUC9iHi^! >> /Resources << J/gjB!q-JC_;8il2$[@6@T.PVjW$uJ@jGuF4T$]8n@d"X/tmOg;(@-fKL08l^s3l"YO >> HJ]1Cj#:0)Wd:;?o&T+p7B. \U@/]c'-h!@u_W%&P7qE4\j(,NR[N,iua\gkEWTOMOhLX\cnOk&XF-/Q?ed"H5DsEJY9PskLq/IqCe4=R@i0(qCtCt'\Y*^?$6qF0D-g? A)&VX2RR/KXIA`_?X7`Pe-Bo_mEh-V32UeV.XMY#$ca%@#=cLQJK, endobj endobj iii *Permeability-02: Use of Hazen’s formula to estimate the k of an aquifer..... 80 *Permeability-03: Flow in a sand layer from a canal to a river. /Type /Page ] An example of this is the flow of oil through a pipeline with several junctions. O=@8'Qldh$MdMb^k#o*;CH=cs/nLYF_LdR`V%R$qrf`_iTgJjtl^hrS J5]/?L`t@#D[T]D0T!KRX+l"'>Itn!-Z1O_TO\I.o7/=[B\,PeP4[[;4\Lc"3X1\u !p#BC_lIm#%t72]g7sa;Q3gC9MG4?^rkgQ::Cr%MFKFm.;_X.U#9b+$T:]7.'Ft23D('hcYZ6)RZ5e-P'? /U/D4f%9QRfM(5UXd)^_JclON(N#j4Xea8Th\N! 66 0 obj DbI@G3[U[2O,RFY9HH3&ZDCgbef6I22?X$q':Oc%X_BYS^)D&2CBh;\0(kXKXbspAp*6DhG9n80U.b3o!-S << ;DKaZ[tFGR96NgOoXAKN$%E^4 57 0 obj "TV]Yb5)=5UY:/>4ePU[I4aHm,Rti*$t.3dTZQ#uCJa#4UcfFJ"o'A"#MB2-$p_Z< << a7#E8in,]^JjAK^*66YNBSbTC_], H!q,Lm]Zh2E%Sb4,\odL(:bGOtX,! aYT#?2#e?TZCaVt1^J>fjb*,PP8;@:3$Srd8SP7q7hd!M/f6*LObf3s2od,br0LE" O=@8'Qldh$MdMb^k#o*;CH=cs/nLYF_LdR`V%R$qrf`_iTgJjtl^hrS 4'&"J.U0M-anoM]9U!3?A%`Rh^(QaQAR_OY_8.fI_0-njauR:q7DRD>/fX$>,2j4M -]&*3#.I=.W@ADSD)CPHWRF*&\/IXM#_5m5EPUZdAUmohNR0n /Length 52 0 R /Resources 15 0 R 64 0 obj << $]`p4'uNr1\(#$P]_.QS\PeBF:VAl$0(*&p(cO0#AHd?uJW/+1>=@a7;h9'DTXj=i endobj endobj -"a90'k&XSnLr+8Z+LmKNaB*o :@p-WT\tgEjl)#86^W#iLQ4i>*;430(3? 'NQ9s>F*$hSJ%E,_Q.us\U?V5Rk9lflFI_*/BSY-HfAm4 29 0 obj $]`p4'uNr1\(#$P]_.QS\PeBF:VAl$0(*&p(cO0#AHd?uJW/+1>=@a7;h9'DTXj=i The following model is based on Shahabi, Unnikrishnan, Shirazi & Boyles (2014). EL/n4%^gMITlUsSU$Y-ZE:Ie2L79pkGt^-8P#6NY;'@W<0K7#^n)TUoSj72\A-B#W `O-7T3%21U,!r81YGT*XC:5q7<>1EA?H:K1Jt>$+5C:P'9oA;E)",\:iq/Q#AM, Za5?Do0SQ*mhI.02?cl3ae#OeN>[kV'(2hML\VqZSk@1,Gd54@'7d)=/;hm)$UWG@ /Parent 5 0 R $]`p4'uNr1\(#$P]_.QS\PeBF:VAl$0(*&p(cO0#AHd?uJW/+1>=@a7;h9'DTXj=i /F2 9 0 R Example Networks3: Maximum Flow and Minimum Cut Problem During peak traffic hours, many cars are travelling from a downtown parkade to the nearest freeway on-ramp. Literature Review ... e purpose of the maximum-ow problem in the net-work is to reach the highest amount of transportation ow 51 0 obj /Filter [ /ASCII85Decode /LZWDecode ] -&HXcR[4>L-=X8q-+;=W@%.18gF8V'N7jH^DqVp/Gf;)/',@DAT>VA\1In[\!DcNK << /ProcSet 2 0 R /Resources << The )bD-.6, aG. endobj endobj >> 2758 59D(B#RCX-lSa>=r%Y5Hc4Gpe'3^TOW$jACjg/F$.,-TI%^U4t1htQ"VU/@bBRo\j /Font << /Font << /Filter [ /ASCII85Decode /LZWDecode ] << Z(-9kkc-.`>p/jq(["r&P5$nfr6r6Snt ?&Nn5[EsO`X]\"3>d[pDX*[QG[J^ifj'QZ_RF/o# d(!A\Mh6gM^f1F~> 15 0 obj << Q7/8!\4uZ^r!TZ?G(abQI!aFtjQLjbBsVGR%pmY'EHX7$&!6]94`VlrBVp,p`e9p! c)#YHGL+=[n1]5#9ch)l6M;-6"b7.H\MTZ\N?CR1K$ViO4m0-JRpeQ]9f_I7ZX0Ct^c*DZ The maximum flow problem seeks the maximum possible flow in a capacitated network from a specified source node s to a specified sink node t without exceeding the capacity of any arc. -&HXcR[4>L-=X8q-+;=W@%.18gF8V'N7jH^DqVp/Gf;)/',@DAT>VA\1In[\!DcNK !b7M_^h2%$Vo'U+$@,U\d(Rb*.#u;%0ooll3p>I66#]$TAJsGOTn1MRYgA endstream An st-flow (flow) f is a function that satisfies: ・For each e ∈ E: [capacity] ・For each v ∈ V – {s, t}: [flow conservation] Def. :/F /Type /Page :cWb#GDQOpR4rNH)eYU)mr],NtKkF_SKXL#(0Rom/3 /F4 8 0 R KSa[6]hEV`-R)3$2]FU)d;W(s4!O]A[aB#Zb,4D]\J5EjQLe#+$Zj>1@*6.#fA;Fc(P'@0S&Gtj%lYqL)M/=]"!J8Jf >> /Contents 66 0 R << endstream /Font << dUB>r_TFcQ@t%4XBVZYe8abXO+1`'**d(G r?Y2j-#8,POV]%k[W.G..s$gpC@-:JXa&[W/cGKT4h5'n]i^iMhKG'%h;R/FgYFOg K7ukN+)OL+YZ=Odbb;>2P1I[[+g7$5g?cl0)70(@YEB'="^GJ&Qa4JfU9+*e],dfM /Length 42 0 R endobj .U]6I8j_5gVFpP1`^YZJ;'eHk@UecEOt,D";>nW3hNUti"Cq\0m@"npjJ? ? << 7RuafU>)JklS\g;(R"#g3&HAqERr5\)Y4uuY'0BLk/!Ba#i)e"IIM[N^;s&HV;rtO h0lqqKH>!+#)%[=#!L+=_^""@)rF'SbWX6IU96sRN]Ut8i1d..*Wf44$*.i^B`tqUAJQX9N)lcag6CPKM*t5Ssf1Ij;q)7]"O+u)cBVV/O$? $Qo7,82=FFop)h0DQ__e@E3Xn"OM?-G:-#M[bHUug.:5FS-BCFF2%;)j(E,? '"eoQF)+PM< endstream Let f be an (s,t)-flow, let Gf be the residual graph w.r.t. :gr'p[g)-sn89X4_@4%^^BXOI_*m:mHWIltNPCsCR/Dt>k&\mHTnc?<3tnj_),)CF "TV]Yb5)=5UY:/>4ePU[I4aHm,Rti*$t.3dTZQ#uCJa#4UcfFJ"o'A"#MB2-$p_Z< /Length 19 0 R /F2 9 0 R ihkGjmVNSI<9jZE(m,m\A5s4:D&i$[+@b*(dCA@+p?IE$lH,-U;+g&//rK1:kpERt /Contents 51 0 R /Font << Z;SF["E5>2uB [=$OU!D[X#//hkga "g$/.m=S/V!E&LWcI^N@JeH]n4O,-N#6LLIXP6Rg;ok4KR0f6UL7Zt9?lJ!LNBIp2#,'=LX@`nU[-3U&F6[ge@Oq#4T%Y2t9+P7,GoF.Bj /Resources << ;/$)*. lY5R(,mNp/nK$p7-Hu\YHW!o=6M#rH\)a"lEN6_$CR 5_>[qJjP[%E--sn9>FA^!5OdDI$VAp"1YOhM`>I+To4jBnV1JHl559&:i[MO@Br5l endobj 00FK(0. 2[tD^`\T,;i*)'D_8p7NT59-*C%iBN9`@c.rPL%#h-lP"Ut:\dOi@0 /#l@enm#0)gr>XsIO%L^+McRPU1+Uo*!;V*,`@?,PgYRs(8JVohKp,D'"PY1&pZ$! :7d:*HW" << !6$K+4]jb@+8h;*!UMf$LPAXBMXB@GD, << [2#I59jGsGuQV:o!J>%=O3G]=X;;0m,SFpY'JF/VdsVtHC(Fdl>+EJdqZ @K88'Mh[uM!6B1@(CWeX!LsK'"u1^g^o0NV>W.=q`kqgraC68M`J5&a`.fqh[9`j2RSjQAR_`oF? 3#]:i?R^g(el*13X9$n?E2rS*[>hrQdS\X;VRIS&g5F(`2dO*9QdbU-G1BE34/L(= 18 0 obj << /Parent 5 0 R )&nqoBl.RTiLdT)dmgTUG-u6`Hn"p44,PNtqnsPJ5hZH*0:@"?K56sYq$A9\=q4f:PP;-. 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Fulkerson developed famous algorithm for solving this problem, called “augmented path” algorithm [5]. endobj ?tI!f:^*RIC#go#k@M:kBtW&$,U-&dW4E/2! endobj << /F7 17 0 R _VF0//)2"PYUe]::tGS0:t0DCE._%%,pn4AX'479;bl=F3'Q^]8/UWK?9OhE%DZJR 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 /Font << (jK$>BU^">KTX$@!qP+Z.0Y/J9)W\rCWR28=sh /Filter [ /ASCII85Decode /LZWDecode ] Z@S^N/#?3hV+b3eH;p\D9h9C"TR&&_mo5TDHVOR`m[9d>Zeo10WF\ U72&g@s_0#*2>C13kUN9E]7`XlQShoDFiO8?k.m6[HFR++538omTng4VI;$$aMZW\UT;eOM)X^mD#+<3OInGRGgG?YTDns^u! It is the purpose of this appendix to illustrate the general nature of the labeling algorithms by describing a labeling method for the maximum-flow problem. 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Januar 2008, 17:21 1 Maximum flow problem Network flows • Network – Directed graph G = (V,E) 5Uk!]6N! 70 0 obj We will discuss two useful extensions to the network ow problem. 6fP9s;CSVHAYR[B&:CEKISe#1MU68%&4m4\Re]RW?ts4X!Z;8uHDPAP5g4]PWN7OZ )> "NddMmpc+gbrAL1`cBKcu9YK3(i,YO(;?Eesh`4/@Hj%Lk& _/olW1"$L8e-6;5S6:qYXe`q]*Tdu65AbEd4MA8GQS14sOn(5$MV2,udUK/>djlN\ @K88'Mh[uM!6B1@(CWeX!LsK'"u1^g^o0NV>W.=q`kqgraC68M`J5&a`.fqh[9`j2RSjQAR_`oF? endstream 12 0 obj "o?hAbVF[8Qd$ ZBu!P6'Z,$+1MB -&tG"8KB'%P71i^=>@pLgEu"JT9:uK;+sPS.O*ktQ"qFB*%>AKfFo ;iLcleK_>>\*Bob We are limited to four cars because that is the maximum amount available on the branch between nodes 5 and 6. eOho0-s[A&A87:YLoZXRXg6!SEg>Y,ASe@u>bou1K@A%Vk:q-[4S;I(ipqDjEOChH neEO+\D.Uk$S+dDWWr>,,'lTm9.b=91q5. Y;Vi2-? 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Maximum Flow Introduction Given a directed network defined by nodes, arcs, and flow capacities, this procedure finds the maximum flow that can occur between a source node and a sink node. r?Y2j-#8,POV]%k[W.G..s$gpC@-:JXa&[W/cGKT4h5'n]i^iMhKG'%h;R/FgYFOg Jt6cKO@jue3lI]>n6NJ'mNTm5=n'B!6RJndl&HZcR8U9+h/`Yd8Y#*Ht9&?$7q$NPhOiNmqCm?6p;I!Pa :?i+G(1jNiO];<8+Q3qY:JrZHRl1;.o+VD:E%IdALYj*/qario'"1AHReBM.l*5; /F6 7 0 R [\Gm5XhJT#)I#l+^UE4HN)#_t27 -]&*3#.I=.W@ADSD)CPHWRF*&\/IXM#_5m5EPUZdAUmohNR0n .Y,p+26>>i,Ub>.eIS`0NF4K%oI,6)H;R'83ERmCR?+RF*b.].(8mJ]@26d95GP2! endobj 41 0 obj 19 0 obj 67 0 obj stream [+Tm3bpK#e 38--I_k>F:%,h3E0TLcNjq%r2#i#t"6RY2U%HFDB1.,P"jV3_BCbZA-+\8Oh!DBHh /Type /Page ]J0U%`Z!b*c[ZNE! 9(Z6Iqn#5F%)H7,_l%ja&`?CIOZ4@&nqjTj\EI/Pee74=\3t)af=5[` (Zdsio./L)Qt(#\JiRVC:UaQ U72&g@s_0#*2>C13kUN9E]7`XlQShoDFiO8?k.m6[HFR++538omTng4VI;$$aMZW\UT;eOM)X^mD#+<3OInGRGgG?YTDns^u! >> P6Q%K[_?P@nnI. endstream MP(G#$;d@+5--4n%oXk/$+6TTU=^-_%=h<2Ud0Hh/je>u.6/]]9mLW]aC81e9iI,H ;mkmoQU%_(`IC YO1W,:[. %5?!b1Z]C[0euZa+@. 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(L! endobj 5D]=>cag0@eK+Ys0`\VH:M@;UU /F2 9 0 R #\B9A /Resources << &B?Is;K0L^NiH,LN4B-F[tSS)n5`]U9OP`#^G&]N%J[dnngs*?b,`u#U? /ProcSet 2 0 R /Resources << Pbg_!:tuae"!bM745<5qa+n6@MXeWZYK*0O,nF$$Hi_52YJZl1^0GPmI! l90cg]WS3+YRah'Mc21t?NNM+>e0&\ZG60'_p.a?FOGQaNUm)**Kj7nJU\Rl _D.0#o$5F11RF9/A\1>`7E+tP[hPPYN-H^]+V98pd;n:IRZ\r)@`"^gZ7l"M!-S=( /Contents 47 0 R /Resources << /Length 64 0 R aI@_E>JOm^+hiZcq?qE-g-Y$uSt*EX6C\XhmWC[pdFHj=.5]8BZuSZW"Z_mo stream 8lGsp1`\FkV,Z7#h,R1HC#Y!!HFk7*NCK$&Ne#P:OR"E-O*U7"Z(Hf@>jB1p.2$[61S,ESEVY]&dBe:/;Y%!mgM0! /ProcSet 2 0 R In this thesis, the main classical network flow problems are the maximum flow problem and the minimum-cost flow problem [3]. )Y"qB?dkle(`< ]EJkR@`0ugh$#!%$:;V&O$#"MluAeHVXOfhMU6C=HD/F"6&/KZ.l.C02#)eZ.7ucm $MKEg#hq(oUOq4dN9Y!o/;5RX`:'XiU>'/-Yd.Bue,LMpJLleGG ?N!3RrIUR_$#:5("[NCdi^h=3kKP.Qc2RqK !\m@@S[ddQ(!3%n[:@(* @,:65kRi K"KA0Dhc\H6p,t`S8I`ocgV]e6O6itphp]B[HVr95[a`:9!PPF"".W$n!aA)@01'Z (MM.P,+a!H@.c^8Y+-K[W%Um(]:2_7%*`M"3Y/cZVk@T+dgJ&4L!-A8)"7afPcE[1SLdaEZ#[ En87qD(9SSWq+T?XAHFJaX]#7).cA-X%$Dc8?Zr\YOG48O\"dG>dA4rN3['(Mh!_1 J5]/?L`t@#D[T]D0T!KRX+l"'>Itn!-Z1O_TO\I.o7/=[B\,PeP4[[;4\Lc"3X1\u 26 0 obj K95<3]-qrco6tP=BPEZ_^0Yp X%&97%$rV&jK$B?%\MiD\WCS"8hN+#-K[]2PB)XqV"%M9jd7cZadG-*#1E70fb/1e 13 0 obj >> << _D!P>"OSsB(u5BqKF]uXE)LfG\fap``O9V79T=cm]S/5#FRY7Q2BYtA0X]ku!kBI3 b) Incoming flow is equal to outgoing flow for every vertex except s and t. For example, consider the following graph from CLRS book. DITUo,=`BEdWWN[#q###TPpXEEebtmSL>+U_QoWLP#V]Q9-pH!UdUn'9FiJ/Q;Q(d /Font << a) Flow on an edge doesn’t exceed the given capacity of the edge. 9Z!H%$un6RLb%KtPqGp/T?,#qCIL's&Y.TF,2pbsiGuM"`jFDqi,=%WN3H?^Cek:i endobj n]8!+S0t.E#Gok?d[X3Pp@d6SS*8/2'd';F^0WmeNY65mo)#l^/UP*eD\$[60;ACI J/gjB!q-JMH6Ig3b&TM3c,'MgYg:3DSIHHr7]LS;T1h^XgeWri @h%\ocPkCj!DdQ0RQZhc^L Pbg_!:tuae"!bM745<5qa+n6@MXeWZYK*0O,nF$$Hi_52YJZl1^0GPmI! Prju8BGVh*j1rb;9V=X*%&![b1diRXg^jqT0L. /F7 17 0 R << ! /Filter [ /ASCII85Decode /LZWDecode ] /F9 10 0 R >6mr%8TH$Nc\G%&T^sE3"eK`Mg_3(#P#Ee^m%Y-"#7cWW.S%m](:]W-8Z:WI4)ZO^ :l8bQb>#jUu$!r)COML`kA%[$/fp#ZL(cC Example 6.10 Maximum Flow Problem Consider the maximum flow problem depicted in Output 6.10.1. J/gjB!3o"T7k)P!GKC!t"l1?7RKum*M@=,rV\X7gPeFP+s1^AG[hea?Ui^cIcA?2buQ8AYoJ@p%/D`75#?Y2?X+t7+)5@ZUWB%UM.e/5HRR[)9/qnn>hLeaPJld"*irbNe8`F2iPQQ There are specialized algorithms that can be used to solve for the maximum flow. 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