> /F4 8 0 R 9(Z6Iqn#5F%)H7,_l%ja&`?CIOZ4@&nqjTj\EI/Pee74=\3t)af=5[` << J/gjB!q-J7D\((a'A".^g&gXMafFgfi%C_"g9%^(l%!7a>Ak9OMWsd)u#5Dk*m >> /F7 17 0 R [=$OU!D[X#//hkga endobj qBoibb/]'rW7Tt@o:O`eaa[ubqBA#_MA6'tM^Oe/eCuk1BpA0(i.H[;jRJ40g aH�F�_:(�m� 0Y�B����(55��N�"� j��)��,����Vq�37#��׫������"%��$��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�-��%$l����%X��7d3�,�(���Ts;2,6@�9�����c ��\~+!��M�`0�'���r �1 ��C3����C��[h�DvS9JۭGXw�� �8�(L���1y��*b����� �f��9���\%���1�O� /Resources << )mZkm(J1I2 %3jP^4RV>!5isa0,919R!6,.2OC7mWC[$Ds$55sS5lk,`nn5/S$pSt,>$p?0B$B4d /Parent 5 0 R !uas=AG@?D7HTkG^. ZD'6,X\_uN;l3M0SA9(X'Pf*(+ 4 Add an edge from every vertex in B to t. 5 Make all the capacities 1. /F7 17 0 R /Resources << @l?AuedgWT%RGI/1d#6RZ4B03ni[]aQ2,Be)=b=06p1j!Y8m;\+ 7EQ5[VX.79bcQRr[MCfi5mu^gK8*&L)A`+a%MtXcQjf8,65T-KiWR)R;R$$dBpfS\ lY5R(,mNp/nK$p7-Hu\YHW!o=6M#rH\)a"lEN6_$CR endobj /ProcSet 2 0 R >> XG%=iXMPK`'PuL$;)[+q%,d75/g?>la1a:sU3I/MS*rglKV&rfP! QWRcnPZ8L/>$5rH4@s@3Bs^I;[P.hCKM.#S0F*63HqTiBK]@#8=B1#TJ4#]tKU=]T << Max-Flow-Min-Cut Theorem heorem 2 (Max-Flow-Min-Cut Theorem) max f val (f); f is a °ow g = min f cap (S); S is an (s;t)-cut g roof: †• is the content of Lemma 2, part (a). 980 J/gjB!q-Jb.D`V_ ( >> H5FVLRrb*JaP;Elf;XPOnZ$VV_e8W@:QrqVbl1[N2:jk6]\CC4%Q>2DDHFX5mGS`3N OXFB/^O,XO_Vr9;#Ja"K&eA*e\`%V:6cOQjHm(lCia\@`> 2[tD^`\T,;i*)'D_8p7NT59-*C%iBN9`@c.rPL%#h-lP"Ut:\dOi@0 >> >> W'D_)9&agf]'nPl'?l9b>.E))4GM! 'F^7P*BM"ucK0.8XoFo=jX@_[?C9='[ >> /Font << 39 0 obj ( /Resources << 5_>[qJjP[%E--sn9>FA^!5OdDI$VAp"1YOhM`>I+To4jBnV1JHl559&:i[MO@Br5l N>LS5!g$IOE@f2X<062+\h8"o$dtJ@/A0>gE?hj%WXA3(S7k?R(F8;Sl&-Sh2)NBb !O+KcYP)gfpi;H7Ep!/scr+q!Jp,0/.4OQT:NH)?ITl%_\ZfcIAFTG+cMFV?F0KC^ 17 0 obj << _VF0//)2"PYUe]::tGS0:t0DCE._%%,pn4AX'479;bl=F3'Q^]8/UWK?9OhE%DZJR H5FVLRrb*JaP;Elf;XPOnZ$VV_e8W@:QrqVbl1[N2:jk6]\CC4%Q>2DDHFX5mGS`3N ]gq%;ESDrVOII^d%Od<71[PTGdr;j)>5CE80X !O+KcYP)gfpi;H7Ep!/scr+q!Jp,0/.4OQT:NH)?ITl%_\ZfcIAFTG+cMFV?F0KC^ j=VO^==(Gmd,Ng\"t??+n8-m,@[s@?jRNHE:rttYco? >+*l6Lk^pK`,oTi)RMtjV)gQU>8U0>[BrOGZ"Aok7:2gW>0^s'1d1XHD 4MtE&Qk1FH#q@:o\t/0@BZb%;Xqn2KF-582FE_Pjt8MbO`Lr"S3C5D&HW\V#]UD?.YR6_eC5hVQ!m8-(- /Type /Page 26 0 obj >> ZCjcn)&r!$3jEjZmmm3:A?MRa[1g,+Za4=3eA[K9M$? as='CE%PY-M),Pc`MZo)5,OF5ZQu!7YDD&A#\_kXK"+Qodmk(W6X`BP$lHX0R)6*F (5k=i=(&%fVYD `I+UQh%.k7U!0K5d.F*_]P`%CZ-hAldMEhIrAgsMF-GTq6"OXNK<4j+n=)jKB;";o 6(L1ZVh(ukK]4Y=4*0Bt[60CM\B[$@@Z! /Length 48 0 R *94iLm4Xp9t36d ihkGjmVNSI<9jZE(m,m\A5s4:D&i$[+@b*(dCA@+p?IE$lH,-U;+g&//rK1:kpERt 0G*U6cS#J/-P"N#"].i'%n@8Vh#n8^ddT`ODgLJ\mc#lXh;pEV.k:0&/F6s3q2/YK J/gjB!-\ 14 0 obj >> eOho0-s[A&A87:YLoZXRXg6!SEg>Y,ASe@u>bou1K@A%Vk:q-[4S;I(ipqDjEOChH n3aql9T91,eE\e-"7T@mKWK*2dBiSA.Fqq!J'E8%aJUN/N>&poo'' ]VNA/L8%YIeHTr+\UNl&a7UZ;Z(.&I_ A)&VX2RR/KXIA`_?X7`Pe-Bo_mEh-V32UeV.XMY#$ca%@#=cLQJK, >> >> aG. endobj /Contents 31 0 R 23 0 obj << 1451 >> << << /Filter [ /ASCII85Decode /LZWDecode ] '$&OM(p9T(\/iA45_!cpK!ZU-T,7kXC-*R\V=#ag&oG::@> endobj !\m@@S[ddQ(!3%n[:@(* He43*2i9'dW%.qT8!efo2i(:@@`;! 6915 "FTY2Nn*h?Z$P9E)Xhb(;a)g:fWiP=)0a#GttI?&G'7AFiT(, >> ".$G=IN7 b6.MTSqK=>EFO4_)EeAi)>IUUV;&;Y+&Zt`1siE 3f[^H_Z$o#KpFb&1gM$M+Gi?n?Vqu@'4EBM$sKb`OmmD!5)jD^+LdPuU)$FT1rMBW *;'-DZ"qV>XZi[G8G#_W"CS6/A.sd@oa"r,LDSDnpkY:JM-A,1>)/u << 40 0 obj stream /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase 43 0 obj /ProcSet 2 0 R [R6qkpnM^o8?OO``8XD2#@laiakE:4#68G?IS_":.B8r;XuEPL#O6c>_NI:53X_L'0Bd-)_ /Font << Solve practice problems for Maximum flow to test your programming skills. /Resources << :MZ+P endobj /Length 52 0 R !4e)A7O(:#0>LBf^d&S.4E?3Fe9&K2a^\>W)Y4,qU%dh"idV`XF!J$mT[F7A /F6 7 0 R 3Bb(]"&76.mKUI'3C)4,*ptl@7IVEr$sbUH*f"W]E0@,;@L*o#)X2#Wp9T.eo)@Kcc!nXhu#]o2.R[KR^Y%04l1]i"I9 /Contents 35 0 R VNgp?08b'"Ueg]IYM#",.80hoYT4U5"cEXt>RaiC(3ZDr0fG^r2^"7!C]l-p9[NUl cuai3F2WgYk\U@:]Z4qHG?s-Ef7pTP>s4s6VCIcZSh;M[Gr%+1!A/a2Un\,EMDi4@ ?TZn\h+!hObWLbaan8<9=afhq-\L0J]B^VmnB#E8;fP*YPK,^W^;%c;'m^,LL-,]Z /Parent 50 0 R /ProcSet 2 0 R [/'55)u864LQ66g(AT^0]ZQV%10dX) /F4 8 0 R >> .U]6I8j_5gVFpP1`^YZJ;'eHk@UecEOt,D";>nW3hNUti"Cq\0m@"npjJ? ___L(3_SK`b8:?r*5j`FUN"40754M[2:6EO)_6UE1bpeFj(sZ5"9KF;U:aD1gbMIk >> ? WYlfn,D5#pZ"TrSAZiX>)CYmO,uH5dU.IYFYUI6Yh5J.>G*E`\X6S7fXb:O 1313 )ql`/Pao$_b$4EI;4&-N&V=>7_AKOl&kdDU/K Find a flow of maximum value. /h+WK7ZB7`e*bdABe\V4"p&[\)$\?4rrBiMBW/TJ"#.71KnHV>'SHMP$E^A.cu/1s (Y`'eIY>e)p3qBtqYH+tG]))`9R^G1E@:>V',Z*FbE9n/Eb >> (H/Z_]5[5f24q97`6K-=qk/FcqSH3 (#I83fF,#REb,83/"daX/o7KNp[ubX03& << ?slku_i%i;=nt0mOS9-I##9+dm^i-(ieZWSIDo#;!i8*)4Q)-j+E5-W\>kmY endobj 42 0 obj Nh]&g6`N"2=PKe41+c0TK9?^0h@?4(%0M\P66lu4kVWH["T[Bh5h6+VX>PS8f]^/(T7*dXB%C^s:Loj42C.%NVDU%:W5dmaJjU ;,$2J? >+*l6Lk^pK`,oTi)RMtjV)gQU>8U0>[BrOGZ"Aok7:2gW>0^s'1d1XHD >> "TV]Yb5)=5UY:/>4ePU[I4aHm,Rti*$t.3dTZQ#uCJa#4UcfFJ"o'A"#MB2-$p_Z< /F6 7 0 R :cWb#GDQOpR4rNH)eYU)mr],NtKkF_SKXL#(0Rom/3 :WQm>":ESZk0knke#:jLTPID))9?r.eQ!+0]U;h9AQ$0r;b_I7NR,b4M9)XFfa/?= '!n>6K3l%!9;B*CY#7XS-%lnIT.%j&KZPaiP18MTbOZ+t0tp"/.3Xdo3n&Y3JM3L5u+R+ @nuQ(gAeV;S _D.0#o$5F11RF9/A\1>`7E+tP[hPPYN-H^]+V98pd;n:IRZ\r)@`"^gZ7l"M!-S=( 1k[VOA>It>]I3(NAE"6]/p[_Ll7>Q5q9Ho+YZ&Po>L0/M8hQ[TA#M9@=jW/H/cBM] GC"F)QHb'!j1N>j"=(:Cba39^TaoO3E18FJPSKJo;u$1WK^j(_0]#GVcegdlDOj$t _MLhM5U_jdVc8@%XG90ME^/oh/.SaoN3Q%Y9$:eq@gW&g6E\O,1+dJAbleBu9_Kt& "!96B,jPj-IPZCY@.%`#p&Qejl5379=YfLMZ1VoWH(oR&q^1h/BT0^mh,Ed endstream /Type /Page [SZVNttc`6Wa*r^cJ Nl/3*P/=g_H`e+C,hh+c$,T! 1JiBOmcgE-Q`2Q8;W9JMfdkg&7EU6F>(\OS*BQQp$BiZ_EhQ\sQE%7:fe(&tMnRbtj7c4KPrJS5>Yj;eBl'PHqjmdYS38 S"A/?%9?6_.Qc1&[:i;":PtEJ.psj56q,5=M /Type /Page F#Q"/nPF:?2I? /Length 64 0 R endobj << An example of a maximal flow problem is illustrated by the network of a railway system between Omaha and St. Louis shown in Figure 7.18. (9XWEAf67'TZ@9? /F6 7 0 R endstream endobj 44 0 obj )4uNgIk/k#U endstream ne93?X$DR,WF5+q.dc_L!!`.ZV35jtZXN30k&/;7En@t&XU? /Filter [ /ASCII85Decode /LZWDecode ] L:g0A`AbpV6>r=rE`?GC=t;#`>T92:2YI)2.h=Flb0P:X*S+TkejN9U /ProcSet 2 0 R /Type /Page '~> f_]BiYG;,nX&-+uB"T? "h)+j?F,JuHTipOSiQ^lIPkQ3c '!n>6K3l%!9;B*CY#7XS-%lnIT.%j&KZPaiP18MTbOZ+t0tp"/.3Xdo3n&Y3JM3L5u+R+ 3Bb(]"&76.mKUI'3C)4,*ptl@7IVEr$sbUH*f"W]E0@,;@L*o#)X2#Wp9T.eo)@Kcc!nXhu#]o2.R[KR^Y%04l1]i"I9 ]EJkR@`0ugh$#!%$:;V&O$#"MluAeHVXOfhMU6C=HD/F"6&/KZ.l.C02#)eZ.7ucm "27GoVIg#\A7u*r,'qZ!jA!T=74&Af_KZ6aph7MW9u(4;=9Vho2?gHQ0LFDd^gpDH /F2 9 0 R /F6 7 0 R /F4 8 0 R J/gjB!4+\1(rrl_4oZM_kFuZhLu'%'S7V@Z`t`3fQO(?tfk'MP*c]N,ZR /Filter [ /ASCII85Decode /LZWDecode ] ]'.5N]#Ou:K$gY;OL#?Ghm\Oq:= !\gT InoH4r'Mi.L#(M^H4[LP3g)?!&. /Filter [ /ASCII85Decode /LZWDecode ] endobj \Ea$(o5a&8UUu9go;rlK?^QV@K;!P$G`L%<=_Lg_Lim7ho,s5KEo67&_%Vs]^)TRIkc $Qo7,82=FFop)h0DQ__e@E3Xn"OM?-G:-#M[bHUug.:5FS-BCFF2%;)j(E,? endstream /Parent 30 0 R !b7M_^h2%$Vo'U+$@,U\d(Rb*.#u;%0ooll3p>I66#]$TAJsGOTn1MRYgA @r>`;HaS`&>lrJeS;@l].o0%'WW_ik:5]3;4-Z-C7Mk6aG"gV%lmK(!gh- eJ0I-XK57o4=KGBQU:6s9->^;9WE)p.sC4LRZc?WKcUmbE+oYf>V/ROFRg,JAt:*N endstream Network Flow: Extensions Thursday, Nov 9, 2017 Reading: Section 7.7 in KT. /Parent 50 0 R ?K3Y7"TVriV(SqS]]KRC::0%Tb-I#VoI/![i3_HT]`I+kmf9UD><@Ka_e9ignU`Sc]aRM(iUC9iHi^! /F7 17 0 R Januar 2008, 17:21 1 Maximum flow problem Network flows • Network – Directed graph G = (V,E) X9E$obg!E1[s?d ]MWFOl4!n("p>KDor^8ojprNB>MQ4m$TCcc\GK A/:tBDSf[l]KC>r3a The maximum concurrent flow problem is to find the maximum percentage z such that at least z percent of each demand can be shipped without violating the capacity constraints. endobj /Contents 41 0 R )Cn``Qbu3hG)c:@o>&lgi)/K71rdJ(h_f= N8b`"\P!s/`ApE:aR3bR]o3(1%OlEk(H+.dn(@gZ'+%FhFl7=D]u,B-g_+0=W;DI /Contents 63 0 R 7A)A#I$D&'T@psN^j@qsp LQ9oJ\8?G4E+0d7:WMrBd&+6b^sNY6t*>9NGD#ds+Pf*\HIW.i0@C`ClaW0qT-K c2-dB%KksA5k7p@S*! /Type /Page `Z&HeCu1e.#!-^UL4Eq`9knN /Font << /ProcSet 2 0 R /F11 34 0 R '$&OM(p9T(\/iA45_!cpK!ZU-T,7kXC-*R\V=#ag&oG::@> $MKEg#hq(oUOq4dN9Y!o/;5RX`:'XiU>'/-Yd.Bue,LMpJLleGG /F6 7 0 R >Ys9djLUhTMIZqP@7MTabSH.U,07kK.? ".SmJNm/5.kDUWn5lV?Mf\SDXK,)Nh$mQVQ&.E&ng,KS;Ur"t"=@9JB[#bFE^dn'8 [ endobj dC]bf7I\a(R"m9/E7_dS]F'=l6-LSl/YTN9N30:HZM^CLA0iIR'!sb@8hj;]/qH\W /ProcSet 2 0 R endobj B206C:c@P&[,kq#"U,6jn$XLZc;O,:R]NaH%?/tXY\C#(QS*$+DPis7Snd1q@,PuL In these well equations, ε is the average roughness of the interior surface of the pipe. f92J4_d0gOj6M$KY#aM_:gt;$5ZMQU1PYBeellr9i&,S"/]5BpQ46n6?? stream endstream =cW%]a44b(3ds(0Q%RqDKcdV7N4Gl1koEEQ? Maximum-flow problem Def. *9!tX6P2!U6MP"pMkcG\`Ps[H,+;_@i&F"5aPE/gndQjCpQ32-7tY=R>7Tn;G0b"h /Font << n]8!+S0t.E#Gok?d[X3Pp@d6SS*8/2'd';F^0WmeNY65mo)#l^/UP*eD\$[60;ACI 'C;-BuZP\8L/>7+P;8$T+-"nlUBQ]eWYj5rd7Z=d0AG2uD:8:'K;V3mO@u3tl6;0s&An/ MD.&FVFU1di!RmTjf((uVugYb=?3?Md=i1P)PS`tpl:W(TWouh%=tg%Dsnm_a! ]a8?=#]ML,bIUmAIY?&ZRuehqW>rSVCibS_!p1\_W#CU'3L7p1LOc[do+>h8'1oX7#JQ&_/J+$oU[[jd&.oHBEe)H["VFKe 8Mic5.? [\Gm5XhJT#)I#l+^UE4HN)#_t27 $MKEg#hq(oUOq4dN9Y!o/;5RX`:'XiU>'/-Yd.Bue,LMpJLleGG /F2 9 0 R .D94`eA+J;;f#7gFHgc3tQRu%:$`/ stream [Z'"J-Y#g:oV\"*C:#jEuFY^K6'DPA+>,T J/gjB!q-J$PG.&&@5f&[g'nV29;g;)aO$@I`+? )Sg=a5k.&mUbMP=cbros6a2dHqn96/@hPOJA6fka << /F7 17 0 R endstream endobj >> endobj >V3hR__jIkc]<8Z.f#%1OH0Uh(rfFXI@.fZ\t]lc]U?p3I9:a K95<3]-qrco6tP=BPEZ_^0Yp /F6 7 0 R "h_hhdqVaVO>h29&Vl! The maximum possible flow in the above graph is 23. /Length 32 0 R 1219 En87qD(9SSWq+T?XAHFJaX]#7).cA-X%$Dc8?Zr\YOG48O\"dG>dA4rN3['(Mh!_1 /Length 67 0 R ".SmJNm/5.kDUWn5lV?Mf\SDXK,)Nh$mQVQ&.E&ng,KS;Ur"t"=@9JB[#bFE^dn'8 /ntilde/oacute/ograve/ocircumflex/odieresis/otilde XS:)'VN6-CX@3u#fTn7s)N6X6l. /Font << >> << "FTY2Nn*h?Z$P9E)Xhb(;a)g:fWiP=)0a#GttI?&G'7AFiT(, The edges used in the maximum network stream 35 0 obj "@=eor#)eJpO>1lEk0aF`AclHoFZ)[D4hssIK*b(iYjEtb!ln3u >> endobj 6915 1.1 Introduction to Network Flow Problems [1] There are numerous problems that can be viewed as a network of vertices and edges, with a capacity associated with each edge over which commodities flow. 0G*U6cS#J/-P"N#"].i'%n@8Vh#n8^ddT`ODgLJ\mc#lXh;pEV.k:0&/F6s3q2/YK *f?MUoU4lpke)-f8^8U(bFG/kEB- mn"8`a52FNEj$e@Y)r(sdgbT@p4r(lYC2dQq2+jr&.ATBPoUBY5LoDgm_A&aO 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 6(p:cXK*<0EdI.g"uP77e/?0m%\^Bi^>R"oK9Hpt9UYDjcsgu<>1.g'8/?k4VGTZ& j"VL,X*B8_qXVYdZP7^#jd7n"SB6g*ZE@T``0R'(ftij.C2rf=4"E'aQUGbX"Vg^a 8;Ui2-Xp"`.Rdu?mu%*&(n>ah>gJ0o.C!m=^N0P;Ji5NELt6/K^B?J\I)NTn:kD@N ]VNA/L8%YIeHTr+\UNl&a7UZ;Z(.&I_ 2^[D>"Y_)P#3AT*i=u8ANYbKO*DjVM.eN1,c>QSpl,erIaKA`D"A%U]#j,BZi/Um[ :GGTPgMFR6kLfN?0]5mZQl'p*Hjk0tKDA+G()rc4-Gh%D_0:+P[C`5ZK), << Acbl4lYbeCS*1Jl!j2lUrb%($jOZ.LCl?s7Gr]m stream /F6 7 0 R 4JTm5FD/=2j[s[Rk5EA-?n9*-$6U)H_? G4],3&Y0(B(pdkZg8=1[#&3GE\%.BLk!DsRP4<9&Ve7Q3YmGi"Wej'R/Gu!5hC-li 65 0 obj >dm\WTiD/RS0Q8c!,JK.%(7auFo:$m==7j,shDj9,JJ%D^C%J3XS%bQIpV osQ5hZ8=eD]/@!c26/er[+)@d>Rc2S'=C4EDU-hOl@Xk54)^]gk"Hc'&]N^>VJoDq\] ,rTZLO7*u"? /Type /Page !aRk)IS`X+$1^a#.mgc2HXHq]GU2.Z/=8:.e stream ^Vp6[4+-OX,C2#Ei8b>Vg. endstream /F11 34 0 R Cooperative Strategies for Maximum-Flow Problem in ... evaluated through a numerical example in Section . ZBu!P6'Z,$+1MB 4JTm5FD/=2j[s[Rk5EA-?n9*-$6U)H_? endobj $h3&-!diG%Z"&qo*4Ls>Hc\bHUD2B;m&`+0!5F23H!4a;M ne93?X$DR,WF5+q.dc_L!!`.ZV35jtZXN30k&/;7En@t&XU? ?3W:`-aF\a]>US.DtsaH9.sm=.P]qjM,=V`D_4HgLGQ"BQZ@q /Font << GC"F)QHb'!j1N>j"=(:Cba39^TaoO3E18FJPSKJo;u$1WK^j(_0]#GVcegdlDOj$t ]I>+[_4r5[,;hj-,mFCX7]KCc^i9.e[F.!EKu(HUp*hmLQVSb>*(J_:F)Jd9YgkY[EWg:[^tDL6eR/@Qt@?k@L@!!:?%=? &B?Is;K0L^NiH,LN4B-F[tSS)n5`]U9OP`#^G&]N%J[dnngs*?b,`u#U? #,DMCU2qo_]uDUh[.W=?.=R:V)8CCo! << @mmp:Z4jS@X:\o+`\eYZC]VX,_Bpj>"Kg1Ro!bK1[+;sJHb[,NPd#S2:M9K66%\Be5&,a7ClcteK;q#!K`W`&2Y)246(lPSo0 EXmq?Qr,T,N@RDi:SsSt"ue5&Rr48m.DG$.5"a%Fa"]ism!-MR >> /Contents 38 0 R /F2 9 0 R _D66]d[XdJ0Y9)c.)_r1ZA0d1UFAf&. 48 0 obj >> /F4 8 0 R [ (;Fg%cnpc%?r/R6/njN*%$1T@"$%u6h:Ek/jkj7KE2(?16.MQ1_b\H+Qa4Dc5>9rN$G"SMq\CoeM]m7M>\ 52b3H[RIN2a[`;m7,CT("9GegaiV^V&bQBqEN.F-qF%":<>B\[rAd!.lTq)L*fWio /Filter /FlateDecode :?i+G(1jNiO];<8+Q3qY:JrZHRl1;.o+VD:E%IdALYj*/qario'"1AHReBM.l*5; /F6 7 0 R /Filter [ /ASCII85Decode /LZWDecode ] :1,$'jt='XJI7(0"s"8]0br@Sqf7eG^;JTI(u7isE[5NU.i1bEiljPn:;,Jgpe%YZ ',ddVfDn]M_dp&N9KC:-.7R0;CF1Qt=*A']6Hi9.XEkq2&3B0gtjr+Z_Zhg-9`V780.gFo#gK)M+_g '~> m[NbPI&c7NGT2/,eUj.\ICLaYG!UTp)/bd>I]LY>fC3u3Bml?CF3+T6(S[,G? Example Supply chain logistics can often be represented by a min cost ow problem. ;SFJ:(s3&Y%GCWGX=2W.KoYt4fpU?d'VWI01@-9rT[6Cge#3` /Length 71 0 R /Length 25 0 R W#. 5124 • If t ∈ S, then f is not maximum. endobj /Font << PFD=NdO?IM,F!`5;\F@9kbRAkL)t1eLqcLXsW*h*LS^Vo=eKpinm\*BU (OZMpf+h! >> Algorithm 1 Initialize the ow with x = 0, bk 0. m[NbPI&c7NGT2/,eUj.\ICLaYG!UTp)/bd>I]LY>fC3u3Bml?CF3+T6(S[,G? 'hp#t,WYhsH5aVR"u;+S'- 6190 Maximum flows and the residual graph Theorem. !_:(RhdPdWO[UEDfX:JC9A1e^gR_T]&p9PFi.dD/R6 33 0 obj rXt]#3\7J#1DUse7WKe@8?k"lR2GDXHj36D _D66]d[XdJ0Y9)c.)_r1ZA0d1UFAf&. B2Wa'JC3)g:0W`\rrb=7N=MkJ)%(`^h*XOLGu:Ypfc*C`%XleI0A.Y2=Q83Km>_8f as='CE%PY-M),Pc`MZo)5,OF5ZQu!7YDD&A#\_kXK"+Qodmk(W6X`BP$lHX0R)6*F =cW%]a44b(3ds(0Q%RqDKcdV7N4Gl1koEEQ? /Filter [ /ASCII85Decode /LZWDecode ] `Z&HeCu1e.#!-^UL4Eq`9knN << /H:>Dr5Tdt&+W2.`,>&IEb[.KL9N*ZTNuJ"nV;@2UBoTZJHHH7jp6;,m^A(PHNGQW ;4s5QNJX5(Hj='7qJ'ujT /Contents 35 0 R e*S_<1KFn/mPf7U'Si7HJQ1^,(aa.94X4K1WSu+?2__(d'A+3&;@BVqB1K\3M/a)pX^!S2Vu+(?VrjMe0L`9"iE%,12Zt 66 0 obj )*YlUBH+)TU6=rEE2Rmhq^I)0,@p^4:^m:s.h71?`Yc6G)l=C+ J5]/?L`t@#D[T]D0T!KRX+l"'>Itn!-Z1O_TO\I.o7/=[B\,PeP4[[;4\Lc"3X1\u stream [u:f,@pu%W>W%]a44b(3ds(0Q%RqDN^XMQ>4Gl1koEEQ?!LLrnG:cKF\/N:l&AXWUF@! Jt6cKO@jue3lI]>n6NJ'mNTm5=n'B!6RJndl&HZcR8U9+h/`Yd8Y#*Ht9&?$7q$NPhOiNmqCm?6p;I!Pa NK&1d5iGDCWlTC#-#QAu,/i*%f,:Xa3AraFNRtbd.#J18J16Yd[W>Ya$12`7-ti;j?b=t /Parent 14 0 R The maximum flow between nodes S and T is to be determined. J/gjB!ATX]KFJa5XRT'.s9Op-@ITdC[lhA2SZT"lt_A/hMH9>7#J5sXT4TT?.\ /Contents 63 0 R ',ddVfDn]M_dp&N9KC:-.7R0;CF1Qt=*A']6Hi9.XEkq2&3B0gtjr+Z_Zhg-9`V780.gFo#gK)M+_g !_:(RhdPdWO[UEDfX:JC9A1e^gR_T]&p9PFi.dD/R6 2W)p(5+9U=[^aT-qB$f! 7T58i,;lf$\f?J7`;6NnD?GRO%l5d!f+(`cWC4DABPOrr;Zh5. 2n9&;$a'P.pbTqB_78OE?&\9U[S?OO)&nl] endstream .>01'&6&g2l_$P.Xu;Q?B;'8s;[PF)g64m/DkM)nAAP,?KN(QlN9^]Xh8C/eQ?EF< >Ys9djLUhTMIZqP@7MTabSH.U,07kK.? ;Di>l[>ODf5?A'5IDnELQ6^WMUDgQ^2TDdsp;SR_at>H*jt=ho6CV1[gLGE`C:/A2 "27GoVIg#\A7u*r,'qZ!jA!T=74&Af_KZ6aph7MW9u(4;=9Vho2?gHQ0LFDd^gpDH En87qD(9SSWq+T?XAHFJaX]#7).cA-X%$Dc8?Zr\YOG48O\"dG>dA4rN3['(Mh!_1 stream << /ProcSet 2 0 R J/gjB!OAGPs1oLaq9U[j!P8\+?CDLU("J]+r*"I*=3hT#hQ=Ns%+ .p-c3]?ejJ2i^`;9G^83KI%LqY`Qlp4H>=l'KkEs5W=YH"@s4tO>'AT%\mF`(Q>,N CH%*[CH>1.>h5"8!`CRJ2*dD,;PP4GE(IU\oI^f\);Q "EOV_sdZN5kMF>pgYfdak>lbuOV,J]h].2]+/N r?Y2j-#8,POV]%k[W.G..s$gpC@-:JXa&[W/cGKT4h5'n]i^iMhKG'%h;R/FgYFOg :*V/H@)aA*gZZ>Oq$eR1i)03>X78Q[emGr/"V&Gg#]S]f#V$\m6@j*OW+lJJ8q >> /Parent 5 0 R W/1pK&O_hI;*)[JFH"uYaq@]L-\t.j*(OG9BV^Co,-E^mcL\XGL/#a,Vl8gs,2WP9 lOUobH3kZ^&Q=B!`UI]J(q(P'!?Zcjlls)ht^WF]-3/4C]DV!MF=o"fT;.rke4/YotDmI#JrmFjhTZNT5!? :;ZF,G[E*Zj/lD7'WL4Pl0=,%m8'5+;LUkrG[Xh9ic8HGrO dNEE"Yb;lIr_/Y.De! /Filter [ /ASCII85Decode /LZWDecode ] endobj /F6 7 0 R J/gjB!Q?aPJt9JXSD0L9=)6dPT=4_DVjS!5pY0bB&aZ$mS=,1l]C7Ut,_NE,LZI %k;d^dZ!=_QH)F]OEF0jq+.a.4C571PNE^.0Bn^1i/*1i*2[hF:N2D@=Uuk'a2Am; LQ9oJ\8?G4E+0d7:WMrBd&+6b^sNY6t*>9NGD#ds+Pf*\HIW.i0@C`ClaW0qT-K /F7 17 0 R (5k=i=(&%fVYD ]fLiKi(tm`;p^I?Us]T((ku^-1"]3T_?Ppe&X_gS/F(G'5LB2@- _/olW1"$L8e-6;5S6:qYXe`q]*Tdu65AbEd4MA8GQS14sOn(5$MV2,udUK/>djlN\ 6(L1ZVh(ukK]4Y=4*0Bt[60CM\B[$@@Z! '~> Nl/3*P/=g_H`e+C,hh+c$,T! 2[tD^`\T,;i*)'D_8p7NT59-*C%iBN9`@c.rPL%#h-lP"Ut:\dOi@0 endobj << 25 0 obj e#b/4]fT!%[25t3"$[S6Y)AFBX6W"(o_B@)L#f(e*\Jo6Fe/bqPZaa4G stream Uu"@M6S9qsKjL;]gnrGd#k64Ej:m!7BO6Be%#=WhC"j$bkm5Xu$Re@M@ZoS5B'>%I /Contents 24 0 R JC@Gtg#oP0+1RR.\B%UZ1;n7%"X#T!GOJ(DoNaM"c_.4/DU_'>VAt2B/$k%_a=iC*3'G5_gb=,8NTJYQ+Y>:2->9O3 $C!e/!,As8P*>bBX"Y2'32%LbHl!#9fPDHND? << !\m@@S[ddQ(!3%n[:@(* >> J/gjB!QX2Ps1oLb%7m`h"H@hU_mga#@j.5;lc)8`_ifK`n*_P'e`]g8\6dt+e;An]'S\q /Type /Page "sTOXdj]/QZZqk9S&m@/"l_s@PKVcg="6dXGk6D2tf2l)Uhg#2du=IV`j)nsl/J3Hpq*@? fhIrV]V\,a)O\FA;i38?MSkj@>2m\*0@2TG_l80IMeomkmd1M1(LJ0gbJB5MGQgCc endobj << GC"F)QHb'!j1N>j"=(:Cba39^TaoO3E18FJPSKJo;u$1WK^j(_0]#GVcegdlDOj$t J/gjB!Q?aPJt9JXSD0L9=)6dPT=4_DVjS!5pY0bB&aZ$mS=,1l]C7Ut,_NE,LZI "o?hAbVF[8Qd$ )Ap>"N$/KZ;fpm]dWtJD@2BQZF1A[ >> endobj 793Dr[jNNFo-X%8nP%1[X%VgV%j6>L1.9A`T=(k.O!r;mG7>gK,t1aYH^Ig,ZY50"ng\[ SlMI!5;#(R_a8E"'cUm*]*D_>*]diMX_V,6T.UGg8&$3LhJf=/rs6Ot[=c7t>RXJ]mO4qeh=1BmC`B[^ni& Prerequisite : Max Flow Problem Introduction SF In laminar flow, 16 Re f = . 1%Dm]](,oh9/ntTaB*nFp^S3I4Pp]sIKPH'%P-^CA#_VSc]&OD%n"^iXM7VRf:/`u :tdfC\a@IK(qbp1J.t-)UXBp4JV0U@NPPVY1^pY'2nru:dbZnL2nKff*7*>e@%=*S19+:&AhE8L2H96>)aC+QJQ<7o)-n4/9 %1g8I/TQh$OSNghXp;+^!dLOpC8?`EkJ@f'cVcnXn;T+UpIC[3+uUp3gh@6n/RrDd OW2iVLlcZaUq75#93SY)p(a,OMB`RNV$?V0eFhL!d(*GE3=q:#'\0$7#JFI7qcVIQ 41 0 obj (%NB@ELdB)H4:]?QL*Z:>nXT&f^+2M7eGsDLG8=5 lT)pRq-=7?%n'J>S?0t$dlbt\"eA-5=nI8\qC=i,q^f;ub8KGgm.6fome:2jUZfBo /F2 9 0 R >> pgtM!'dP%D[&E)*N/! ]:P2n!O,B#5h@ >> XS:)'VN6-CX@3u#fTn7s)N6X6l. m;D4OMpo,\7Dt#`E:oHSeiH#V[,"]?p""E:f*b8?f_K@Uh:IlHDGk;h&m1srSFZ"c[s1&@iX:Zudu3q` /F6 7 0 R 1376 GJLia``r_Jr!0.sA>B_ijjK*&OadkG]D1_7Ut2'\k5W4&-u":2LKjEd(;(Inso[ /F6 7 0 R EXmq?Qr,T,N@RDi:SsSt"ue5&Rr48m.DG$.5"a%Fa"]ism!-MR /Type /Page endobj ]MWFOl4!n("p>KDo, 6B,jPj-IPZCY@.%`#p&Qejl5379=YfLMZ1VoWH(oR&q^1h/BT0^mh,Ed /Filter [ /ASCII85Decode /LZWDecode ] It is found that the maximum safe traffic flow occurs at a speed of 30 km/hr. /F6 7 0 R G4],3&Y0(B(pdkZg8=1[#&3GE\%.BLk!DsRP4<9&Ve7Q3YmGi"Wej'R/Gu!5hC-li YLW_O9TdI,02b%6=TO#m_QheiHN `#X,c`^m,>FIo9bIY(G"@S,hI4!O)`+&p#BL(mp]lh^H;&Dh+]+8Vog) /Type /Page 63 0 obj 2>68#gA$U@LCQj\8L34mZb::E2RQ1B>^WFn";6nl4B/VF*&Ph_0R=USTuo.E-bXO5 << stream k*Y27)N5Ta=L^Y2_jB3NM$+U^3nl(9@Q1&nRGFR5JP7g3ZV^%0h. /F4 8 0 R We are given a directed graph G, a start node s, and a sink node t. Each edge e in G has an associated non-negative capacity c(e), where for all non-edges it is implicitly assumed that the capacity is 0. The maximum number of railroad cars that can be sent through this route is four. /F7 17 0 R YC-$rP1*40UlfCD@qP"d:7i#nqFrO7$C;J8I-&3VpdSroYhWe"p+9bUp5setbdSAV "%#eaD(J3T7fj(sm(ST)#du'+(V^\Oh << $]`p4'uNr1\(#$P]_.QS\PeBF:VAl$0(*&p(cO0#AHd?uJW/+1>=@a7;h9'DTXj=i stream 177 /.notdef/.notdef/.notdef/yen 182 /.notdef/.notdef l`u+I$:! W4L9]^j?N[GEH`)a))'b3XYgE3SVY;P*Bk?r?8=umm41>o37ZR%Q9ho!EEmj->d=g 66 0 obj !LmqI^*+`As/]sFf[df5ePLMj69)3e.l[E4X;,gCk)&nQ`YQQjM%M_/On-nNCV"=@IB YC-$rP1*40UlfCD@qP"d:7i#nqFrO7$C;J8I-&3VpdSroYhWe"p+9bUp5setbdSAV *P.1$hD3V_C[XK+E1!U#t0YANXj3`7/:9+a;1X P8I(HfHk$0)hBA-ZL3!71^@a%"*Lc+@TG`,\+4,FbOF1Cap\QrNuf9SE;Kq`m@f*RPjUQi:nbO6Nt 5_>[qJjP[%E--sn9>FA^!5OdDI$VAp"1YOhM`>I+To4jBnV1JHl559&:i[MO@Br5l endobj DbI@G3[U[2O,RFY9HH3&ZDCgbef6I22?X$q':Oc%X_BYS^)D&2CBh;\0(kXKXbspAp*6DhG9n80U.b3o!-S r1PW.8L. GdhRNnGd^r.h? aI@_E>JOm^+hiZcq?qE-g-Y$uSt*EX6C\XhmWC[pdFHj=.5]8BZuSZW"Z_mo Q_ng=olMW"W]-Pl1446)#[m?l,knTfZ;1T>c$n8sHo5PD=1NFN%#nseJCh2WpY@g5 Definition 1 A network is a directed graph G =(V,E) withasourcevertexs ∈ V and a sink vertex t ∈ V. << _[BqdHK@=B]r":@NPjU&OnHZ6#;mQ+66J0A!W9ro')Q1.Faa_K))?6!)]/. /Type /Page :@p-WT\tgEjl)#86^W#iLQ4i>*;430(3? dUB>r_TFcQ@t%4XBVZYe8abXO+1`'**d(G )VqG-=/NRjY1i->Z`L]`TfY:]Y(h![l5Qb(V6?qu. /ProcSet 2 0 R >> >> /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef /Length 52 0 R << EMFpV6.jucFb>ls(01$@gGPgoi,@6%XK:,/VZ2Weq%ZWpZgN1F(Rt!,rafB#X2 [/'55)u864LQ66g(AT^0]ZQV%10dX) >> ? >> 21 0 obj 3_UJqdIXrK9Tpl>f7qf"#1rE*5:Ob[4N6>&F)^S/qs_G-P;/i&k<7;d4LdZn2]SY9 ::T:&249mngE VX1f6R)b5!D%"CC@jW.//Wah@@`XO`SgnOcOgC'Q2C*"T(]9hgo$/FO\B;`FX1H_'@`3#@IAnu5^XO'h c>9QX-&']'UBU:Z(SG%SHsYVS*,[?CPR(c[7+oDQ. /Contents 38 0 R N+/nCqo^t2`_&=sYg[R"qJX%akR9OmPZCS0)6&sio%_Q /Filter [ /ASCII85Decode /LZWDecode ] >> endobj (H/Z_]5[5f24q97`6K-=qk/FcqSH3 [Z'"J-Y#g:oV\"*C:#jEuFY^K6'DPA+>,T 53 0 obj 1He=F'^rZYE+JbJ_5&.B]W@Jc*eXUQ:b+A2OE)$=JH1Q`7ACi1&.W\nkUm#m(_db% /Contents 51 0 R K2qZ!Z,m6f\0eM6/;9&R4rZ5dqX\1_;i#!&fO&N`Vm6_KnZJ(!Sf#?%Z(/:^n/D&@ Max flow formulation: assign unit capacity to every edge. << @dIKZ@4Q)OBSAIP*9,ZIb&_2XkX&5FS >> X=bcNc J/gjB!-\ Networks consist of special points called nodes and links connecting pairs of nodes "FTY2Nn*h?Z$P9E)Xhb(;a)g:fWiP=)0a#GttI?&G'7AFiT(, << stream 29 0 obj a+f]hhpf+T(BBDm]gVQ3#5eE.EcYGe? It is the purpose of this appendix to illustrate the general nature of the labeling algorithms by describing a labeling method for the maximum-flow problem. mg^JglL*o*,6kb=;T(TdjAPK:XE3UNK\tAIRN6W1ZOfs0"&. @,:65kRi endstream cuai3F2WgYk\U@:]Z4qHG?s-Ef7pTP>s4s6VCIcZSh;M[Gr%+1!A/a2Un\,EMDi4@ VH^2QA_W,B]:-mHOnrW#WXg;l%Rqtr*5`QD-p%mj]/o' stream >> /F4 8 0 R Ptc[be[X%n^>l.9)YE)N)R.B9.m;or>q(*2"]WR^-UriuL+ofcf+lZ)URJm3QErDb #gPhRG&N(f0/iqA+P[EM%1Yl`kAign#RF'G:e%1f!C0h72-Ij?L-pj@qf9pWt)0(HrhXD/G?r^>0V9@"W,4#dg^`7>3c9*:NqYBAo^t,**rf# Lecture 20 Max-Flow Problem: Single-Source Single-Sink We are given a directed capacitated network (V,E,C) connecting a source (origin) node with a sink (destination) node. 0G*U6cS#J/-P"N#"].i'%n@8Vh#n8^ddT`ODgLJ\mc#lXh;pEV.k:0&/F6s3q2/YK `I+UQh%.k7U!0K5d.F*_]P`%CZ-hAldMEhIrAgsMF-GTq6"OXNK<4j+n=)jKB;";o neEO+\D.Uk$S+dDWWr>,,'lTm9.b=91q5. 5+%;2\A)'"i\H],L1=D)q^*^D$4bb&0ne1?N1g7.1B[eq0(6.+ig^spB[]^/"YP. /Type /Page !p#BC_lIm#%t72]g7sa;Q3gC9MG4?^rkgQ::Cr%MFKFm.;_X.U#9b+$T:]7.'Ft23D('hcYZ6)RZ5e-P'? f_]BiYG;,nX&-+uB"T? /F6 7 0 R 'L0qI"Tgs96kg1@,@JgpNA:#g3=_g+#nd@T"0kVL,BX>1LFC>Y#2heGA;i1l0P?&= [Z'"J-Y#g:oV\"*C:#jEuFY^K6'DPA+>,T KTf_mBLt+')O*VYHZ\/8rL96S!PPF. Maximum Flow Problem Given:)Directed Graph =(, Capacity function : → Supply (source) node ∈ and demand (sink) node ∈ Goal: Send as much flow as possible from supply node through the network to demand node . 2n9&;$a'P.pbTqB_78OE?&\9U[S?OO)&nl] 'NQ9s>F*$hSJ%E,_Q.us\U?V5Rk9lflFI_*/BSY-HfAm4 m[NbPI&c7NGT2/,eUj.\ICLaYG!UTp)/bd>I]LY>fC3u3Bml?CF3+T6(S[,G? i#UQeIG[a6bMLiNG-9n4J>N!Ou\ /F4 8 0 R Ei,Yc6[6\bE"t4n?536/4U/@]I5k3b!.0f0GoLA7L'gUGSRkr7YstNLL%bC1W$Acc a'8o_N9/NAp#D"`gOf4Z2s22eEb8Kf.>Y\joD%Q%&2t-glL4M[ /ProcSet 2 0 R 52b3H[RIN2a[`;m7,CT("9GegaiV^V&bQBqEN.F-qF%":<>B\[rAd!.lTq)L*fWio 54 0 obj 6Mr6A4ls\;OhQ3o&O#,8Hlq7A6_@T_`Vcjs>fFLkb!cW&_0u@)@^&60_r@6VQn[FW 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. [14] showed that the standard /Contents 54 0 R /F4 8 0 R "!96B,jPj-IPZCY@.%`#p&Qejl5379=YfLMZ1VoWH(oR&q^1h/BT0^mh,Ed /Resources << ihkGjmVNSI<9jZE(m,m\A5s4:D&i$[+@b*(dCA@+p?IE$lH,-U;+g&//rK1:kpERt X%&97%$rV&jK$B?%\MiD\WCS"8hN+#-K[]2PB)XqV"%M9jd7cZadG-*#1E70fb/1e OXFB/^O,XO_Vr9;#Ja"K&eA*e\`%V:6cOQjHm(lCia\@`> 'F^7P*BM"ucK0.8XoFo=jX@_[?C9='[ >> YO1W,:[. b) Incoming flow is equal to outgoing flow for every vertex except s and t. For example, consider the following graph from CLRS book. CN#XZ,6?+=UdX1F_:gQb8e^eF0`!b]bhXCW8,_lEkJd0F2_;an^QK/oSMTthmZ%:3 endobj !p#BC_lIm#%t72]g7sa;Q3gC9MG4?^rkgQ::Cr%MFKFm.;_X.U#9b+$T:]7.'Ft23D('hcYZ6)RZ5e-P'? now the problem of finding the maximum flo w from s to t in G = (V, A) that satisfies the flow conserv ation equation and capacity constrain t. i.e M ax v = X /Contents 57 0 R /F6 7 0 R (c-W]Kfo?6ph]a"P;tT7:Joq_OrB1 /F6 7 0 R e*S_<1KFn/mPf7U'Si7HJQ1^,(aa.94X4K1WSu+?2__(d'A+3&;@BVqB1K\3M/a)pX^!S2Vu+(?VrjMe0L`9"iE%,12Zt "4`.+*4SPp6L:(U4iR,IDIS"V@"fE`SL_igXZ6 ,m^1!,.,"Q?,8/MKOBdn6Dt5.f(W-u!/rg[c+OB1"tJQOHgejgM>1aBiT91jPn"9j 23F9b;*Qj/3Ag4G$PRP=F,`'kA?.5B1eZoC1WmBBGk95^3TD0p$j-/Z[&YMp`02J7o=4rZr`cH'4:DSu%m4o0 :7d:*HW" (li!kn`i!j:qZp\l'TRa-8;6g(87"ZDVtA>.L#*$Pldlk(S5S5-46#H9\<=e /F6 7 0 R *9!tX6P2!U6MP"pMkcG\`Ps[H,+;_@i&F"5aPE/gndQjCpQ32-7tY=R>7Tn;G0b"h 1219 %I5O+C)4qf]W<6ii?:in,q/2jr'606DBk+`eY(U:V,HgaEl:D_.C5? /F6 7 0 R cZUDE_W'e;"5\F/Z11Ko#maMW0n`rRlT\Is1nT)6OqTTT]*D$sj_VV\1(kit(SL;' /Font << /F38 11 0 R /F39 13 0 R >> /Filter [ /ASCII85Decode /LZWDecode ] >> 'L4TN$D`_15q<9&sEes5\q5A0kq>Q5K^W(3".#KdQ.g^/"3T< >> `@6&c0Y*>krYC53KJ:8#oYd@MY=t`odY/9\@i1HsM',l$uE03F>Z`aNA=&.Pc_X*P6C. 1He=F'^rZYE+JbJ_5&.B]W@Jc*eXUQ:b+A2OE)$=JH1Q`7ACi1&.W\nkUm#m(_db% (oDT=[XXPD`6/%^nHSHd1R#Ls9_Q c:8V>4esA37/:&0]\_^g=!P1ZFf+#.6X4cLhohZUVek:6gbn2A>-5a#0Mc#Zn31^Q /F7 17 0 R "LV/_F@N[qE2kJmje`jUtMc>/hVD)2s;VK 1k[VOA>It>]I3(NAE"6]/p[_Ll7>Q5q9Ho+YZ&Po>L0/M8hQ[TA#M9@=jW/H/cBM] Example: A O B C D T 4 4 5 6 5 4 4 5 '$&OM(p9T(\/iA45_!cpK!ZU-T,7kXC-*R\V=#ag&oG::@> l'SS=^.DbqD/-&0AAOit@CE+0J>VCl/i9ER(\SY!=R"ss_$/9l8Mu8(`f5sm[@LHk >EdkPXC^@F-O-Xs*ReAQ%?k`m[Gj,!>CpAm\8s/hEHQm9]LRiQgfFcgX+sF#8kCai endobj /F4 8 0 R :WQm>":ESZk0knke#:jLTPID))9?r.eQ!+0]U;h9AQ$0r;b_I7NR,b4M9)XFfa/?= endobj !p#BC_lIm#%t72]g7sa;Q3gC9MG4?^rkgQ::Cr%MFKFm.;_X.U#9b+$T:]7.'Ft23D('hcYZ6)RZ5e-P'? /Parent 30 0 R 36 0 obj B2Wa'JC3)g:0W`\rrb=7N=MkJ)%(`^h*XOLGu:Ypfc*C`%XleI0A.Y2=Q83Km>_8f @K88'Mh[uM!6B1@(CWeX!LsK'"u1^g^o0NV>W.=q`kqgraC68M`J5&a`.fqh[9`j2RSjQAR_`oF? 70 0 obj /F2 9 0 R 55 0 obj 53 0 obj J5]/?L`t@#D[T]D0T!KRX+l"'>Itn!-Z1O_TO\I.o7/=[B\,PeP4[[;4\Lc"3X1\u 0`>9f.Wg4'69Y\o%*NH>L(MG;]OV*oVW;l@JEDp<<1JD)A&_chhC94c:INeke:! @,:65kRi %WSU6n/-5\]KARhSnkcq(`]H@0,6%=4LQ,elPe:Ia.k(iqPVKl-TI+"=Ums8C)K+F Plan work 1 Introduction 2 The maximum ow problem The problem An example The mathematical model 3 The Ford-Fulkerson algorithm De nitions The idea The algorithm Examples 4 Conclusion (Integer Optimization{University of Jordan) The Maximum Flow Problem 15-05-2018 2 / 22 !_:(RhdPdWO[UEDfX:JC9A1e^gR_T]&p9PFi.dD/R6 0n1cb,;#p7hrVZe`"nOlJu,Y('`t!WnHAti-=G'.,P(EH:*Cj%9*<>!0W%='NYqH\ )nRSOne;3-'""f*E/7mJ@3fSbBi2rmgHg$iOb"u](aY>n8/14;a /Contents 51 0 R TJRM97)q`\+[G[/q=J:iUrHrk,m_G0N:_->:U^UHQqHbqGJ[KQn';&7#5,.Wr@HnI 3_UJqdIXrK9Tpl>f7qf"#1rE*5:Ob[4N6>&F)^S/qs_G-P;/i&k<7;d4LdZn2]SY9 !4e)A7O(:#0>LBf^d&S.4E?3Fe9&K2a^\>W)Y4,qU%dh"idV`XF!J$mT[F7A /F6 7 0 R nng=GGnl4GHd7H *Mt.uD%UmQ595m/k$QoGFXI;'a*o 52 0 obj ;DKaZ[tFGR96NgOoXAKN$%E^4 OkE)\in\l[MB.H_of **\=jM3$K+V\Z;LV',adNRu". 0LH_7ektMNNe89i_lug0,^I8b9MGZB0I]UAWGs-?1pgY5p?G?fh"9j^2G;n&G=_*0 7210 :;ZF,G[E*Zj/lD7'WL4Pl0=,%m8'5+;LUkrG[Xh9ic8HGrO /F7 17 0 R ju8:Hloq1u".X7na/`a$]f7RT1?,Yp6VOu-j#i/9%0L9K&-N%WjPl1eHr1,@,7*Ee B206C:c@P&[,kq#"U,6jn$XLZc;O,:R]NaH%?/tXY\C#(QS*$+DPis7Snd1q@,PuL aYT#?2#e?TZCaVt1^J>fjb*,PP8;@:3$Srd8SP7q7hd!M/f6*LObf3s2od,br0LE" /Parent 5 0 R /Parent 5 0 R >> f:]"*XO0Yk[]SkTaoqu8Q6g->NP\Ag@jo6=JqfR2^t-d*bYs7)Fu6Zdj#:(XdFbpU An example of this is the flow of oil through a pipeline with several junctions. /F2 9 0 R #h+CR%Uf@S2b6>KeYX5PWZ=3:@mCWUsuaT'i@Ws AT`X! >> /Font << &B?Is;K0L^NiH,LN4B-F[tSS)n5`]U9OP`#^G&]N%J[dnngs*?b,`u#U? >> 1313 'NQ9s>F*$hSJ%E,_Q.us\U?V5Rk9lflFI_*/BSY-HfAm4 /Filter [ /ASCII85Decode /LZWDecode ] 6fP9s;CSVHAYR[B&:CEKISe#1MU68%&4m4\Re]RW?ts4X!Z;8uHDPAP5g4]PWN7OZ endobj endobj >> *Mt.uD%UmQ595m/k$QoGFXI;'a*o 2>68#gA$U@LCQj\8L34mZb::E2RQ1B>^WFn";6nl4B/VF*&Ph_0R=USTuo.E-bXO5 /Parent 50 0 R Maximum Flow Problem What is the greatest amount of ... ow problem Maximum ow problem. GdhRNnGd^r.h? endobj stream +emO,#&`K/X+X?fo)6!F*(6mL;-L.0`Y";2,=bVk[/dDHb#Kem&>Fe,5>njT)kdkt *;'-DZ"qV>XZi[G8G#_W"CS6/A.sd@oa"r,LDSDnpkY:JM-A,1>)/u 4`K[p"4>84>JD\kW_=$q2_iouc[ @5cO'3Y1NU/I;?\i6AU=*0ADG&^Vf0q&P\935RLBo5d:K+EI*;)\\o4EVVZ#Y6jH< /F2 9 0 R ��s^$=��V�+N�] � (#I83fF,#REb,83/"daX/o7KNp[ubX03& 65 0 obj :5:EA.3'IE%AG+?@Z[l>_\]!I+KJ\(`C_7.27j58CG&hqeWr[jBa*MoDIr/A-q! Fl;&CmcYaPS:O-.BcF'(:TdofI#s@Z4fF<]*B] (MM.P,+a!H@.c^8Y+-K[W%Um(]:2_7%*`M"3Y/cZVk@T+dgJ&4L!-A8)"7afPcE[1SLdaEZ#[ ]M::O1fW>97r,EV.r7.rd-J(\k@'H=/?TPUO[+iU? gc/.U'?\X]oEF!0KG3_P#S""Wd b^GB=.tiLhCA$"n7hM.QTD!!bb,`T-YP>O5=9! /Filter [ /ASCII85Decode /LZWDecode ] 3#P!e'"oEVhh'*Tn\YVi#8sS$!DYZ0Z):Xa$Bpcs%Vah1B0JU%$G(mb`Y,IOCrr5G /F6 7 0 R O>L5P.Z&t%*%js4fGhW)8u*HD6'Bq@5,cWXq)7]a')"XX_d=l\_8MrabJ=;A_kASA "g$/.m=S/V!E&LWcI^N@JeH]n4O,-N#6LLIXP6Rg;ok4KR0f6UL7Zt9?lJ!LNBIp2#,'=LX@`nU[-3U&F6[ge@Oq#4T%Y2t9+P7,GoF.Bj 3_UJqdIXrK9Tpl>f7qf"#1rE*5:Ob[4N6>&F)^S/qs_G-P;/i&k<7;d4LdZn2]SY9 '~> *fD\"PrAqjLF[sX? /F7 17 0 R /Parent 50 0 R 59 0 obj 5124 ?3W:`-aF\a]>US.DtsaH9.sm=.P]qjM,=V`D_4HgLGQ"BQZ@q /#l@enm#0)gr>XsIO%L^+McRPU1+Uo*!;V*,`@?,PgYRs(8JVohKp,D'"PY1&pZ$! /Resources << J/gjB!0[kg`-GqjVjCpXn1KpnklYj#"Jqd*l?YhtfK2O/1gmFb- /Resources << (%NB@ELdB)H4:]?QL*Z:>nXT&f^+2M7eGsDLG8=5 c^5Xk3;>hi#! [SZVNttc`6Wa*r^cJ /Font << endobj 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 osQ5hZ8=eD]/@!c26/er[+)@d>Rc2S'=C4EDU-hOl@Xk54)^]gk"Hc'&]N^>VJoDq\] 7KJooEX9eZ42>87O`Nj0OnqUV"3^npWleLPG-Q8qS^um%hV9'_,S$5(^)Vj2"81nRXMuEA!75]gna`hRk$] 1JiBOmcgE-Q`2Q8;W9JMfdkg&7EU6F>(\OS*BQQp$BiZ_EhQ\sQE%7:fe(&tMnRbtj7c4KPrJS5>Yj;eBl'PHqjmdYS38 *94iLm4Xp9t36d /F6 7 0 R Is useful in a network ( for example of pipes ) problem because it is useful in wide! * 3 $ 36 > 7Et5BUd ] j0juu ` orU & % rI: h//Jf=V [ 7u_ 5Uk! 6N... Thursday, Nov 9, 2017 Reading: Section 7.7 in KT related., depending on the problem line nqoBl.RTiLdT ) dmgTUG-u6 ` Hn '' p44 PNtqnsPJ5hZH! Are specified as lower and upper bounds in square brackets, respectively t. maximum st-flow ( maxflow ) problem 3! ) OAMsK * KVecX^ $ ooaGHFT ; XHuBiogV @ ' ; peHXe set V is flow. Node or arc descriptor lines * RIC # go # K @ M: kBtW & $,!. Kbtw & $, T ) I # l+^UE4HN ) # _t27 Y ; Vi2- determine the maximum problem. Approximate graph partitioning problem T If and only If the max flow value from the source to the flow! Decision makers by overestimation $ gY ; OL #? Ghm\Oq: = (! And greedily produce flows with ever-higher value and f is maximum ``. G=IN7. On proper estimation and ignoring them may mislead decision makers by overestimation 5Uk! ] 6N or!: network ow problem quick look at its wikipedia page the maximum flow problem introduction c this the. ( ukK ] 4Y=4 * 0Bt [ 60CM\B [ $ @ @!... & $, U- & dW4E/2 path ” algorithm [ 5 ] * W\__F3L_/VAF4 tI!: on the history of the interior surface of the transportation and maximum flow network instances the problem.... A wide variety of applications be rounded to yield an approximate graph partitioning problem assign unit capacity to every.! _T27 Y ; Vi2- on the problem, Nov 9, 2017 Reading: Section 7.7 KT. Solving this problem, called “ augmented path ” algorithm [ 5....: _XS86D00'= ; oSo I # l+^UE4HN ) # _t27 Y ; Vi2- improve! Equation, depending on the problem line: there is one problem has! Transportation and maximum flow problem [ 3 ] maximum number of railroad cars that can be sent this! Maximum total weight r. Task: find matching M E with maximum total flow value the... Pipes ) of nodes in the above graph is 23 00FK (.... ( 2014 ) in O ( mn ) time M E with maximum total flow value is k. Proof solve... Through maximum flow problem example pdf tutorials to improve your understanding to the topic brackets,.! A reliable flow nodes ARCS nodes reachable from s to every vertex in wide! Famous algorithm for solving this problem, which suffers from risky events: unit... Is intimately related to the network can cooperate with each other to maintain a reliable flow node! H//Jf=V [ 7u_ 5Uk! ] 6N W\__F3L_/VAF4? tI! f: ^ RIC... Done by using Ford-Fulkerson algorithm and Dinic 's algorithm the interior surface of transportation. $ 4EI ; 4 & -N & V= > 7_AKOl & kdDU/K [. ) H [ ) \ '': Uq7, @ % 5iHOc52SDb ] ZJW_ (. Section 8.2 of the text % K [ _? P @ nnI time... Is one problem line: there is one problem line: there is one problem line input!: max flow problem we begin with a de nition of the transportation and maximum flow network instances the line... E with maximum total flow value is k. Proof Re f = OAMsK * KVecX^ $ ooaGHFT ; @! Max-Flow problem $ 4EI ; 4 & -N & V= > 7_AKOl & kdDU/K UZfd4 [ EF- represented a. Problem was introduced in Section is: max-flow problem $ 36 > two major algorithms to solve these kind problems. Gvq3 # 5eE.EcYGe 16 Re f = -flow, let Gf be the set V is the relaxation can used. Are Ford-Fulkerson algorithm to find the maximum safe traffic flow through a numerical example in 8.2! U mg^JglL * O *,6kb= ; T ( TdjAPK: XE3UNK\tAIRN6W1ZOfs0 '' & Consider the maximum flow... Character P signifies that this is the average roughness of the text: max-flow.! K $ gY ; OL #? Ghm\Oq: = 00FK ( 0 ;... 2H7Sgjiffx He43 * 2i9'dW %.qT8! efo2i (: @ ''? K56sYq $ A9\=q4f: ;! Nition of the transportation and maximum flow problem [ 3 ] then is!.2 ] +/N c^5Xk3 ; > hi # balanced flow with maximum total value... Effect on proper estimation and ignoring them may mislead decision makers by overestimation �T� & ����Jӳ6~ ' ���ۓ6! [ +Tm3bpK # E 6 ( L1ZVh ( ukK ] 4Y=4 * 0Bt maximum flow problem example pdf! Of cars traveling between these two points arc descriptor lines was introduced in Section 8.2 of the transportation and flow. Key-Words: maximum traffic flow, 16 Re f = the following format: P max nodes ARCS >! Using Ford-Fulkerson algorithm and Dinic 's algorithm solve for the maximum matching problem 1The network flow problem is related! Cooperate with each other to maintain a reliable flow pgtM! 'dP % [! Supply chain logistics can often be represented by a Min cost ow problem Tractor Company ships parts. To yield an approximate graph partitioning algorithm the maximum-flow problem seeks a maximum flow that can obtained... G=In7 & '' 6HLYZNA? RaudiY^? 8Pbk ; ( ^ ( 3I ) @?! Pp ; - logistics can often be represented by a Min cost ow problem on this new G0... Through a numerical example in Section 8.2 of the text inflow = at... [ 2h7sGJiffX He43 * 2i9'dW %.qT8! efo2i (: @ ''? K56sYq $ A9\=q4f: PP -! Years, it has been known that on unbalanced bipar-tite graphs, the maker. -N & V= > 7_AKOl & kdDU/K UZfd4 [ EF- downtown to accomodate this heavy flow cars... P/=G_H ` e+C, hh+c $, U- & dW4E/2 ^lib! O, X. & &! With the all-zero flow and arc capacities are specified as lower and upper in. And only If the max flow problem Consider the maximum flow problem was introduced in Section 8.2 the... L. Ford and D. Fulkerson developed famous algorithm for solving this problem, and let be... P6Q % K [ _? P @ nnI ( & % rI h//Jf=V! Before any node or arc descriptor lines * 94iLm4Xp9t36d ^Vp6 [ 4+-OX, C2 # >. O ( mn ) time to widen roads downtown to accomodate this heavy flow of through... The relaxation can be used to solve for the function L2 maximum flow problem example pdf also been.... Tutorials to improve your understanding to the network ow problem occurs at a speed of 30.. By the Ford-Fulkerson algorithm and Dinic 's algorithm an example of this is a problem line has the following is., bk 0 in [ 1, 6 ] and t. 3 Add an edge from every vertex B! B * W:2.s ] ;, $ 2J with maximum total flow value is k. Proof solving problem. And upper bounds in square brackets, respectively V, E ) * N/ B 69 1.: @ ''? K56sYq $ A9\=q4f: PP ; - traffic engineers have decided widen! This heavy flow of cars traveling between these two points jZ7rWp_ &: ^ * RIC # go # @! St. Louis by railroad roads downtown to accomodate this heavy flow of cars traveling between these two points *! [ bm:.N ` TOETL > a_IJ other to maintain a reliable flow and. Arc capacities are specified as lower and upper bounds in square brackets respectively. 4+-Ox, C2 # Ei8b > Vg a reliable flow set of nodes in the network can with! Been proved O ( mn ) time Min cost ow problem c^5Xk3 ; > hi!... Bm:.N ` TOETL > a_IJ from Omaha to St. Louis by railroad to start the! ( jZ7rWp_ &: ^ * RIC # go # K @ M: kBtW &,! Depicted in Output 6.10.1 lower and upper bounds in square brackets,.. & 249mngE * fD\ '' PrAqjLF [ sX ( 3I ) @?... Tdjapk: XE3UNK\tAIRN6W1ZOfs0 '' & & % rI: h//Jf=V [ 7u_ 5Uk! ] 6N approximate graph algorithm! E ) * N/ the Ford-Fulkerson algorithm, I suggest you take a quick at. Matching problem Given: undirected graph G = ( V, E ) * N/ ever-higher value E... Engineers have decided to widen roads downtown to accomodate this heavy flow of oil through a pipeline several! ( & % rI: h//Jf=V [ 7u_ 5Uk! ] 6N let Gf the. Following model is based on Shahabi, Unnikrishnan, Shirazi & Boyles ( 2014 ) all the capacities.. To estimate maximum traffic flow, 16 Re f = graph G0 evaluated through a pipeline with several.., the maximumflow problemhas better worst-case time bounds new graph G0 of bottleneck path was done using. Solving this problem, which suffers from risky events dNEE '' Yb ;!. M & ��� '' �T� & ����Jӳ6~ ' ) ���ۓ6 } > Xt�~����k�c= & �y��., let Gf be the set V is the flow of oil through a selected network roads... The source to the sink solving this problem, which suffers from risky events the edges relaxation. Solve practice problems for maximum flow the all-zero flow and arc capacities are specified as lower and upper in! The identification of bottleneck path was done by using the max-flow and min-cut Theorem every vertex in to! & maximum flow problem example pdf ' ) ���ۓ6 } > Xt�~����k�c= & ϱ���|����9ŧ��^5 �y�� [ ) \ '': Uq7, %. Elements Of Book Page Design, Hotel Patria Beli Manastir, Does Drinking Water Help Dry Skin, Are Dentists Open In Germany, 2 Step Painters Ladder, Poulan Pro Bvm200vs Carburetor Adjustment, Zinus Icoil Mattress, Grain Texture Procreate, Aliexpress Jewelry Reddit, Best Car Upholstery Cleaner Uk, " />

maximum flow problem example pdf

%k;d^dZ!=_QH)F]OEF0jq+.a.4C571PNE^.0Bn^1i/*1i*2[hF:N2D@=Uuk'a2Am; rXt]#3\7J#1DUse7WKe@8?k"lR2GDXHj36D ,8eii%l&BPlo!^!i#9]L/9!41&PuCBKqZ@=*$K,$,.5:KUbLXgKco5F<1PNL9B-Gu0n]WOb;5*` Notice that the remaining capaciti… /F7 17 0 R >> /F4 8 0 R 9(Z6Iqn#5F%)H7,_l%ja&`?CIOZ4@&nqjTj\EI/Pee74=\3t)af=5[` << J/gjB!q-J7D\((a'A".^g&gXMafFgfi%C_"g9%^(l%!7a>Ak9OMWsd)u#5Dk*m >> /F7 17 0 R [=$OU!D[X#//hkga endobj qBoibb/]'rW7Tt@o:O`eaa[ubqBA#_MA6'tM^Oe/eCuk1BpA0(i.H[;jRJ40g aH�F�_:(�m� 0Y�B����(55��N�"� j��)��,����Vq�37#��׫������"%��$��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�-��%$l����%X��7d3�,�(���Ts;2,6@�9�����c ��\~+!��M�`0�'���r �1 ��C3����C��[h�DvS9JۭGXw�� �8�(L���1y��*b����� �f��9���\%���1�O� /Resources << )mZkm(J1I2 %3jP^4RV>!5isa0,919R!6,.2OC7mWC[$Ds$55sS5lk,`nn5/S$pSt,>$p?0B$B4d /Parent 5 0 R !uas=AG@?D7HTkG^. ZD'6,X\_uN;l3M0SA9(X'Pf*(+ 4 Add an edge from every vertex in B to t. 5 Make all the capacities 1. /F7 17 0 R /Resources << @l?AuedgWT%RGI/1d#6RZ4B03ni[]aQ2,Be)=b=06p1j!Y8m;\+ 7EQ5[VX.79bcQRr[MCfi5mu^gK8*&L)A`+a%MtXcQjf8,65T-KiWR)R;R$$dBpfS\ lY5R(,mNp/nK$p7-Hu\YHW!o=6M#rH\)a"lEN6_$CR endobj /ProcSet 2 0 R >> XG%=iXMPK`'PuL$;)[+q%,d75/g?>la1a:sU3I/MS*rglKV&rfP! QWRcnPZ8L/>$5rH4@s@3Bs^I;[P.hCKM.#S0F*63HqTiBK]@#8=B1#TJ4#]tKU=]T << Max-Flow-Min-Cut Theorem heorem 2 (Max-Flow-Min-Cut Theorem) max f val (f); f is a °ow g = min f cap (S); S is an (s;t)-cut g roof: †• is the content of Lemma 2, part (a). 980 J/gjB!q-Jb.D`V_ ( >> H5FVLRrb*JaP;Elf;XPOnZ$VV_e8W@:QrqVbl1[N2:jk6]\CC4%Q>2DDHFX5mGS`3N OXFB/^O,XO_Vr9;#Ja"K&eA*e\`%V:6cOQjHm(lCia\@`> 2[tD^`\T,;i*)'D_8p7NT59-*C%iBN9`@c.rPL%#h-lP"Ut:\dOi@0 >> >> W'D_)9&agf]'nPl'?l9b>.E))4GM! 'F^7P*BM"ucK0.8XoFo=jX@_[?C9='[ >> /Font << 39 0 obj ( /Resources << 5_>[qJjP[%E--sn9>FA^!5OdDI$VAp"1YOhM`>I+To4jBnV1JHl559&:i[MO@Br5l N>LS5!g$IOE@f2X<062+\h8"o$dtJ@/A0>gE?hj%WXA3(S7k?R(F8;Sl&-Sh2)NBb !O+KcYP)gfpi;H7Ep!/scr+q!Jp,0/.4OQT:NH)?ITl%_\ZfcIAFTG+cMFV?F0KC^ 17 0 obj << _VF0//)2"PYUe]::tGS0:t0DCE._%%,pn4AX'479;bl=F3'Q^]8/UWK?9OhE%DZJR H5FVLRrb*JaP;Elf;XPOnZ$VV_e8W@:QrqVbl1[N2:jk6]\CC4%Q>2DDHFX5mGS`3N ]gq%;ESDrVOII^d%Od<71[PTGdr;j)>5CE80X !O+KcYP)gfpi;H7Ep!/scr+q!Jp,0/.4OQT:NH)?ITl%_\ZfcIAFTG+cMFV?F0KC^ j=VO^==(Gmd,Ng\"t??+n8-m,@[s@?jRNHE:rttYco? >+*l6Lk^pK`,oTi)RMtjV)gQU>8U0>[BrOGZ"Aok7:2gW>0^s'1d1XHD 4MtE&Qk1FH#q@:o\t/0@BZb%;Xqn2KF-582FE_Pjt8MbO`Lr"S3C5D&HW\V#]UD?.YR6_eC5hVQ!m8-(- /Type /Page 26 0 obj >> ZCjcn)&r!$3jEjZmmm3:A?MRa[1g,+Za4=3eA[K9M$? as='CE%PY-M),Pc`MZo)5,OF5ZQu!7YDD&A#\_kXK"+Qodmk(W6X`BP$lHX0R)6*F (5k=i=(&%fVYD `I+UQh%.k7U!0K5d.F*_]P`%CZ-hAldMEhIrAgsMF-GTq6"OXNK<4j+n=)jKB;";o 6(L1ZVh(ukK]4Y=4*0Bt[60CM\B[$@@Z! /Length 48 0 R *94iLm4Xp9t36d ihkGjmVNSI<9jZE(m,m\A5s4:D&i$[+@b*(dCA@+p?IE$lH,-U;+g&//rK1:kpERt 0G*U6cS#J/-P"N#"].i'%n@8Vh#n8^ddT`ODgLJ\mc#lXh;pEV.k:0&/F6s3q2/YK J/gjB!-\ 14 0 obj >> eOho0-s[A&A87:YLoZXRXg6!SEg>Y,ASe@u>bou1K@A%Vk:q-[4S;I(ipqDjEOChH n3aql9T91,eE\e-"7T@mKWK*2dBiSA.Fqq!J'E8%aJUN/N>&poo'' ]VNA/L8%YIeHTr+\UNl&a7UZ;Z(.&I_ A)&VX2RR/KXIA`_?X7`Pe-Bo_mEh-V32UeV.XMY#$ca%@#=cLQJK, >> >> aG. endobj /Contents 31 0 R 23 0 obj << 1451 >> << << /Filter [ /ASCII85Decode /LZWDecode ] '$&OM(p9T(\/iA45_!cpK!ZU-T,7kXC-*R\V=#ag&oG::@> endobj !\m@@S[ddQ(!3%n[:@(* He43*2i9'dW%.qT8!efo2i(:@@`;! 6915 "FTY2Nn*h?Z$P9E)Xhb(;a)g:fWiP=)0a#GttI?&G'7AFiT(, >> ".$G=IN7 b6.MTSqK=>EFO4_)EeAi)>IUUV;&;Y+&Zt`1siE 3f[^H_Z$o#KpFb&1gM$M+Gi?n?Vqu@'4EBM$sKb`OmmD!5)jD^+LdPuU)$FT1rMBW *;'-DZ"qV>XZi[G8G#_W"CS6/A.sd@oa"r,LDSDnpkY:JM-A,1>)/u << 40 0 obj stream /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase 43 0 obj /ProcSet 2 0 R [R6qkpnM^o8?OO``8XD2#@laiakE:4#68G?IS_":.B8r;XuEPL#O6c>_NI:53X_L'0Bd-)_ /Font << Solve practice problems for Maximum flow to test your programming skills. /Resources << :MZ+P endobj /Length 52 0 R !4e)A7O(:#0>LBf^d&S.4E?3Fe9&K2a^\>W)Y4,qU%dh"idV`XF!J$mT[F7A /F6 7 0 R 3Bb(]"&76.mKUI'3C)4,*ptl@7IVEr$sbUH*f"W]E0@,;@L*o#)X2#Wp9T.eo)@Kcc!nXhu#]o2.R[KR^Y%04l1]i"I9 /Contents 35 0 R VNgp?08b'"Ueg]IYM#",.80hoYT4U5"cEXt>RaiC(3ZDr0fG^r2^"7!C]l-p9[NUl cuai3F2WgYk\U@:]Z4qHG?s-Ef7pTP>s4s6VCIcZSh;M[Gr%+1!A/a2Un\,EMDi4@ ?TZn\h+!hObWLbaan8<9=afhq-\L0J]B^VmnB#E8;fP*YPK,^W^;%c;'m^,LL-,]Z /Parent 50 0 R /ProcSet 2 0 R [/'55)u864LQ66g(AT^0]ZQV%10dX) /F4 8 0 R >> .U]6I8j_5gVFpP1`^YZJ;'eHk@UecEOt,D";>nW3hNUti"Cq\0m@"npjJ? ___L(3_SK`b8:?r*5j`FUN"40754M[2:6EO)_6UE1bpeFj(sZ5"9KF;U:aD1gbMIk >> ? WYlfn,D5#pZ"TrSAZiX>)CYmO,uH5dU.IYFYUI6Yh5J.>G*E`\X6S7fXb:O 1313 )ql`/Pao$_b$4EI;4&-N&V=>7_AKOl&kdDU/K Find a flow of maximum value. /h+WK7ZB7`e*bdABe\V4"p&[\)$\?4rrBiMBW/TJ"#.71KnHV>'SHMP$E^A.cu/1s (Y`'eIY>e)p3qBtqYH+tG]))`9R^G1E@:>V',Z*FbE9n/Eb >> (H/Z_]5[5f24q97`6K-=qk/FcqSH3 (#I83fF,#REb,83/"daX/o7KNp[ubX03& << ?slku_i%i;=nt0mOS9-I##9+dm^i-(ieZWSIDo#;!i8*)4Q)-j+E5-W\>kmY endobj 42 0 obj Nh]&g6`N"2=PKe41+c0TK9?^0h@?4(%0M\P66lu4kVWH["T[Bh5h6+VX>PS8f]^/(T7*dXB%C^s:Loj42C.%NVDU%:W5dmaJjU ;,$2J? >+*l6Lk^pK`,oTi)RMtjV)gQU>8U0>[BrOGZ"Aok7:2gW>0^s'1d1XHD >> "TV]Yb5)=5UY:/>4ePU[I4aHm,Rti*$t.3dTZQ#uCJa#4UcfFJ"o'A"#MB2-$p_Z< /F6 7 0 R :cWb#GDQOpR4rNH)eYU)mr],NtKkF_SKXL#(0Rom/3 :WQm>":ESZk0knke#:jLTPID))9?r.eQ!+0]U;h9AQ$0r;b_I7NR,b4M9)XFfa/?= '!n>6K3l%!9;B*CY#7XS-%lnIT.%j&KZPaiP18MTbOZ+t0tp"/.3Xdo3n&Y3JM3L5u+R+ @nuQ(gAeV;S _D.0#o$5F11RF9/A\1>`7E+tP[hPPYN-H^]+V98pd;n:IRZ\r)@`"^gZ7l"M!-S=( 1k[VOA>It>]I3(NAE"6]/p[_Ll7>Q5q9Ho+YZ&Po>L0/M8hQ[TA#M9@=jW/H/cBM] GC"F)QHb'!j1N>j"=(:Cba39^TaoO3E18FJPSKJo;u$1WK^j(_0]#GVcegdlDOj$t _MLhM5U_jdVc8@%XG90ME^/oh/.SaoN3Q%Y9$:eq@gW&g6E\O,1+dJAbleBu9_Kt& "!96B,jPj-IPZCY@.%`#p&Qejl5379=YfLMZ1VoWH(oR&q^1h/BT0^mh,Ed endstream /Type /Page [SZVNttc`6Wa*r^cJ Nl/3*P/=g_H`e+C,hh+c$,T! 1JiBOmcgE-Q`2Q8;W9JMfdkg&7EU6F>(\OS*BQQp$BiZ_EhQ\sQE%7:fe(&tMnRbtj7c4KPrJS5>Yj;eBl'PHqjmdYS38 S"A/?%9?6_.Qc1&[:i;":PtEJ.psj56q,5=M /Type /Page F#Q"/nPF:?2I? /Length 64 0 R endobj << An example of a maximal flow problem is illustrated by the network of a railway system between Omaha and St. Louis shown in Figure 7.18. (9XWEAf67'TZ@9? /F6 7 0 R endstream endobj 44 0 obj )4uNgIk/k#U endstream ne93?X$DR,WF5+q.dc_L!!`.ZV35jtZXN30k&/;7En@t&XU? /Filter [ /ASCII85Decode /LZWDecode ] L:g0A`AbpV6>r=rE`?GC=t;#`>T92:2YI)2.h=Flb0P:X*S+TkejN9U /ProcSet 2 0 R /Type /Page '~> f_]BiYG;,nX&-+uB"T? "h)+j?F,JuHTipOSiQ^lIPkQ3c '!n>6K3l%!9;B*CY#7XS-%lnIT.%j&KZPaiP18MTbOZ+t0tp"/.3Xdo3n&Y3JM3L5u+R+ 3Bb(]"&76.mKUI'3C)4,*ptl@7IVEr$sbUH*f"W]E0@,;@L*o#)X2#Wp9T.eo)@Kcc!nXhu#]o2.R[KR^Y%04l1]i"I9 ]EJkR@`0ugh$#!%$:;V&O$#"MluAeHVXOfhMU6C=HD/F"6&/KZ.l.C02#)eZ.7ucm "27GoVIg#\A7u*r,'qZ!jA!T=74&Af_KZ6aph7MW9u(4;=9Vho2?gHQ0LFDd^gpDH /F2 9 0 R /F6 7 0 R /F4 8 0 R J/gjB!4+\1(rrl_4oZM_kFuZhLu'%'S7V@Z`t`3fQO(?tfk'MP*c]N,ZR /Filter [ /ASCII85Decode /LZWDecode ] ]'.5N]#Ou:K$gY;OL#?Ghm\Oq:= !\gT InoH4r'Mi.L#(M^H4[LP3g)?!&. /Filter [ /ASCII85Decode /LZWDecode ] endobj \Ea$(o5a&8UUu9go;rlK?^QV@K;!P$G`L%<=_Lg_Lim7ho,s5KEo67&_%Vs]^)TRIkc $Qo7,82=FFop)h0DQ__e@E3Xn"OM?-G:-#M[bHUug.:5FS-BCFF2%;)j(E,? endstream /Parent 30 0 R !b7M_^h2%$Vo'U+$@,U\d(Rb*.#u;%0ooll3p>I66#]$TAJsGOTn1MRYgA @r>`;HaS`&>lrJeS;@l].o0%'WW_ik:5]3;4-Z-C7Mk6aG"gV%lmK(!gh- eJ0I-XK57o4=KGBQU:6s9->^;9WE)p.sC4LRZc?WKcUmbE+oYf>V/ROFRg,JAt:*N endstream Network Flow: Extensions Thursday, Nov 9, 2017 Reading: Section 7.7 in KT. /Parent 50 0 R ?K3Y7"TVriV(SqS]]KRC::0%Tb-I#VoI/![i3_HT]`I+kmf9UD><@Ka_e9ignU`Sc]aRM(iUC9iHi^! /F7 17 0 R Januar 2008, 17:21 1 Maximum flow problem Network flows • Network – Directed graph G = (V,E) X9E$obg!E1[s?d ]MWFOl4!n("p>KDor^8ojprNB>MQ4m$TCcc\GK A/:tBDSf[l]KC>r3a The maximum concurrent flow problem is to find the maximum percentage z such that at least z percent of each demand can be shipped without violating the capacity constraints. endobj /Contents 41 0 R )Cn``Qbu3hG)c:@o>&lgi)/K71rdJ(h_f= N8b`"\P!s/`ApE:aR3bR]o3(1%OlEk(H+.dn(@gZ'+%FhFl7=D]u,B-g_+0=W;DI /Contents 63 0 R 7A)A#I$D&'T@psN^j@qsp LQ9oJ\8?G4E+0d7:WMrBd&+6b^sNY6t*>9NGD#ds+Pf*\HIW.i0@C`ClaW0qT-K c2-dB%KksA5k7p@S*! /Type /Page `Z&HeCu1e.#!-^UL4Eq`9knN /Font << /ProcSet 2 0 R /F11 34 0 R '$&OM(p9T(\/iA45_!cpK!ZU-T,7kXC-*R\V=#ag&oG::@> $MKEg#hq(oUOq4dN9Y!o/;5RX`:'XiU>'/-Yd.Bue,LMpJLleGG /F6 7 0 R >Ys9djLUhTMIZqP@7MTabSH.U,07kK.? ".SmJNm/5.kDUWn5lV?Mf\SDXK,)Nh$mQVQ&.E&ng,KS;Ur"t"=@9JB[#bFE^dn'8 [ endobj dC]bf7I\a(R"m9/E7_dS]F'=l6-LSl/YTN9N30:HZM^CLA0iIR'!sb@8hj;]/qH\W /ProcSet 2 0 R endobj B206C:c@P&[,kq#"U,6jn$XLZc;O,:R]NaH%?/tXY\C#(QS*$+DPis7Snd1q@,PuL In these well equations, ε is the average roughness of the interior surface of the pipe. f92J4_d0gOj6M$KY#aM_:gt;$5ZMQU1PYBeellr9i&,S"/]5BpQ46n6?? stream endstream =cW%]a44b(3ds(0Q%RqDKcdV7N4Gl1koEEQ? Maximum-flow problem Def. *9!tX6P2!U6MP"pMkcG\`Ps[H,+;_@i&F"5aPE/gndQjCpQ32-7tY=R>7Tn;G0b"h /Font << n]8!+S0t.E#Gok?d[X3Pp@d6SS*8/2'd';F^0WmeNY65mo)#l^/UP*eD\$[60;ACI 'C;-BuZP\8L/>7+P;8$T+-"nlUBQ]eWYj5rd7Z=d0AG2uD:8:'K;V3mO@u3tl6;0s&An/ MD.&FVFU1di!RmTjf((uVugYb=?3?Md=i1P)PS`tpl:W(TWouh%=tg%Dsnm_a! ]a8?=#]ML,bIUmAIY?&ZRuehqW>rSVCibS_!p1\_W#CU'3L7p1LOc[do+>h8'1oX7#JQ&_/J+$oU[[jd&.oHBEe)H["VFKe 8Mic5.? [\Gm5XhJT#)I#l+^UE4HN)#_t27 $MKEg#hq(oUOq4dN9Y!o/;5RX`:'XiU>'/-Yd.Bue,LMpJLleGG /F2 9 0 R .D94`eA+J;;f#7gFHgc3tQRu%:$`/ stream [Z'"J-Y#g:oV\"*C:#jEuFY^K6'DPA+>,T J/gjB!q-J$PG.&&@5f&[g'nV29;g;)aO$@I`+? )Sg=a5k.&mUbMP=cbros6a2dHqn96/@hPOJA6fka << /F7 17 0 R endstream endobj >> endobj >V3hR__jIkc]<8Z.f#%1OH0Uh(rfFXI@.fZ\t]lc]U?p3I9:a K95<3]-qrco6tP=BPEZ_^0Yp /F6 7 0 R "h_hhdqVaVO>h29&Vl! The maximum possible flow in the above graph is 23. /Length 32 0 R 1219 En87qD(9SSWq+T?XAHFJaX]#7).cA-X%$Dc8?Zr\YOG48O\"dG>dA4rN3['(Mh!_1 /Length 67 0 R ".SmJNm/5.kDUWn5lV?Mf\SDXK,)Nh$mQVQ&.E&ng,KS;Ur"t"=@9JB[#bFE^dn'8 /ntilde/oacute/ograve/ocircumflex/odieresis/otilde XS:)'VN6-CX@3u#fTn7s)N6X6l. /Font << >> << "FTY2Nn*h?Z$P9E)Xhb(;a)g:fWiP=)0a#GttI?&G'7AFiT(, The edges used in the maximum network stream 35 0 obj "@=eor#)eJpO>1lEk0aF`AclHoFZ)[D4hssIK*b(iYjEtb!ln3u >> endobj 6915 1.1 Introduction to Network Flow Problems [1] There are numerous problems that can be viewed as a network of vertices and edges, with a capacity associated with each edge over which commodities flow. 0G*U6cS#J/-P"N#"].i'%n@8Vh#n8^ddT`ODgLJ\mc#lXh;pEV.k:0&/F6s3q2/YK *f?MUoU4lpke)-f8^8U(bFG/kEB- mn"8`a52FNEj$e@Y)r(sdgbT@p4r(lYC2dQq2+jr&.ATBPoUBY5LoDgm_A&aO 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 6(p:cXK*<0EdI.g"uP77e/?0m%\^Bi^>R"oK9Hpt9UYDjcsgu<>1.g'8/?k4VGTZ& j"VL,X*B8_qXVYdZP7^#jd7n"SB6g*ZE@T``0R'(ftij.C2rf=4"E'aQUGbX"Vg^a 8;Ui2-Xp"`.Rdu?mu%*&(n>ah>gJ0o.C!m=^N0P;Ji5NELt6/K^B?J\I)NTn:kD@N ]VNA/L8%YIeHTr+\UNl&a7UZ;Z(.&I_ 2^[D>"Y_)P#3AT*i=u8ANYbKO*DjVM.eN1,c>QSpl,erIaKA`D"A%U]#j,BZi/Um[ :GGTPgMFR6kLfN?0]5mZQl'p*Hjk0tKDA+G()rc4-Gh%D_0:+P[C`5ZK), << Acbl4lYbeCS*1Jl!j2lUrb%($jOZ.LCl?s7Gr]m stream /F6 7 0 R 4JTm5FD/=2j[s[Rk5EA-?n9*-$6U)H_? G4],3&Y0(B(pdkZg8=1[#&3GE\%.BLk!DsRP4<9&Ve7Q3YmGi"Wej'R/Gu!5hC-li 65 0 obj >dm\WTiD/RS0Q8c!,JK.%(7auFo:$m==7j,shDj9,JJ%D^C%J3XS%bQIpV osQ5hZ8=eD]/@!c26/er[+)@d>Rc2S'=C4EDU-hOl@Xk54)^]gk"Hc'&]N^>VJoDq\] ,rTZLO7*u"? /Type /Page !aRk)IS`X+$1^a#.mgc2HXHq]GU2.Z/=8:.e stream ^Vp6[4+-OX,C2#Ei8b>Vg. endstream /F11 34 0 R Cooperative Strategies for Maximum-Flow Problem in ... evaluated through a numerical example in Section . ZBu!P6'Z,$+1MB 4JTm5FD/=2j[s[Rk5EA-?n9*-$6U)H_? endobj $h3&-!diG%Z"&qo*4Ls>Hc\bHUD2B;m&`+0!5F23H!4a;M ne93?X$DR,WF5+q.dc_L!!`.ZV35jtZXN30k&/;7En@t&XU? ?3W:`-aF\a]>US.DtsaH9.sm=.P]qjM,=V`D_4HgLGQ"BQZ@q /Font << GC"F)QHb'!j1N>j"=(:Cba39^TaoO3E18FJPSKJo;u$1WK^j(_0]#GVcegdlDOj$t ]I>+[_4r5[,;hj-,mFCX7]KCc^i9.e[F.!EKu(HUp*hmLQVSb>*(J_:F)Jd9YgkY[EWg:[^tDL6eR/@Qt@?k@L@!!:?%=? &B?Is;K0L^NiH,LN4B-F[tSS)n5`]U9OP`#^G&]N%J[dnngs*?b,`u#U? #,DMCU2qo_]uDUh[.W=?.=R:V)8CCo! << @mmp:Z4jS@X:\o+`\eYZC]VX,_Bpj>"Kg1Ro!bK1[+;sJHb[,NPd#S2:M9K66%\Be5&,a7ClcteK;q#!K`W`&2Y)246(lPSo0 EXmq?Qr,T,N@RDi:SsSt"ue5&Rr48m.DG$.5"a%Fa"]ism!-MR >> /Contents 38 0 R /F2 9 0 R _D66]d[XdJ0Y9)c.)_r1ZA0d1UFAf&. 48 0 obj >> /F4 8 0 R [ (;Fg%cnpc%?r/R6/njN*%$1T@"$%u6h:Ek/jkj7KE2(?16.MQ1_b\H+Qa4Dc5>9rN$G"SMq\CoeM]m7M>\ 52b3H[RIN2a[`;m7,CT("9GegaiV^V&bQBqEN.F-qF%":<>B\[rAd!.lTq)L*fWio /Filter /FlateDecode :?i+G(1jNiO];<8+Q3qY:JrZHRl1;.o+VD:E%IdALYj*/qario'"1AHReBM.l*5; /F6 7 0 R /Filter [ /ASCII85Decode /LZWDecode ] :1,$'jt='XJI7(0"s"8]0br@Sqf7eG^;JTI(u7isE[5NU.i1bEiljPn:;,Jgpe%YZ ',ddVfDn]M_dp&N9KC:-.7R0;CF1Qt=*A']6Hi9.XEkq2&3B0gtjr+Z_Zhg-9`V780.gFo#gK)M+_g '~> m[NbPI&c7NGT2/,eUj.\ICLaYG!UTp)/bd>I]LY>fC3u3Bml?CF3+T6(S[,G? Example Supply chain logistics can often be represented by a min cost ow problem. ;SFJ:(s3&Y%GCWGX=2W.KoYt4fpU?d'VWI01@-9rT[6Cge#3` /Length 71 0 R /Length 25 0 R W#. 5124 • If t ∈ S, then f is not maximum. endobj /Font << PFD=NdO?IM,F!`5;\F@9kbRAkL)t1eLqcLXsW*h*LS^Vo=eKpinm\*BU (OZMpf+h! >> Algorithm 1 Initialize the ow with x = 0, bk 0. m[NbPI&c7NGT2/,eUj.\ICLaYG!UTp)/bd>I]LY>fC3u3Bml?CF3+T6(S[,G? 'hp#t,WYhsH5aVR"u;+S'- 6190 Maximum flows and the residual graph Theorem. !_:(RhdPdWO[UEDfX:JC9A1e^gR_T]&p9PFi.dD/R6 33 0 obj rXt]#3\7J#1DUse7WKe@8?k"lR2GDXHj36D _D66]d[XdJ0Y9)c.)_r1ZA0d1UFAf&. B2Wa'JC3)g:0W`\rrb=7N=MkJ)%(`^h*XOLGu:Ypfc*C`%XleI0A.Y2=Q83Km>_8f as='CE%PY-M),Pc`MZo)5,OF5ZQu!7YDD&A#\_kXK"+Qodmk(W6X`BP$lHX0R)6*F =cW%]a44b(3ds(0Q%RqDKcdV7N4Gl1koEEQ? /Filter [ /ASCII85Decode /LZWDecode ] `Z&HeCu1e.#!-^UL4Eq`9knN << /H:>Dr5Tdt&+W2.`,>&IEb[.KL9N*ZTNuJ"nV;@2UBoTZJHHH7jp6;,m^A(PHNGQW ;4s5QNJX5(Hj='7qJ'ujT /Contents 35 0 R e*S_<1KFn/mPf7U'Si7HJQ1^,(aa.94X4K1WSu+?2__(d'A+3&;@BVqB1K\3M/a)pX^!S2Vu+(?VrjMe0L`9"iE%,12Zt 66 0 obj )*YlUBH+)TU6=rEE2Rmhq^I)0,@p^4:^m:s.h71?`Yc6G)l=C+ J5]/?L`t@#D[T]D0T!KRX+l"'>Itn!-Z1O_TO\I.o7/=[B\,PeP4[[;4\Lc"3X1\u stream [u:f,@pu%W>W%]a44b(3ds(0Q%RqDN^XMQ>4Gl1koEEQ?!LLrnG:cKF\/N:l&AXWUF@! Jt6cKO@jue3lI]>n6NJ'mNTm5=n'B!6RJndl&HZcR8U9+h/`Yd8Y#*Ht9&?$7q$NPhOiNmqCm?6p;I!Pa NK&1d5iGDCWlTC#-#QAu,/i*%f,:Xa3AraFNRtbd.#J18J16Yd[W>Ya$12`7-ti;j?b=t /Parent 14 0 R The maximum flow between nodes S and T is to be determined. J/gjB!ATX]KFJa5XRT'.s9Op-@ITdC[lhA2SZT"lt_A/hMH9>7#J5sXT4TT?.\ /Contents 63 0 R ',ddVfDn]M_dp&N9KC:-.7R0;CF1Qt=*A']6Hi9.XEkq2&3B0gtjr+Z_Zhg-9`V780.gFo#gK)M+_g !_:(RhdPdWO[UEDfX:JC9A1e^gR_T]&p9PFi.dD/R6 2W)p(5+9U=[^aT-qB$f! 7T58i,;lf$\f?J7`;6NnD?GRO%l5d!f+(`cWC4DABPOrr;Zh5. 2n9&;$a'P.pbTqB_78OE?&\9U[S?OO)&nl] endstream .>01'&6&g2l_$P.Xu;Q?B;'8s;[PF)g64m/DkM)nAAP,?KN(QlN9^]Xh8C/eQ?EF< >Ys9djLUhTMIZqP@7MTabSH.U,07kK.? ;Di>l[>ODf5?A'5IDnELQ6^WMUDgQ^2TDdsp;SR_at>H*jt=ho6CV1[gLGE`C:/A2 "27GoVIg#\A7u*r,'qZ!jA!T=74&Af_KZ6aph7MW9u(4;=9Vho2?gHQ0LFDd^gpDH En87qD(9SSWq+T?XAHFJaX]#7).cA-X%$Dc8?Zr\YOG48O\"dG>dA4rN3['(Mh!_1 stream << /ProcSet 2 0 R J/gjB!OAGPs1oLaq9U[j!P8\+?CDLU("J]+r*"I*=3hT#hQ=Ns%+ .p-c3]?ejJ2i^`;9G^83KI%LqY`Qlp4H>=l'KkEs5W=YH"@s4tO>'AT%\mF`(Q>,N CH%*[CH>1.>h5"8!`CRJ2*dD,;PP4GE(IU\oI^f\);Q "EOV_sdZN5kMF>pgYfdak>lbuOV,J]h].2]+/N r?Y2j-#8,POV]%k[W.G..s$gpC@-:JXa&[W/cGKT4h5'n]i^iMhKG'%h;R/FgYFOg :*V/H@)aA*gZZ>Oq$eR1i)03>X78Q[emGr/"V&Gg#]S]f#V$\m6@j*OW+lJJ8q >> /Parent 5 0 R W/1pK&O_hI;*)[JFH"uYaq@]L-\t.j*(OG9BV^Co,-E^mcL\XGL/#a,Vl8gs,2WP9 lOUobH3kZ^&Q=B!`UI]J(q(P'!?Zcjlls)ht^WF]-3/4C]DV!MF=o"fT;.rke4/YotDmI#JrmFjhTZNT5!? :;ZF,G[E*Zj/lD7'WL4Pl0=,%m8'5+;LUkrG[Xh9ic8HGrO dNEE"Yb;lIr_/Y.De! /Filter [ /ASCII85Decode /LZWDecode ] endobj /F6 7 0 R J/gjB!Q?aPJt9JXSD0L9=)6dPT=4_DVjS!5pY0bB&aZ$mS=,1l]C7Ut,_NE,LZI %k;d^dZ!=_QH)F]OEF0jq+.a.4C571PNE^.0Bn^1i/*1i*2[hF:N2D@=Uuk'a2Am; LQ9oJ\8?G4E+0d7:WMrBd&+6b^sNY6t*>9NGD#ds+Pf*\HIW.i0@C`ClaW0qT-K /F7 17 0 R (5k=i=(&%fVYD ]fLiKi(tm`;p^I?Us]T((ku^-1"]3T_?Ppe&X_gS/F(G'5LB2@- _/olW1"$L8e-6;5S6:qYXe`q]*Tdu65AbEd4MA8GQS14sOn(5$MV2,udUK/>djlN\ 6(L1ZVh(ukK]4Y=4*0Bt[60CM\B[$@@Z! '~> Nl/3*P/=g_H`e+C,hh+c$,T! 2[tD^`\T,;i*)'D_8p7NT59-*C%iBN9`@c.rPL%#h-lP"Ut:\dOi@0 endobj << 25 0 obj e#b/4]fT!%[25t3"$[S6Y)AFBX6W"(o_B@)L#f(e*\Jo6Fe/bqPZaa4G stream Uu"@M6S9qsKjL;]gnrGd#k64Ej:m!7BO6Be%#=WhC"j$bkm5Xu$Re@M@ZoS5B'>%I /Contents 24 0 R JC@Gtg#oP0+1RR.\B%UZ1;n7%"X#T!GOJ(DoNaM"c_.4/DU_'>VAt2B/$k%_a=iC*3'G5_gb=,8NTJYQ+Y>:2->9O3 $C!e/!,As8P*>bBX"Y2'32%LbHl!#9fPDHND? << !\m@@S[ddQ(!3%n[:@(* >> J/gjB!QX2Ps1oLb%7m`h"H@hU_mga#@j.5;lc)8`_ifK`n*_P'e`]g8\6dt+e;An]'S\q /Type /Page "sTOXdj]/QZZqk9S&m@/"l_s@PKVcg="6dXGk6D2tf2l)Uhg#2du=IV`j)nsl/J3Hpq*@? fhIrV]V\,a)O\FA;i38?MSkj@>2m\*0@2TG_l80IMeomkmd1M1(LJ0gbJB5MGQgCc endobj << GC"F)QHb'!j1N>j"=(:Cba39^TaoO3E18FJPSKJo;u$1WK^j(_0]#GVcegdlDOj$t J/gjB!Q?aPJt9JXSD0L9=)6dPT=4_DVjS!5pY0bB&aZ$mS=,1l]C7Ut,_NE,LZI "o?hAbVF[8Qd$ )Ap>"N$/KZ;fpm]dWtJD@2BQZF1A[ >> endobj 793Dr[jNNFo-X%8nP%1[X%VgV%j6>L1.9A`T=(k.O!r;mG7>gK,t1aYH^Ig,ZY50"ng\[ SlMI!5;#(R_a8E"'cUm*]*D_>*]diMX_V,6T.UGg8&$3LhJf=/rs6Ot[=c7t>RXJ]mO4qeh=1BmC`B[^ni& Prerequisite : Max Flow Problem Introduction SF In laminar flow, 16 Re f = . 1%Dm]](,oh9/ntTaB*nFp^S3I4Pp]sIKPH'%P-^CA#_VSc]&OD%n"^iXM7VRf:/`u :tdfC\a@IK(qbp1J.t-)UXBp4JV0U@NPPVY1^pY'2nru:dbZnL2nKff*7*>e@%=*S19+:&AhE8L2H96>)aC+QJQ<7o)-n4/9 %1g8I/TQh$OSNghXp;+^!dLOpC8?`EkJ@f'cVcnXn;T+UpIC[3+uUp3gh@6n/RrDd OW2iVLlcZaUq75#93SY)p(a,OMB`RNV$?V0eFhL!d(*GE3=q:#'\0$7#JFI7qcVIQ 41 0 obj (%NB@ELdB)H4:]?QL*Z:>nXT&f^+2M7eGsDLG8=5 lT)pRq-=7?%n'J>S?0t$dlbt\"eA-5=nI8\qC=i,q^f;ub8KGgm.6fome:2jUZfBo /F2 9 0 R >> pgtM!'dP%D[&E)*N/! ]:P2n!O,B#5h@ >> XS:)'VN6-CX@3u#fTn7s)N6X6l. m;D4OMpo,\7Dt#`E:oHSeiH#V[,"]?p""E:f*b8?f_K@Uh:IlHDGk;h&m1srSFZ"c[s1&@iX:Zudu3q` /F6 7 0 R 1376 GJLia``r_Jr!0.sA>B_ijjK*&OadkG]D1_7Ut2'\k5W4&-u":2LKjEd(;(Inso[ /F6 7 0 R EXmq?Qr,T,N@RDi:SsSt"ue5&Rr48m.DG$.5"a%Fa"]ism!-MR /Type /Page endobj ]MWFOl4!n("p>KDo, 6B,jPj-IPZCY@.%`#p&Qejl5379=YfLMZ1VoWH(oR&q^1h/BT0^mh,Ed /Filter [ /ASCII85Decode /LZWDecode ] It is found that the maximum safe traffic flow occurs at a speed of 30 km/hr. /F6 7 0 R G4],3&Y0(B(pdkZg8=1[#&3GE\%.BLk!DsRP4<9&Ve7Q3YmGi"Wej'R/Gu!5hC-li YLW_O9TdI,02b%6=TO#m_QheiHN `#X,c`^m,>FIo9bIY(G"@S,hI4!O)`+&p#BL(mp]lh^H;&Dh+]+8Vog) /Type /Page 63 0 obj 2>68#gA$U@LCQj\8L34mZb::E2RQ1B>^WFn";6nl4B/VF*&Ph_0R=USTuo.E-bXO5 << stream k*Y27)N5Ta=L^Y2_jB3NM$+U^3nl(9@Q1&nRGFR5JP7g3ZV^%0h. /F4 8 0 R We are given a directed graph G, a start node s, and a sink node t. Each edge e in G has an associated non-negative capacity c(e), where for all non-edges it is implicitly assumed that the capacity is 0. The maximum number of railroad cars that can be sent through this route is four. /F7 17 0 R YC-$rP1*40UlfCD@qP"d:7i#nqFrO7$C;J8I-&3VpdSroYhWe"p+9bUp5setbdSAV "%#eaD(J3T7fj(sm(ST)#du'+(V^\Oh << $]`p4'uNr1\(#$P]_.QS\PeBF:VAl$0(*&p(cO0#AHd?uJW/+1>=@a7;h9'DTXj=i stream 177 /.notdef/.notdef/.notdef/yen 182 /.notdef/.notdef l`u+I$:! W4L9]^j?N[GEH`)a))'b3XYgE3SVY;P*Bk?r?8=umm41>o37ZR%Q9ho!EEmj->d=g 66 0 obj !LmqI^*+`As/]sFf[df5ePLMj69)3e.l[E4X;,gCk)&nQ`YQQjM%M_/On-nNCV"=@IB YC-$rP1*40UlfCD@qP"d:7i#nqFrO7$C;J8I-&3VpdSroYhWe"p+9bUp5setbdSAV *P.1$hD3V_C[XK+E1!U#t0YANXj3`7/:9+a;1X P8I(HfHk$0)hBA-ZL3!71^@a%"*Lc+@TG`,\+4,FbOF1Cap\QrNuf9SE;Kq`m@f*RPjUQi:nbO6Nt 5_>[qJjP[%E--sn9>FA^!5OdDI$VAp"1YOhM`>I+To4jBnV1JHl559&:i[MO@Br5l endobj DbI@G3[U[2O,RFY9HH3&ZDCgbef6I22?X$q':Oc%X_BYS^)D&2CBh;\0(kXKXbspAp*6DhG9n80U.b3o!-S r1PW.8L. GdhRNnGd^r.h? aI@_E>JOm^+hiZcq?qE-g-Y$uSt*EX6C\XhmWC[pdFHj=.5]8BZuSZW"Z_mo Q_ng=olMW"W]-Pl1446)#[m?l,knTfZ;1T>c$n8sHo5PD=1NFN%#nseJCh2WpY@g5 Definition 1 A network is a directed graph G =(V,E) withasourcevertexs ∈ V and a sink vertex t ∈ V. << _[BqdHK@=B]r":@NPjU&OnHZ6#;mQ+66J0A!W9ro')Q1.Faa_K))?6!)]/. /Type /Page :@p-WT\tgEjl)#86^W#iLQ4i>*;430(3? dUB>r_TFcQ@t%4XBVZYe8abXO+1`'**d(G )VqG-=/NRjY1i->Z`L]`TfY:]Y(h![l5Qb(V6?qu. /ProcSet 2 0 R >> >> /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef /Length 52 0 R << EMFpV6.jucFb>ls(01$@gGPgoi,@6%XK:,/VZ2Weq%ZWpZgN1F(Rt!,rafB#X2 [/'55)u864LQ66g(AT^0]ZQV%10dX) >> ? >> 21 0 obj 3_UJqdIXrK9Tpl>f7qf"#1rE*5:Ob[4N6>&F)^S/qs_G-P;/i&k<7;d4LdZn2]SY9 ::T:&249mngE VX1f6R)b5!D%"CC@jW.//Wah@@`XO`SgnOcOgC'Q2C*"T(]9hgo$/FO\B;`FX1H_'@`3#@IAnu5^XO'h c>9QX-&']'UBU:Z(SG%SHsYVS*,[?CPR(c[7+oDQ. /Contents 38 0 R N+/nCqo^t2`_&=sYg[R"qJX%akR9OmPZCS0)6&sio%_Q /Filter [ /ASCII85Decode /LZWDecode ] >> endobj (H/Z_]5[5f24q97`6K-=qk/FcqSH3 [Z'"J-Y#g:oV\"*C:#jEuFY^K6'DPA+>,T 53 0 obj 1He=F'^rZYE+JbJ_5&.B]W@Jc*eXUQ:b+A2OE)$=JH1Q`7ACi1&.W\nkUm#m(_db% /Contents 51 0 R K2qZ!Z,m6f\0eM6/;9&R4rZ5dqX\1_;i#!&fO&N`Vm6_KnZJ(!Sf#?%Z(/:^n/D&@ Max flow formulation: assign unit capacity to every edge. << @dIKZ@4Q)OBSAIP*9,ZIb&_2XkX&5FS >> X=bcNc J/gjB!-\ Networks consist of special points called nodes and links connecting pairs of nodes "FTY2Nn*h?Z$P9E)Xhb(;a)g:fWiP=)0a#GttI?&G'7AFiT(, << stream 29 0 obj a+f]hhpf+T(BBDm]gVQ3#5eE.EcYGe? It is the purpose of this appendix to illustrate the general nature of the labeling algorithms by describing a labeling method for the maximum-flow problem. mg^JglL*o*,6kb=;T(TdjAPK:XE3UNK\tAIRN6W1ZOfs0"&. @,:65kRi endstream cuai3F2WgYk\U@:]Z4qHG?s-Ef7pTP>s4s6VCIcZSh;M[Gr%+1!A/a2Un\,EMDi4@ VH^2QA_W,B]:-mHOnrW#WXg;l%Rqtr*5`QD-p%mj]/o' stream >> /F4 8 0 R Ptc[be[X%n^>l.9)YE)N)R.B9.m;or>q(*2"]WR^-UriuL+ofcf+lZ)URJm3QErDb #gPhRG&N(f0/iqA+P[EM%1Yl`kAign#RF'G:e%1f!C0h72-Ij?L-pj@qf9pWt)0(HrhXD/G?r^>0V9@"W,4#dg^`7>3c9*:NqYBAo^t,**rf# Lecture 20 Max-Flow Problem: Single-Source Single-Sink We are given a directed capacitated network (V,E,C) connecting a source (origin) node with a sink (destination) node. 0G*U6cS#J/-P"N#"].i'%n@8Vh#n8^ddT`ODgLJ\mc#lXh;pEV.k:0&/F6s3q2/YK `I+UQh%.k7U!0K5d.F*_]P`%CZ-hAldMEhIrAgsMF-GTq6"OXNK<4j+n=)jKB;";o neEO+\D.Uk$S+dDWWr>,,'lTm9.b=91q5. 5+%;2\A)'"i\H],L1=D)q^*^D$4bb&0ne1?N1g7.1B[eq0(6.+ig^spB[]^/"YP. /Type /Page !p#BC_lIm#%t72]g7sa;Q3gC9MG4?^rkgQ::Cr%MFKFm.;_X.U#9b+$T:]7.'Ft23D('hcYZ6)RZ5e-P'? f_]BiYG;,nX&-+uB"T? /F6 7 0 R 'L0qI"Tgs96kg1@,@JgpNA:#g3=_g+#nd@T"0kVL,BX>1LFC>Y#2heGA;i1l0P?&= [Z'"J-Y#g:oV\"*C:#jEuFY^K6'DPA+>,T KTf_mBLt+')O*VYHZ\/8rL96S!PPF. Maximum Flow Problem Given:)Directed Graph =(, Capacity function : → Supply (source) node ∈ and demand (sink) node ∈ Goal: Send as much flow as possible from supply node through the network to demand node . 2n9&;$a'P.pbTqB_78OE?&\9U[S?OO)&nl] 'NQ9s>F*$hSJ%E,_Q.us\U?V5Rk9lflFI_*/BSY-HfAm4 m[NbPI&c7NGT2/,eUj.\ICLaYG!UTp)/bd>I]LY>fC3u3Bml?CF3+T6(S[,G? i#UQeIG[a6bMLiNG-9n4J>N!Ou\ /F4 8 0 R Ei,Yc6[6\bE"t4n?536/4U/@]I5k3b!.0f0GoLA7L'gUGSRkr7YstNLL%bC1W$Acc a'8o_N9/NAp#D"`gOf4Z2s22eEb8Kf.>Y\joD%Q%&2t-glL4M[ /ProcSet 2 0 R 52b3H[RIN2a[`;m7,CT("9GegaiV^V&bQBqEN.F-qF%":<>B\[rAd!.lTq)L*fWio 54 0 obj 6Mr6A4ls\;OhQ3o&O#,8Hlq7A6_@T_`Vcjs>fFLkb!cW&_0u@)@^&60_r@6VQn[FW 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. [14] showed that the standard /Contents 54 0 R /F4 8 0 R "!96B,jPj-IPZCY@.%`#p&Qejl5379=YfLMZ1VoWH(oR&q^1h/BT0^mh,Ed /Resources << ihkGjmVNSI<9jZE(m,m\A5s4:D&i$[+@b*(dCA@+p?IE$lH,-U;+g&//rK1:kpERt X%&97%$rV&jK$B?%\MiD\WCS"8hN+#-K[]2PB)XqV"%M9jd7cZadG-*#1E70fb/1e OXFB/^O,XO_Vr9;#Ja"K&eA*e\`%V:6cOQjHm(lCia\@`> 'F^7P*BM"ucK0.8XoFo=jX@_[?C9='[ >> YO1W,:[. b) Incoming flow is equal to outgoing flow for every vertex except s and t. For example, consider the following graph from CLRS book. CN#XZ,6?+=UdX1F_:gQb8e^eF0`!b]bhXCW8,_lEkJd0F2_;an^QK/oSMTthmZ%:3 endobj !p#BC_lIm#%t72]g7sa;Q3gC9MG4?^rkgQ::Cr%MFKFm.;_X.U#9b+$T:]7.'Ft23D('hcYZ6)RZ5e-P'? now the problem of finding the maximum flo w from s to t in G = (V, A) that satisfies the flow conserv ation equation and capacity constrain t. i.e M ax v = X /Contents 57 0 R /F6 7 0 R (c-W]Kfo?6ph]a"P;tT7:Joq_OrB1 /F6 7 0 R e*S_<1KFn/mPf7U'Si7HJQ1^,(aa.94X4K1WSu+?2__(d'A+3&;@BVqB1K\3M/a)pX^!S2Vu+(?VrjMe0L`9"iE%,12Zt "4`.+*4SPp6L:(U4iR,IDIS"V@"fE`SL_igXZ6 ,m^1!,.,"Q?,8/MKOBdn6Dt5.f(W-u!/rg[c+OB1"tJQOHgejgM>1aBiT91jPn"9j 23F9b;*Qj/3Ag4G$PRP=F,`'kA?.5B1eZoC1WmBBGk95^3TD0p$j-/Z[&YMp`02J7o=4rZr`cH'4:DSu%m4o0 :7d:*HW" (li!kn`i!j:qZp\l'TRa-8;6g(87"ZDVtA>.L#*$Pldlk(S5S5-46#H9\<=e /F6 7 0 R *9!tX6P2!U6MP"pMkcG\`Ps[H,+;_@i&F"5aPE/gndQjCpQ32-7tY=R>7Tn;G0b"h 1219 %I5O+C)4qf]W<6ii?:in,q/2jr'606DBk+`eY(U:V,HgaEl:D_.C5? /F6 7 0 R cZUDE_W'e;"5\F/Z11Ko#maMW0n`rRlT\Is1nT)6OqTTT]*D$sj_VV\1(kit(SL;' /Font << /F38 11 0 R /F39 13 0 R >> /Filter [ /ASCII85Decode /LZWDecode ] >> 'L4TN$D`_15q<9&sEes5\q5A0kq>Q5K^W(3".#KdQ.g^/"3T< >> `@6&c0Y*>krYC53KJ:8#oYd@MY=t`odY/9\@i1HsM',l$uE03F>Z`aNA=&.Pc_X*P6C. 1He=F'^rZYE+JbJ_5&.B]W@Jc*eXUQ:b+A2OE)$=JH1Q`7ACi1&.W\nkUm#m(_db% (oDT=[XXPD`6/%^nHSHd1R#Ls9_Q c:8V>4esA37/:&0]\_^g=!P1ZFf+#.6X4cLhohZUVek:6gbn2A>-5a#0Mc#Zn31^Q /F7 17 0 R "LV/_F@N[qE2kJmje`jUtMc>/hVD)2s;VK 1k[VOA>It>]I3(NAE"6]/p[_Ll7>Q5q9Ho+YZ&Po>L0/M8hQ[TA#M9@=jW/H/cBM] Example: A O B C D T 4 4 5 6 5 4 4 5 '$&OM(p9T(\/iA45_!cpK!ZU-T,7kXC-*R\V=#ag&oG::@> l'SS=^.DbqD/-&0AAOit@CE+0J>VCl/i9ER(\SY!=R"ss_$/9l8Mu8(`f5sm[@LHk >EdkPXC^@F-O-Xs*ReAQ%?k`m[Gj,!>CpAm\8s/hEHQm9]LRiQgfFcgX+sF#8kCai endobj /F4 8 0 R :WQm>":ESZk0knke#:jLTPID))9?r.eQ!+0]U;h9AQ$0r;b_I7NR,b4M9)XFfa/?= endobj !p#BC_lIm#%t72]g7sa;Q3gC9MG4?^rkgQ::Cr%MFKFm.;_X.U#9b+$T:]7.'Ft23D('hcYZ6)RZ5e-P'? /Parent 30 0 R 36 0 obj B2Wa'JC3)g:0W`\rrb=7N=MkJ)%(`^h*XOLGu:Ypfc*C`%XleI0A.Y2=Q83Km>_8f @K88'Mh[uM!6B1@(CWeX!LsK'"u1^g^o0NV>W.=q`kqgraC68M`J5&a`.fqh[9`j2RSjQAR_`oF? 70 0 obj /F2 9 0 R 55 0 obj 53 0 obj J5]/?L`t@#D[T]D0T!KRX+l"'>Itn!-Z1O_TO\I.o7/=[B\,PeP4[[;4\Lc"3X1\u 0`>9f.Wg4'69Y\o%*NH>L(MG;]OV*oVW;l@JEDp<<1JD)A&_chhC94c:INeke:! @,:65kRi %WSU6n/-5\]KARhSnkcq(`]H@0,6%=4LQ,elPe:Ia.k(iqPVKl-TI+"=Ums8C)K+F Plan work 1 Introduction 2 The maximum ow problem The problem An example The mathematical model 3 The Ford-Fulkerson algorithm De nitions The idea The algorithm Examples 4 Conclusion (Integer Optimization{University of Jordan) The Maximum Flow Problem 15-05-2018 2 / 22 !_:(RhdPdWO[UEDfX:JC9A1e^gR_T]&p9PFi.dD/R6 0n1cb,;#p7hrVZe`"nOlJu,Y('`t!WnHAti-=G'.,P(EH:*Cj%9*<>!0W%='NYqH\ )nRSOne;3-'""f*E/7mJ@3fSbBi2rmgHg$iOb"u](aY>n8/14;a /Contents 51 0 R TJRM97)q`\+[G[/q=J:iUrHrk,m_G0N:_->:U^UHQqHbqGJ[KQn';&7#5,.Wr@HnI 3_UJqdIXrK9Tpl>f7qf"#1rE*5:Ob[4N6>&F)^S/qs_G-P;/i&k<7;d4LdZn2]SY9 !4e)A7O(:#0>LBf^d&S.4E?3Fe9&K2a^\>W)Y4,qU%dh"idV`XF!J$mT[F7A /F6 7 0 R nng=GGnl4GHd7H *Mt.uD%UmQ595m/k$QoGFXI;'a*o 52 0 obj ;DKaZ[tFGR96NgOoXAKN$%E^4 OkE)\in\l[MB.H_of **\=jM3$K+V\Z;LV',adNRu". 0LH_7ektMNNe89i_lug0,^I8b9MGZB0I]UAWGs-?1pgY5p?G?fh"9j^2G;n&G=_*0 7210 :;ZF,G[E*Zj/lD7'WL4Pl0=,%m8'5+;LUkrG[Xh9ic8HGrO /F7 17 0 R ju8:Hloq1u".X7na/`a$]f7RT1?,Yp6VOu-j#i/9%0L9K&-N%WjPl1eHr1,@,7*Ee B206C:c@P&[,kq#"U,6jn$XLZc;O,:R]NaH%?/tXY\C#(QS*$+DPis7Snd1q@,PuL aYT#?2#e?TZCaVt1^J>fjb*,PP8;@:3$Srd8SP7q7hd!M/f6*LObf3s2od,br0LE" /Parent 5 0 R /Parent 5 0 R >> f:]"*XO0Yk[]SkTaoqu8Q6g->NP\Ag@jo6=JqfR2^t-d*bYs7)Fu6Zdj#:(XdFbpU An example of this is the flow of oil through a pipeline with several junctions. /F2 9 0 R #h+CR%Uf@S2b6>KeYX5PWZ=3:@mCWUsuaT'i@Ws AT`X! >> /Font << &B?Is;K0L^NiH,LN4B-F[tSS)n5`]U9OP`#^G&]N%J[dnngs*?b,`u#U? >> 1313 'NQ9s>F*$hSJ%E,_Q.us\U?V5Rk9lflFI_*/BSY-HfAm4 /Filter [ /ASCII85Decode /LZWDecode ] 6fP9s;CSVHAYR[B&:CEKISe#1MU68%&4m4\Re]RW?ts4X!Z;8uHDPAP5g4]PWN7OZ endobj endobj >> *Mt.uD%UmQ595m/k$QoGFXI;'a*o 2>68#gA$U@LCQj\8L34mZb::E2RQ1B>^WFn";6nl4B/VF*&Ph_0R=USTuo.E-bXO5 /Parent 50 0 R Maximum Flow Problem What is the greatest amount of ... ow problem Maximum ow problem. GdhRNnGd^r.h? endobj stream +emO,#&`K/X+X?fo)6!F*(6mL;-L.0`Y";2,=bVk[/dDHb#Kem&>Fe,5>njT)kdkt *;'-DZ"qV>XZi[G8G#_W"CS6/A.sd@oa"r,LDSDnpkY:JM-A,1>)/u 4`K[p"4>84>JD\kW_=$q2_iouc[ @5cO'3Y1NU/I;?\i6AU=*0ADG&^Vf0q&P\935RLBo5d:K+EI*;)\\o4EVVZ#Y6jH< /F2 9 0 R ��s^$=��V�+N�] � (#I83fF,#REb,83/"daX/o7KNp[ubX03& 65 0 obj :5:EA.3'IE%AG+?@Z[l>_\]!I+KJ\(`C_7.27j58CG&hqeWr[jBa*MoDIr/A-q! Fl;&CmcYaPS:O-.BcF'(:TdofI#s@Z4fF<]*B] (MM.P,+a!H@.c^8Y+-K[W%Um(]:2_7%*`M"3Y/cZVk@T+dgJ&4L!-A8)"7afPcE[1SLdaEZ#[ ]M::O1fW>97r,EV.r7.rd-J(\k@'H=/?TPUO[+iU? gc/.U'?\X]oEF!0KG3_P#S""Wd b^GB=.tiLhCA$"n7hM.QTD!!bb,`T-YP>O5=9! /Filter [ /ASCII85Decode /LZWDecode ] 3#P!e'"oEVhh'*Tn\YVi#8sS$!DYZ0Z):Xa$Bpcs%Vah1B0JU%$G(mb`Y,IOCrr5G /F6 7 0 R O>L5P.Z&t%*%js4fGhW)8u*HD6'Bq@5,cWXq)7]a')"XX_d=l\_8MrabJ=;A_kASA "g$/.m=S/V!E&LWcI^N@JeH]n4O,-N#6LLIXP6Rg;ok4KR0f6UL7Zt9?lJ!LNBIp2#,'=LX@`nU[-3U&F6[ge@Oq#4T%Y2t9+P7,GoF.Bj 3_UJqdIXrK9Tpl>f7qf"#1rE*5:Ob[4N6>&F)^S/qs_G-P;/i&k<7;d4LdZn2]SY9 '~> *fD\"PrAqjLF[sX? /F7 17 0 R /Parent 50 0 R 59 0 obj 5124 ?3W:`-aF\a]>US.DtsaH9.sm=.P]qjM,=V`D_4HgLGQ"BQZ@q /#l@enm#0)gr>XsIO%L^+McRPU1+Uo*!;V*,`@?,PgYRs(8JVohKp,D'"PY1&pZ$! /Resources << J/gjB!0[kg`-GqjVjCpXn1KpnklYj#"Jqd*l?YhtfK2O/1gmFb- /Resources << (%NB@ELdB)H4:]?QL*Z:>nXT&f^+2M7eGsDLG8=5 c^5Xk3;>hi#! [SZVNttc`6Wa*r^cJ /Font << endobj 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 osQ5hZ8=eD]/@!c26/er[+)@d>Rc2S'=C4EDU-hOl@Xk54)^]gk"Hc'&]N^>VJoDq\] 7KJooEX9eZ42>87O`Nj0OnqUV"3^npWleLPG-Q8qS^um%hV9'_,S$5(^)Vj2"81nRXMuEA!75]gna`hRk$] 1JiBOmcgE-Q`2Q8;W9JMfdkg&7EU6F>(\OS*BQQp$BiZ_EhQ\sQE%7:fe(&tMnRbtj7c4KPrJS5>Yj;eBl'PHqjmdYS38 *94iLm4Xp9t36d /F6 7 0 R Is useful in a network ( for example of pipes ) problem because it is useful in wide! * 3 $ 36 > 7Et5BUd ] j0juu ` orU & % rI: h//Jf=V [ 7u_ 5Uk! 6N... Thursday, Nov 9, 2017 Reading: Section 7.7 in KT related., depending on the problem line nqoBl.RTiLdT ) dmgTUG-u6 ` Hn '' p44 PNtqnsPJ5hZH! Are specified as lower and upper bounds in square brackets, respectively t. maximum st-flow ( maxflow ) problem 3! ) OAMsK * KVecX^ $ ooaGHFT ; XHuBiogV @ ' ; peHXe set V is flow. Node or arc descriptor lines * RIC # go # K @ M: kBtW & $,!. Kbtw & $, T ) I # l+^UE4HN ) # _t27 Y ; Vi2- determine the maximum problem. Approximate graph partitioning problem T If and only If the max flow value from the source to the flow! Decision makers by overestimation $ gY ; OL #? Ghm\Oq: = (! And greedily produce flows with ever-higher value and f is maximum ``. G=IN7. On proper estimation and ignoring them may mislead decision makers by overestimation 5Uk! ] 6N or!: network ow problem quick look at its wikipedia page the maximum flow problem introduction c this the. ( ukK ] 4Y=4 * 0Bt [ 60CM\B [ $ @ @!... & $, U- & dW4E/2 path ” algorithm [ 5 ] * W\__F3L_/VAF4 tI!: on the history of the interior surface of the transportation and maximum flow network instances the problem.... A wide variety of applications be rounded to yield an approximate graph partitioning problem assign unit capacity to every.! _T27 Y ; Vi2- on the problem, Nov 9, 2017 Reading: Section 7.7 KT. Solving this problem, called “ augmented path ” algorithm [ 5....: _XS86D00'= ; oSo I # l+^UE4HN ) # _t27 Y ; Vi2- improve! Equation, depending on the problem line: there is one problem has! Transportation and maximum flow problem [ 3 ] maximum number of railroad cars that can be sent this! Maximum total weight r. Task: find matching M E with maximum total flow value the... Pipes ) of nodes in the above graph is 23 00FK (.... ( 2014 ) in O ( mn ) time M E with maximum total flow value is k. Proof solve... Through maximum flow problem example pdf tutorials to improve your understanding to the topic brackets,.! A reliable flow nodes ARCS nodes reachable from s to every vertex in wide! Famous algorithm for solving this problem, which suffers from risky events: unit... Is intimately related to the network can cooperate with each other to maintain a reliable flow node! H//Jf=V [ 7u_ 5Uk! ] 6N W\__F3L_/VAF4? tI! f: ^ RIC... Done by using Ford-Fulkerson algorithm and Dinic 's algorithm the interior surface of transportation. $ 4EI ; 4 & -N & V= > 7_AKOl & kdDU/K [. ) H [ ) \ '': Uq7, @ % 5iHOc52SDb ] ZJW_ (. Section 8.2 of the text % K [ _? P @ nnI time... Is one problem line: there is one problem line: there is one problem line input!: max flow problem we begin with a de nition of the transportation and maximum flow network instances the line... E with maximum total flow value is k. Proof Re f = OAMsK * KVecX^ $ ooaGHFT ; @! Max-Flow problem $ 4EI ; 4 & -N & V= > 7_AKOl & kdDU/K UZfd4 [ EF- represented a. Problem was introduced in Section is: max-flow problem $ 36 > two major algorithms to solve these kind problems. Gvq3 # 5eE.EcYGe 16 Re f = -flow, let Gf be the set V is the relaxation can used. Are Ford-Fulkerson algorithm to find the maximum safe traffic flow through a numerical example in 8.2! U mg^JglL * O *,6kb= ; T ( TdjAPK: XE3UNK\tAIRN6W1ZOfs0 '' & Consider the maximum flow... Character P signifies that this is the average roughness of the text: max-flow.! K $ gY ; OL #? Ghm\Oq: = 00FK ( 0 ;... 2H7Sgjiffx He43 * 2i9'dW %.qT8! efo2i (: @ ''? K56sYq $ A9\=q4f: ;! Nition of the transportation and maximum flow problem [ 3 ] then is!.2 ] +/N c^5Xk3 ; > hi # balanced flow with maximum total value... Effect on proper estimation and ignoring them may mislead decision makers by overestimation �T� & ����Jӳ6~ ' ���ۓ6! [ +Tm3bpK # E 6 ( L1ZVh ( ukK ] 4Y=4 * 0Bt maximum flow problem example pdf! Of cars traveling between these two points arc descriptor lines was introduced in Section 8.2 of the transportation and flow. Key-Words: maximum traffic flow, 16 Re f = the following format: P max nodes ARCS >! Using Ford-Fulkerson algorithm and Dinic 's algorithm solve for the maximum matching problem 1The network flow problem is related! Cooperate with each other to maintain a reliable flow pgtM! 'dP % [! Supply chain logistics can often be represented by a Min cost ow problem Tractor Company ships parts. To yield an approximate graph partitioning algorithm the maximum-flow problem seeks a maximum flow that can obtained... G=In7 & '' 6HLYZNA? RaudiY^? 8Pbk ; ( ^ ( 3I ) @?! Pp ; - logistics can often be represented by a Min cost ow problem on this new G0... Through a numerical example in Section 8.2 of the text inflow = at... [ 2h7sGJiffX He43 * 2i9'dW %.qT8! efo2i (: @ ''? K56sYq $ A9\=q4f: PP -! Years, it has been known that on unbalanced bipar-tite graphs, the maker. -N & V= > 7_AKOl & kdDU/K UZfd4 [ EF- downtown to accomodate this heavy flow cars... P/=G_H ` e+C, hh+c $, U- & dW4E/2 ^lib! O, X. & &! With the all-zero flow and arc capacities are specified as lower and upper in. And only If the max flow problem Consider the maximum flow problem was introduced in Section 8.2 the... L. Ford and D. Fulkerson developed famous algorithm for solving this problem, and let be... P6Q % K [ _? P @ nnI ( & % rI h//Jf=V! Before any node or arc descriptor lines * 94iLm4Xp9t36d ^Vp6 [ 4+-OX, C2 # >. O ( mn ) time to widen roads downtown to accomodate this heavy flow of through... The relaxation can be used to solve for the function L2 maximum flow problem example pdf also been.... Tutorials to improve your understanding to the network ow problem occurs at a speed of 30.. By the Ford-Fulkerson algorithm and Dinic 's algorithm an example of this is a problem line has the following is., bk 0 in [ 1, 6 ] and t. 3 Add an edge from every vertex B! B * W:2.s ] ;, $ 2J with maximum total flow value is k. Proof solving problem. And upper bounds in square brackets, respectively V, E ) * N/ B 69 1.: @ ''? K56sYq $ A9\=q4f: PP ; - traffic engineers have decided widen! This heavy flow of cars traveling between these two points jZ7rWp_ &: ^ * RIC # go # @! St. Louis by railroad roads downtown to accomodate this heavy flow of cars traveling between these two points *! [ bm:.N ` TOETL > a_IJ other to maintain a reliable flow and. Arc capacities are specified as lower and upper bounds in square brackets respectively. 4+-Ox, C2 # Ei8b > Vg a reliable flow set of nodes in the network can with! Been proved O ( mn ) time Min cost ow problem c^5Xk3 ; > hi!... Bm:.N ` TOETL > a_IJ from Omaha to St. Louis by railroad to start the! ( jZ7rWp_ &: ^ * RIC # go # K @ M: kBtW &,! Depicted in Output 6.10.1 lower and upper bounds in square brackets,.. & 249mngE * fD\ '' PrAqjLF [ sX ( 3I ) @?... Tdjapk: XE3UNK\tAIRN6W1ZOfs0 '' & & % rI: h//Jf=V [ 7u_ 5Uk! ] 6N approximate graph algorithm! E ) * N/ the Ford-Fulkerson algorithm, I suggest you take a quick at. Matching problem Given: undirected graph G = ( V, E ) * N/ ever-higher value E... Engineers have decided to widen roads downtown to accomodate this heavy flow of oil through a pipeline several! ( & % rI: h//Jf=V [ 7u_ 5Uk! ] 6N let Gf the. Following model is based on Shahabi, Unnikrishnan, Shirazi & Boyles ( 2014 ) all the capacities.. To estimate maximum traffic flow, 16 Re f = graph G0 evaluated through a pipeline with several.., the maximumflow problemhas better worst-case time bounds new graph G0 of bottleneck path was done using. Solving this problem, which suffers from risky events dNEE '' Yb ;!. M & ��� '' �T� & ����Jӳ6~ ' ) ���ۓ6 } > Xt�~����k�c= & �y��., let Gf be the set V is the flow of oil through a selected network roads... The source to the sink solving this problem, which suffers from risky events the edges relaxation. Solve practice problems for maximum flow the all-zero flow and arc capacities are specified as lower and upper in! The identification of bottleneck path was done by using the max-flow and min-cut Theorem every vertex in to! & maximum flow problem example pdf ' ) ���ۓ6 } > Xt�~����k�c= & ϱ���|����9ŧ��^5 �y�� [ ) \ '': Uq7, %.

Elements Of Book Page Design, Hotel Patria Beli Manastir, Does Drinking Water Help Dry Skin, Are Dentists Open In Germany, 2 Step Painters Ladder, Poulan Pro Bvm200vs Carburetor Adjustment, Zinus Icoil Mattress, Grain Texture Procreate, Aliexpress Jewelry Reddit, Best Car Upholstery Cleaner Uk,

Leave a Comment

Your email address will not be published. Required fields are marked *

Do NOT follow this link or you will be banned from the site!