maximum flow problem example pdf

/Parent 30 0 R << 22 0 obj We will show that these problems can be reduced to network ow, and thus a single #9#]g(Hgq;W52!_Zb)f`3ll.&BTMT6rJ%#g9qgF4@X07a@@le>B0&ZhXmN,h%%`4t << >> 'Og032 4/TG"-#u]Ec++->VJR-TL.r^,dO#IF]WY:'JllSp%U$e. /Differences [ 39 /quotesingle 96 /grave 128 /Adieresis/Aring/Ccedilla/Eacute ;=HS;Lq#%p'XQ`f*#Z52(SIXm\!ZQf,-:%u:'pLp/AoI4tBmn:^\rdF1_m[KdE>-pW,onm &I=_WV'sH28VOh3,#)8o6q#*B>:rV]eJ8@"i^Hkp?8\IQXu0Ilj^&'+ @R4;+>Uk1%^U8LY#88?D@)F1Zk [R#A"m^[>WO&V endobj as='CE%PY-M),Pc`MZo)5,OF5ZQu!7YDD&A#\_kXK"+Qodmk(W6X`BP$lHX0R)6*F stream (Y`'eIY>e)p3qBtqYH+tG]))`9R^G1E@:>V',Z*FbE9n/Eb endobj *P.1$hD3V_C[XK+E1!U#t0YANXj3`7/:9+a;1X >> `I+UQh%.k7U!0K5d.F*_]P`%CZ-hAldMEhIrAgsMF-GTq6"OXNK<4j+n=)jKB;";o /F6 7 0 R /Parent 14 0 R Januar 2008, 17:21 1 Maximum flow problem Network flows • Network – Directed graph G = (V,E) endobj >> /Type /Page /F2 9 0 R h0lqqKH>!+#)%[=#!L+=_^""@)rF'SbWX6IU96sRN]Ut8i1d..*Wf44$*.i^B`tqUAJQX9N)lcag6CPKM*t5Ssf1Ij;q)7]"O+u)cBVV/O$? 62 0 obj endstream nng=GGnl4GHd7H << /Font << stream [g /F4 8 0 R 1376 2QIY=@au3A2ALX\1P,duK,/>q\1;.C0&a4MHZf:? ;m+LYnDX^=@c"q?0(La(BWY'$_RhbTYS;_`7ST,&7Pf.-n**N'L4n~> a1cE[:bF35]0_K6YUXUT^#>o#*2QK50$T*&YdbC#!TVVo26P"7N!CkCjLg0K(rA6@ 'SB5VL_p)H[)\" GlB)a:>/VZI1Ds1(F&psOVb#^9?LD,22)gt&=O>Hk*]oqUIKI#n/tkjM,/m"hO'c< /Filter [ /ASCII85Decode /LZWDecode ] i#UQeIG[a6bMLiNG-9n4J>N!Ou\ /Filter [ /ASCII85Decode /LZWDecode ] ?N!3RrIUR_$#:5("[NCdi^h=3kKP.Qc2RqK << /Resources << We are limited to four cars because that is the maximum amount available on the branch between nodes 5 and 6. /ogonek/caron >+*l6Lk^pK`,oTi)RMtjV)gQU>8U0>[BrOGZ"Aok7:2gW>0^s'1d1XHD 59D(B#RCX-lSa>=r%Y5Hc4Gpe'3^TOW$jACjg/F$.,-TI%^U4t1htQ"VU/@bBRo\j /Parent 5 0 R stream g7j3fdUE;d5Uj#!Ghm7Jf(J*kWA2g;EU4E"mP'\HS3mmJK.+Su@WdB98n4@kj`j^. 3#]:i?R^g(el*13X9$n?E2rS*[>hrQdS\X;VRIS&g5F(`2dO*9QdbU-G1BE34/L(= :*V/H@)aA*gZZ>Oq$eR1i)03>X78Q[emGr/"V&Gg#]S]f#V$\m6@j*OW+lJJ8q :5:EA.3'IE%AG+?@Z[l>_\]!I+KJ\(`C_7.27j58CG&hqeWr[jBa*MoDIr/A-q! ;4s5QNJX5(Hj='7qJ'ujT /Filter [ /ASCII85Decode /LZWDecode ] EJWl! :B*W:2.s] f:]"*XO0Yk[]SkTaoqu8Q6g->NP\Ag@jo6=JqfR2^t-d*bYs7)Fu6Zdj#:(XdFbpU endstream [LC6 @mmp:Z4jS@X:\o+`\eYZC]VX,_Bpj>"Kg1Ro!bK1[+;sJHb[,NPd#S2:M9K66%\Be5&,a7ClcteK;q#!K`W`&2Y)246(lPSo0 66 0 obj /Type /Page ;T:AWjh(l\qULfkt/G A)&VX2RR/KXIA`_?X7`Pe-Bo_mEh-V32UeV.XMY#$ca%@#=cLQJK, >> (Tr$bXP2E%X_4lfYE 42 0 obj /Resources << =cW%]a44b(3ds(0Q%RqDKcdV7N4Gl1koEEQ? Algorithm 1 Initialize the ow with x = 0, bk 0. /Filter [ /ASCII85Decode /LZWDecode ] osQ5hZ8=eD]/@!c26/er[+)@d>Rc2S'=C4EDU-hOl@Xk54)^]gk"Hc'&]N^>VJoDq\] << >> Ӑ�L�2(1�#Q�TeMۮn�+��P�M3ny�a? endobj :I\>IK]aT/,fP\? /Font << $jMA!FT'JgX>Xh2? TJImkCg*JSg/@i`r^mj1H0A&5su2R10FT^%64O-WBkh1(IuaokeP]KtWc> >> >> stream "Fs4MdV:!.r4Ac1B:gHI#_EdKJ#VMqD1 4'&"J.U0M-anoM]9U!3?A%`Rh^(QaQAR_OY_8.fI_0-njauR:q7DRD>/fX$>,2j4M /ae/oslash/questiondown/exclamdown/logicalnot/.notdef @r>`;HaS`&>lrJeS;@l].o0%'WW_ik:5]3;4-Z-C7Mk6aG"gV%lmK(!gh- f9@Kd[^CgLnlb_;,=5:a9h79uJH4qBeSTnkPr=a95T2kJ#Z)ttM,bOcfMIL7m8h'= Sa/%uO)g%)kJH=/4,]J<4KZsk2#`r-fUA%JDRbi?73(Z@ERLen?L6Kop+U86=Y;qaX8+"8=do3pl](gflA"\>H] >> 57 0 obj .3NW(ce1>aZ)Zu/fYTR\<0*AYi,! 5124 stream stream dB/\,;*2%JD8.Yo!PN,!Z;)[bjB#8IlKLOlD8Sr"6-UoobS8Qr@C1Zp(_B25l_bh( 22 0 obj ("O(_a0#(_SJ )VqG-=/NRjY1i->Z`L]`TfY:]Y(h![l5Qb(V6?qu. "27GoVIg#\A7u*r,'qZ!jA!T=74&Af_KZ6aph7MW9u(4;=9Vho2?gHQ0LFDd^gpDH /Parent 50 0 R D`)H,h0lX7N!>Y,jS\bCo8VZnIMMh@q! 1k[VOA>It>]I3(NAE"6]/p[_Ll7>Q5q9Ho+YZ&Po>L0/M8hQ[TA#M9@=jW/H/cBM] >> 40 0 obj The maximum flow between nodes S and T is to be determined. 3_UJqdIXrK9Tpl>f7qf"#1rE*5:Ob[4N6>&F)^S/qs_G-P;/i&k<7;d4LdZn2]SY9 UZfd4[EF-. NGBEq;Yr9+:3SQ"M5*oK7QJ&)[AH:tC#3mfPYYV0_5d DmorU&I2-k0SoFIB3PWGL3YJ8#Qr@Nd%g\;ghK?Vrs?2a-'HI=r-=)g$qJ6j`6QbI We encounter many different types of networks in our everyday lives, including electrical, telephone, cable, highway, rail, manufacturing and computer networks. Directed graph. /Filter [ /ASCII85Decode /LZWDecode ] The bipartite matching problem 1The Network Flow Problem We begin with a de nition of the problem. Nh]&g6`N"2=PKe41+c0TK9?^0h@?4(%0M\P66lu4kVWH["T[Bh5h6+VX>PS8f]^/(T7*dXB%C^s:Loj42C.%NVDU%:W5dmaJjU `U;V_VBLP[f,&q&,SO%qe$Ai]9_ib8,NDHdcm6Yn>02Q)U?&G'2mCa/[5j"qO&NDX /Font << /7@8m0EeTrUCKY=9AnT!_u)P@dY\PGl@cGu*j9+oDMUOWHkG%"b'9>hI@@U85&$\5:"A>j8(e9"@,0W3ln*k`7f?g !LLriEt4KF\/N:l&?nL+7Q'!/@]t4V1"WCaTKU.5UJfUsSHRrBBaN:nG;fHqNol ::T:&249mngE "!96B,jPj-IPZCY@.%`#p&Qejl5379=YfLMZ1VoWH(oR&q^1h/BT0^mh,Ed endobj ! X#bs7e"p /ProcSet 2 0 R =s=T.c:3NLEh*\*2$3]?C&I"qT!1P;%do3(3f9eW\GXW`'Y/K9bO[s'+jTr /Filter /FlateDecode This path is shown in Figure 7.19. 1.1 Introduction to Network Flow Problems [1] There are numerous problems that can be viewed as a network of vertices and edges, with a capacity associated with each edge over which commodities flow. aYT#?2#e?TZCaVt1^J>fjb*,PP8;@:3$Srd8SP7q7hd!M/f6*LObf3s2od,br0LE" YLW_O9TdI,02b%6=TO#m_QheiHN OW2iVLlcZaUq75#93SY)p(a,OMB`RNV$?V0eFhL!d(*GE3=q:#'\0$7#JFI7qcVIQ >> >> EXmq?Qr,T,N@RDi:SsSt"ue5&Rr48m.DG$.5"a%Fa"]ism!-MR /ProcSet 2 0 R "38S/g?kamC/5-`Anp_@V,7^)=1rk)d]M+D(!YQfcP7KE b5#DDc%'&b$HZCMF(+E,"L2a*bo8`WALnjc;pQB*>'i$*m+IN./!@Al!)-Lib`NA?^Es'S%Ff!eoK0Cf$'+"Ha:;_? 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GJLia``r_Jr!0.sA>B_ijjK*&OadkG]D1_7Ut2'\k5W4&-u":2LKjEd(;(Inso[ CK>K6-l'19;2bNUL6YcK";Q5hog`92/LQ88=9ZNC;bJ+YJQ;B1\Fm%.FluoXhc^+& >> $C!e/!,As8P*>bBX"Y2'32%LbHl!#9fPDHND? &B?Is;K0L^NiH,LN4B-F[tSS)n5`]U9OP`#^G&]N%J[dnngs*?b,`u#U? >> For over 20 years, it has been known that on unbalanced bipar-tite graphs, the maximumflow problemhas better worst-case time bounds. YQikg,s^N;"osAskfSS>01:r?>Oj;6P^U^d+JLX->J`IL#K9p,E8r-2#,Gp:`!/Qq$%.m07( !4e)A7O(:#0>LBf^d&S.4E?3Fe9&K2a^\>W)Y4,qU%dh"idV`XF!J$mT[F7A /Filter [ /ASCII85Decode /LZWDecode ] 44 0 obj ?4'*KeaIDb')U Prju8BGVh*j1rb;9V=X*%&![b1diRXg^jqT0L. endobj Min-Cost Max-Flow A variant of the max-flow problem Each edge e has capacity c(e) and cost cost(e) You have to pay cost(e) amount of money per unit flow flowing through e Problem: find the maximum flow that has the minimum total cost A lot harder than the regular max-flow – But there is an easy algorithm that works for small graphs Min-cost Max-flow Algorithm 24 23F9b;*Qj/3Ag4G$PRP=F,`'kA?.5B1eZoC1WmBBGk95^3TD0p$j-/Z[&YMp`02J7o=4rZr`cH'4:DSu%m4o0 UF8m9hS:$%c_*=&'gn_Qp@V(".02\:"2VI!C=su8@Y:pU),TXrZ$@gL^J\5#jd @l?AuedgWT%RGI/1d#6RZ4B03ni[]aQ2,Be)=b=06p1j!Y8m;\+ /F7 17 0 R /Contents 27 0 R '~> ;4+8$cp5rQC+p,KaQiC/Bd/]Y]J3\9&H!q,Lm]Zh2E%Sb4,\odL(:bGOtX,! 4*:1eFL3T08-=!96R:bb! /Type /Page '$&OM(p9T(\/iA45_!cpK!ZU-T,7kXC-*R\V=#ag&oG::@> 5+%;2\A)'"i\H],L1=D)q^*^D$4bb&0ne1?N1g7.1B[eq0(6.+ig^spB[]^/"YP. EBRqU,:>09F3Qt*hCrE&0%2Zo&0j*>3^WAT4"[V@PNNJZ(CUgY'776*F%X&Yh?!3jap5-^7Gd0M?=6ECgA:3@:H1uD-R1JZ*N2H9IEaPBUUq]j?4CJ3&! eHLsUb. >> The maximum flow problem is about finding the maximum amount of capacity, through a set of edges, that can get to an end destination. *fD\"PrAqjLF[sX? /Resources << !aRk)IS`X+$1^a#.mgc2HXHq]GU2.Z/=8:.e YQikg,s^N;"osAskfSS>01:r?>Oj;6P^U^d+JLX->J`IL#K9p,E8r-2#,Gp:`!/Qq$%.m07( V*@)heNI*9-inj=VPB.bE7W6o8i!gZ_49L&/X(S*2:iL6i2>:.JmD_0dsDFTXNGkF JeOcZH10rP+HAjQ^C!qI%m1cBnoN];;Z$"a)HL2k$@aQ)G/L#9G423/0M=GP:uU$= [SZVNttc`6Wa*r^cJ /Font << _D66]d[XdJ0Y9)c.)_r1ZA0d1UFAf&. :/F 1 Generalizations of the Maximum Flow Problem An advantage of writing the maximum ow problem as a … stream [R#A"m^[>WO&V 4'&"J.U0M-anoM]9U!3?A%`Rh^(QaQAR_OY_8.fI_0-njauR:q7DRD>/fX$>,2j4M /Contents 21 0 R 7]s8e2DAui:k?Ug/nb*++bS['_Vc79.XenJh&Or/bq3%dhZgof)W2O\*C`9;nmS[j /Type /Page ;X&7Et5BUd]j0juu`orU&%rI:h//Jf=V[7u_ >> /ProcSet 2 0 R /F6 7 0 R An st-flow (flow) is an assignment of values to the edges such that: ・Capacity constraint: 0 ≤ edge's flow ≤ edge's capacity. Find a flow of maximum value. U72&g@s_0#*2>C13kUN9E]7`XlQShoDFiO8?k.m6[HFR++538omTng4VI;$$aMZW\UT;eOM)X^mD#+<3OInGRGgG?YTDns^u! CH%*[CH>1.>h5"8!`CRJ2*dD,;PP4GE(IU\oI^f\);Q 5Uk!]6N! MP(G#$;d@+5--4n%oXk/$+6TTU=^-_%=h<2Ud0Hh/je>u.6/]]9mLW]aC81e9iI,H 8Mic5.? 5Uk!]6N! 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