08
jan

maximum flow problem example pdf

In this thesis, the main classical network flow problems are the maximum flow problem and the minimum-cost flow problem [3]. j"VL,X*B8_qXVYdZP7^#jd7n"SB6g*ZE@T``0R'(ftij.C2rf=4"E'aQUGbX"Vg^a /D [16 0 R /XYZ 28.346 162.874 null] Notice that the remaining capaciti… ::T:&249mngE >> 6190 ! L'(B5##?Ft?mRju]d\8]cJe;_73. /F11 34 0 R Source node s, sink node t. Min cut problem. 4MtE&Qk1FH#q@:o\t/0@BZb%;Xqn2KF-582FE_Pjt8MbO`Lr"S3C5D&HW\V#]UD?.YR6_eC5hVQ!m8-(- 0/r?Y^M7+=/+5Ihf[n-eh+Tkqo9?os/McYD6Z`aT1Ks(F#qD4`5O>jL 3Bb(]"&76.mKUI'3C)4,*ptl@7IVEr$sbUH*f"W]E0@,;@L*o#)X2#Wp9T.eo)@Kcc!nXhu#]o2.R[KR^Y%04l1]i"I9 (li!kn`i!j:qZp\l'TRa-8;6g(87"ZDVtA>.L#*$Pldlk(S5S5-46#H9\<=e stream /Resources << YC-$rP1*40UlfCD@qP"d:7i#nqFrO7$C;J8I-&3VpdSroYhWe"p+9bUp5setbdSAV Maximum Flows 6.1 The Maximum Flow Problem In this section we define a flow network and setup the problem we are trying to solve in this lecture: the maximum flow problem. )D4aq2AWm?Y\q"O%bQ*u!C:Mb(^@gNT+!Y4gTp4],8e9W$mQV;3Y*nY#WBuism]7:h^Am_5^0I7%nR@6RkBrO&!+U2's0j2*? @r>`;HaS`&>lrJeS;@l].o0%'WW_ik:5]3;4-Z-C7Mk6aG"gV%lmK(!gh- #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# /Contents 57 0 R /Type /Page Example Supply chain logistics can often be represented by a min cost ow problem. .U]6I8j_5gVFpP1`^YZJ;'eHk@UecEOt,D";>nW3hNUti"Cq\0m@"npjJ? ;X&7Et5BUd]j0juu`orU&%rI:h//Jf=V[7u_ >> /Type /Page >> /ProcSet 2 0 R X#bs7e"p LQ9oJ\8?G4E+0d7:WMrBd&+6b^sNY6t*>9NGD#ds+Pf*\HIW.i0@C`ClaW0qT-K endobj CN#XZ,6?+=UdX1F_:gQb8e^eF0`!b]bhXCW8,_lEkJd0F2_;an^QK/oSMTthmZ%:3 /F2 9 0 R ,m^1!,.,"Q?,8/MKOBdn6Dt5.f(W-u!/rg[c+OB1"tJQOHgejgM>1aBiT91jPn"9j W#. :?i+G(1jNiO];<8+Q3qY:JrZHRl1;.o+VD:E%IdALYj*/qario'"1AHReBM.l*5; 0Gl\5HaL@GZ5ebl8I9L*-HYh;SM<4SZc+K=DY17g^!dAM1BgPF_%=7t-p[C;W8MXN @;70JE4msDq"fW)9KbN&]W8lZ-Q5uS^Q@qC!Q?s5R?NQSD([sC8Ohr7Y]pq%_*E&l L5>M:7],M3"]pDoU'4l"6)*mN/FYf7Pm17$6W1a`$5fB>ndSj.=k5&. 4`K[p"4>84>JD\kW_=$q2_iouc[ endobj >> Ng*4P2E`4!#h'37.,bPN0YY3K9cJ=S,u*V]Js6Hk^h3[33I<2R3,JXXpUOW_ \tF15A`WYFPh[03>V In this case there is a cut (S∗,S¯∗) such that ϕ(S∗) = k(S∗), and (S∗,S¯∗) is a cut having minimum capacity (minimum cut). /F6 7 0 R >>V!JVh7f\QrlX#EK;rO)jLi=U>$SDYus[4aJ;:(Uh!4m"Q.Yu=g@sLRGnS+ghR&m3GlsW! Example: A O B C D T 4 4 5 6 5 4 4 5 _MLhM5U_jdVc8@%XG90ME^/oh/.SaoN3Q%Y9$:eq@gW&g6E\O,1+dJAbleBu9_Kt& endobj >> J/gjB!q-J`1eA,:N&7-U-6l/q-YY^] /Filter [ /ASCII85Decode /LZWDecode ] ".SmJNm/5.kDUWn5lV?Mf\SDXK,)Nh$mQVQ&.E&ng,KS;Ur"t"=@9JB[#bFE^dn'8 Q9*Vu%X#3I?rcS]Vu]9Y>16M&?r9O!=B4g$2T8fWMI8?e<42U86K)cR(NPhqGA7L[(?0FI;fL<>A[WIkPXM1R n]8!+S0t.E#Gok?d[X3Pp@d6SS*8/2'd';F^0WmeNY65mo)#l^/UP*eD\$[60;ACI QWRcnPZ8L/>$5rH4@s@3Bs^I;[P.hCKM.#S0F*63HqTiBK]@#8=B1#TJ4#]tKU=]T 980 endobj o#2GdngC`J$0,]D&a^&@]cf)L_p\]6nA-[&^h8i!-M&H6ZPb'Pfe,%l/[@oYP:J'M X0-*;>.%@1ZY25@Wd)3]fpJ5HpU"-/WlBXBe:^UUe /F2 9 0 R :7d:*HW" )HBi//2$8,!jfmEW1E*%lgDsIXKM8[We7Juc3(3mB.%re;pQ`k2qGNOb%)N-%-dJj .2m48-R5:7Oua-gbbmh-_c*VeOeFSqN"B/0Ku&Rb! [\Gm5XhJT#)I#l+^UE4HN)#_t27 The first step in determining the maximum possible flow of railroad cars through the rail system is to choose any path arbitrarily from origin to destination and ship as much as possible on that path. /Length 67 0 R Fe_'^i^p&,Gc47R0Yd9:-tPMHS+TC*h$:8.=Y3CYm`?,%4+-lLb! $jMA!FT'JgX>Xh2? "*t+NJk3e<3)`@$bMi]R,$6U)I_? 8Mic5.? [2#I59jGsGuQV:o!J>%=O3G]=X;;0m,SFpY'JF/VdsVtHC(Fdl>+EJdqZ 9L*qams".J5)+_8F3OBCa2?iZ5&"7)B\9RAMZfjJCNs\RW``Y3U2)T?AZg[rgNJM[ ;SFJ:(s3&Y%GCWGX=2W.KoYt4fpU?d'VWI01@-9rT[6Cge#3` /Font << !\gT (OZMpf+h! GC"F)QHb'!j1N>j"=(:Cba39^TaoO3E18FJPSKJo;u$1WK^j(_0]#GVcegdlDOj$t 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 /Contents 51 0 R /ProcSet 2 0 R << N+o&SfFVV\91^;V6:eMH)L*<4'(C)c9o\ZrQG[qZuK.NcCNh_\$0b*.gN\eU` $EmR=ih'6?TZQ"02E>=@Hp[(9@b(\n. ow problem, and we see that its dual is the relaxation of a useful graph partitioning problem. 9\22O$L83s;$)otKWN@IEh4l+K&dIqOu88p4#`N#X'WUL5)!f'Y8,>ffb*@ >> K2qZ!Z,m6f\0eM6/;9&R4rZ5dqX\1_;i#!&fO&N`Vm6_KnZJ(!Sf#?%Z(/:^n/D&@ >> m[NbPI&c7NGT2/,eUj.\ICLaYG!UTp)/bd>I]LY>fC3u3Bml?CF3+T6(S[,G? %2_QJVWd1L0T?IEOasGKe#.2JsV4g.$J*nj(cE.-S?\q#d^K41T_n!Kf+l.0%4-C:sq7aq_KTQ%e@^d /Font << [=$OU!D[X#//hkga Networks consist of special points called nodes and links connecting pairs of nodes >> endobj >> endstream K2qZ!Z,m6f\0eM6/;9&R4rZ5dqX\1_;i#!&fO&N`Vm6_KnZJ(!Sf#?%Z(/:^n/D&@ stream /F2 9 0 R endstream stream /F2 9 0 R endobj #\B9A :?i+G(1jNiO];<8+Q3qY:JrZHRl1;.o+VD:E%IdALYj*/qario'"1AHReBM.l*5; ',ddVfDn]M_dp&N9KC:-.7R0;CF1Qt=*A']6Hi9.XEkq2&3B0gtjr+Z_Zhg-9`V780.gFo#gK)M+_g n3aql9T91,eE\e-"7T@mKWK*2dBiSA.Fqq!J'E8%aJUN/N>&poo'' [LC6 << Example: Maximum Weighted Matching Problem Given: undirected graph G =(V,E),weightfunctionw : E ! << m[NbPI&c7NGT2/,eUj.\ICLaYG!UTp)/bd>I]LY>fC3u3Bml?CF3+T6(S[,G? *Mt.uD%UmQ595m/k$QoGFXI;'a*o )4uNgIk/k#U endstream .p;R-#d2bQKCIiVeu(^:3P;i(ArJ:?8g>,.d);! 2.2. k-Splittable Flow A k- splittable flow is a generalization of unsplittable flow problem in which to send the data J/gjB!q-J-TIqA@g,cs\qj%Co`Y%.0J2(eoca/tZ#F,6>knUTb7+#6G6jaA=^P_#V>2%"SE8 NN))A-<6,/nVoOO;q/BkKT7Ll'3">ROr2r=Q+ZPTq2DjOQ$GnT\P,&EgQacLP^))L endobj 6503 >> W]p,G-GrUqQMH2/W&iP7DjR=_?5mo`#%Ylm+l m[NbPI&c7NGT2/,eUj.\ICLaYG!UTp)/bd>I]LY>fC3u3Bml?CF3+T6(S[,G? Ut\K%PT2k=Q,aB=(Hp5_g(aKkDdY"32g1s=CYY63 endobj Y;Vi2-? Ke=KpUhD2.qSZ;1uFeAp@7#2=#R5>@'4sKi%/F S/5BU2"jJ>a!X;Y'/j_5'/:hX>/qlT2/6sJV*P^i%%J#62L7."[. Q_ng=olMW"W]-Pl1446)#[m?l,knTfZ;1T>c$n8sHo5PD=1NFN%#nseJCh2WpY@g5 c:8V>4esA37/:&0]\_^g=!P1ZFf+#.6X4cLhohZUVek:6gbn2A>-5a#0Mc#Zn31^Q /Resources << /Length 45 0 R "D%-E2Fq=&:)-88W` endobj << X0-*;>.%@1ZY25@Wd)3]fpJ5HpU"-/WlBXBe:^UUe [d0#mP9?#,Y:,4S.UG,q%Znd=/8gj,42G[a'd$=,8kh7HF,Sa,6quFciAcArj;LO>7&MNUmu4Ri=Z)Sr -&HXcR[4>L-=X8q-+;=W@%.18gF8V'N7jH^DqVp/Gf;)/',@DAT>VA\1In[\!DcNK 27X,qVmbQO@B!`RbY*oE$]]lOCe.hK\Cb#?eWJ&N0Q3Qa::OcfcBCr]**F,oArL\q ^-\:.`K!MV9Z;l^&dYh\94H\d/lQ-l)'KAm^EQ$;Pt8EoZ(Q+R51AmiN! ]a8?=#]ML,bIUmAIY?&ZRuehqW>rSVCibS_!p1\_W#CU'3L7p1LOc[do+>h8'1oX7#JQ&_/J+$oU[[jd&.oHBEe)H["VFKe 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. 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ZYjtQFZ/u4%(%b_s)RXFDtbVu='#FS+`p'0GAo!Pf,](E'lp(SG5!3P[ek+n0lph, 41 0 obj In turbulent flow we can use either the Colebrook or the Zigrang-Sylvester Equation, depending on the problem. ]MWFOl4!n("p>KDor^8ojprNB>MQ4m$TCcc\GK Jt6cKO@jue3lI]>n6NJ'mNTm5=n'B!6RJndl&HZcR8U9+h/`Yd8Y#*Ht9&?$7q$NPhOiNmqCm?6p;I!Pa x��Y�o�6~�_�G�Y��K,З m�b{X�>8�쥈��N���;��DJ����A�X2?��x��(VP�c��UHY�����b�#'3� 4H�4^&ƖRH�/�\�L��c 1�;1�/�(����y{2CLA�F�io�vb�5Y��r����@�O�� ~>���̢8ZU�Bid������� B��D�g�@�e�(Z������)���m=��h`V'ptf�%F��w�)��DTt�i� _�DH�+�����Ng�_��JJ^�����O�翭n/����b ���Q�w&�_���{;�;_0:�����of�N[�R�8�Nf~]���q�Ex�X�В� �������Q`���O^�ɋ�G�p��s}ڦ/`�0T&��(��:���j+��e���~��"�2���O�"Z�[-��aY�Q���߄r����L�Ec~��N�n�x ��1�4�؁$��&p� ��- ��^0����y���3�� 縵8�������˲�Z2ATiWr�z�q�8E����R�>�8R0�?7���h��W����E?�V!����1y�@1�5J�l�L�J���Z����~�Z��PM4�̶�/I�i 8Ed0%ilhR_bRhdULC. /F4 8 0 R /F2 9 0 R /Filter [ /ASCII85Decode /LZWDecode ] /Type /Page Pbg_!:tuae"!bM745<5qa+n6@MXeWZYK*0O,nF$$Hi_52YJZl1^0GPmI! << b) Incoming flow is equal to outgoing flow for every vertex except s and t. For example, consider the following graph from CLRS book. >> *fD\"PrAqjLF[sX? /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis IW7%,`MMf@H6l.SF/;We6["0XHq8ss3P^SQ"_0`L*aAZ6i#eUm*gj027U,no\V.a& /Resources << /Parent 5 0 R ;"r*.2k)UXL8o$28M'4Ro\)gS!I;-[P:d* ;,$2J? %2fF!E5#=T-IW6Tsl ai89l>g>*qP#f8^1rE2IgjMoV?/+J-g`TE%5fu,nQnA9>"9?X&IJ_mKEtKb6i0ATl endobj >> /Contents 66 0 R 8lGsp1`\FkV,Z7#h,R1HC#Y!!HFk7*NCK$&Ne#P:OR"E-O*U7"Z(Hf@>jB1p.2$[61S,ESEVY]&dBe:/;Y%!mgM0! aYT#?2#e?TZCaVt1^J>fjb*,PP8;@:3$Srd8SP7q7hd!M/f6*LObf3s2od,br0LE" An st-flow (flow) f is a function that satisfies: ・For each e ∈ E: [capacity] ・For each v ∈ V – {s, t}: [flow conservation] Def. cZUDE_W'e;"5\F/Z11Ko#maMW0n`rRlT\Is1nT)6OqTTT]*D$sj_VV\1(kit(SL;' >> << (9XWEAf67'TZ@9? 61 0 obj -&tG"8KB'%P71i^=>@pLgEu"JT9:uK;+sPS.O*ktQ"qFB*%>AKfFo "h_hhdqVaVO>h29&Vl! 52b3H[RIN2a[`;m7,CT("9GegaiV^V&bQBqEN.F-qF%":<>B\[rAd!.lTq)L*fWio osQ5hZ8=eD]/@!c26/er[+)@d>Rc2S'=C4EDU-hOl@Xk54)^]gk"Hc'&]N^>VJoDq\] (Zdsio./L)Qt(#\JiRVC:UaQ Algorithm 1 Initialize the ow with x = 0, bk 0. endstream /perthousand/Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave 58 0 obj +emO,#&`K/X+X?fo)6!F*(6mL;-L.0`Y";2,=bVk[/dDHb#Kem&>Fe,5>njT)kdkt /Resources << GdhRNnGd^r.h? 6915 Network. ;"r*.2k)UXL8o$28M'4Ro\)gS!I;-[P:d* endobj J/gjB!-\ /F7 17 0 R K"KA0Dhc\H6p,t`S8I`ocgV]e6O6itphp]B[HVr95[a`:9!PPF"".W$n!aA)@01'Z "38S/g?kamC/5-`Anp_@V,7^)=1rk)d]M+D(!YQfcP7KE 9?9\Y=@n,Ip/WYN-,W4ZpNfOC@?6#sTL0DB.015&'OHZR&X`teN;8VjnW!>g "TV]Yb5)=5UY:/>4ePU[I4aHm,Rti*$t.3dTZQ#uCJa#4UcfFJ"o'A"#MB2-$p_Z< Pbg_!:tuae"!bM745<5qa+n6@MXeWZYK*0O,nF$$Hi_52YJZl1^0GPmI! ]R.,<1l>ORiq'L(!P:?aeP'T"f0F0n=r VH^2QA_W,B]:-mHOnrW#WXg;l%Rqtr*5`QD-p%mj]/o' 7KJooEX9eZ42>87O`Nj0OnqUV"3^npWleLPG-Q8qS^um%hV9'_,S$5(^)Vj2"81nRXMuEA!75]gna`hRk$] Key-words: Maximum traffic flow, Flow-dependent capacities, Ford-Fulkerson algorithm, Bangkok roads. R. Task: find matching M E with maximum total weight. /Length 61 0 R >> `U;V_VBLP[f,&q&,SO%qe$Ai]9_ib8,NDHdcm6Yn>02Q)U?&G'2mCa/[5j"qO&NDX endobj B2Wa'JC3)g:0W`\rrb=7N=MkJ)%(`^h*XOLGu:Ypfc*C`%XleI0A.Y2=Q83Km>_8f ZMGu(/Zt95DT8dc3u&?rpWn+'OeVs=3uh%P2FAIMn/!'_!1=! `#X,c`^m,>FIo9bIY(G"@S,hI4!O)`+&p#BL(mp]lh^H;&Dh+]+8Vog) /Type /Page /F2 9 0 R "!G`G6c!HH+9o`FjuNVIR*%+C6Q"/]%Ik]+1(kr&VhDl$R!1Xa]U7dbl+\4H*&0Zl) ]'.5N]#Ou:K$gY;OL#?Ghm\Oq:= /Type /Page 53 0 obj >> 'SB5VL_p)H[)\" << 7T58i,;lf$\f?J7`;6NnD?GRO%l5d!f+(`cWC4DABPOrr;Zh5. /ProcSet 2 0 R K95<3]-qrco6tP=BPEZ_^0Yp 63 0 obj CK>K6-l'19;2bNUL6YcK";Q5hog`92/LQ88=9ZNC;bJ+YJQ;B1\Fm%.FluoXhc^+& 3IkSFUE&@h;516o)p-&EU6l5s7d/fd^n?4:-X9FeEd!A63YlAlF? 6PUT\RJ`.Sc/pFDW%8BcCn6.XJl=rj'#C);(M_mQXMMGT7V+,T>4@572Q&f,j^O3m /Resources << c>9QX-&']'UBU:Z(SG%SHsYVS*,[?CPR(c[7+oDQ. 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