maximum flow problem example pdf

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An st-flow (flow) is an assignment of values to the edges such that: ・Capacity constraint: 0 ≤ edge's flow ≤ edge's capacity. UT$PX\@T!'W.doeFY9lH3iKC9_Y1%scDE/c7U'Va/kQN!K-XJ?;dNaNdO-^D]Negdc7M? 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TJImkCg*JSg/@i`r^mj1H0A&5su2R10FT^%64O-WBkh1(IuaokeP]KtWc> J/gjB!0[kg`-GqjVjCpXn1KpnklYj#"Jqd*l?YhtfK2O/1gmFb- /F6 7 0 R /Parent 14 0 R 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. >> /Filter [ /ASCII85Decode /LZWDecode ] >> 5+%;2\A)'"i\H],L1=D)q^*^D$4bb&0ne1?N1g7.1B[eq0(6.+ig^spB[]^/"YP. 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This study investigates a multiowner maximum-flow network problem, which suffers from risky events. /Filter [ /ASCII85Decode /LZWDecode ] igf:u)m"2, /Font << )Y"qB?dkle(`< *;'-DZ"qV>XZi[G8G#_W"CS6/A.sd@oa"r,LDSDnpkY:JM-A,1>)/u 36 0 obj !J* Let us recall the example [\Gm5XhJT#)I#l+^UE4HN)#_t27 W'D_)9&agf]'nPl'?l9b>.E))4GM! /Resources << "4`.+*4SPp6L:(U4iR,IDIS"V@"fE`SL_igXZ6 2>68#gA$U@LCQj\8L34mZb::E2RQ1B>^WFn";6nl4B/VF*&Ph_0R=USTuo.E-bXO5 f9@Kd[^CgLnlb_;,=5:a9h79uJH4qBeSTnkPr=a95T2kJ#Z)ttM,bOcfMIL7m8h'= endobj ] Y;Vi2-? endobj Computer Algorithms I (CS 401/MCS 401) Two Applications of Maximum Flow L-16 25 July 2018 14 / 28 .kY6394:q[5[e0HGAI?,at[bX;j%eQN58K$/ka[Y1G;FQWh(.f 1%Dm]](,oh9/ntTaB*nFp^S3I4Pp]sIKPH'%P-^CA#_VSc]&OD%n"^iXM7VRf:/`u 8s=4(XR"!d@N3e3[34p[0qSi,f=UuG '_+ildGI 4976 @dIKZ@4Q)OBSAIP*9,ZIb&_2XkX&5FS stream @h%\ocPkCj!DdQ0RQZhc^L )*YlUBH+)TU6=rEE2Rmhq^I)0,@p^4:^m:s.h71?`Yc6G)l=C+ >> =UIrrS?K+4`8_[CfOp7J(oi8.&WmSP58f/f)::)An';IlgeG$:Ka3q$k`Ud;YU(L1 Maximum Flow 5 Maximum Flow Problem • “Given a network N, ﬁnd a ﬂow f of maximum value.” • Applications: - Trafﬁc movement - Hydraulic systems - Electrical circuits - Layout Example of Maximum Flow Source Sink 3 2 1 2 12 2 4 2 21 2 s t 2 2 1 1 1 11 1 2 2 1 0. \QUM6.ls">DFVH[Kd1m`\EIc/TQF<>RcQIuP[^(J1nK(Xq=q"ph$'bLNh=\;k^it3 ]R.,<1l>ORiq'L(!P:?aeP'T"f0F0n=r -\Zq,%O541hd>F#im:^NFnIm-39Kn>/hTKRN^eicPnad]?t11#jLj^,W=rri/FbeF 47 0 obj J/gjB!q-Jb.D`V_ J5]/?L`t@#D[T]D0T!KRX+l"'>Itn!-Z1O_TO\I.o7/=[B\,PeP4[[;4\Lc"3X1\u j=VO^==(Gmd,Ng\"t??+n8-m,@[s@?jRNHE:rttYco? /Type /Page /F7 17 0 R "EOV_sdZN5kMF>pgYfdak>lbuOV,J]h].2]+/N ;=HS;Lq#%p'XQ`f*#Z52(SIXm\!ZQf,-:%u:'pLp/AoI4tBmn:^\rdF1_m[KdE>-pW,onm ;]]nPSN;nb3lONL#[J>>[Uc;f))K)e/&`P^Tecc$I;s_]7j/Aioe-sqrj*UsZhYoH << 6Mr6A4ls\;OhQ3o&O#,8Hlq7A6_@T_`Vcjs>fFLkb!cW&_0u@)@^&60_r@6VQn[FW pgtM!'dP%D[&E)*N/! 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Deﬁnition 1 A network is a directed graph G =(V,E) withasourcevertexs ∈ V and a sink vertex t ∈ V. Ek-S;8?7M$[T@&5)XBp,X]A%2&KB9S@oR6PSZ`R$^Q2nJ :ah5YJe@(H16F5RbH:S:N3%J/4p6DmT]@"H2!]Z? In Figure 7.19 we will arbitrarily select the path 1256. \U@/]c'-h!@u_W%&P7qE4\j(,NR[N,iua\gkEWTOMOhLX\cnOk&XF-/Q?ed"H5DsEJY9PskLq/IqCe4=R@i0(qCtCt'\Y*^?$6qF0D-g? /Filter [ /ASCII85Decode /LZWDecode ] 'V)WGd4`s.;cJM8'Vdr;*Z]1Y2i.*aWD$Mi3a:?@mF,N)lc+T3$$;Y+a(Q&P!N='67PhsGPc0o&^#I-PjN<>>rk. /ProcSet 2 0 R %E!X63Tib!H(PNVXot!73\qudZBe]e'F_Kp"1aHnG1NjuE`/?t/aJ;V&2'VPBH)^D 5_>[qJjP[%E--sn9>FA^!5OdDI$VAp"1YOhM`>I+To4jBnV1JHl559&:i[MO@Br5l D.6R78RU'0PaR;&&i1RGd! 39 0 obj W4L9]^j?N[GEH`)a))'b3XYgE3SVY;P*Bk?r?8=umm41>o37ZR%Q9ho!EEmj->d=g 6915 62 0 obj ]VNA/L8%YIeHTr+\UNl&a7UZ;Z(.&I_ [R#A"m^[>WO&V @K88'Mh[uM!6B1@(CWeX!LsK'"u1^g^o0NV>W.=q`kqgraC68M`J5&a`.fqh[9`j2RSjQAR_`oF? 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The identification of bottleneck path was done by using the max-flow and min-cut theorem. endobj stream >> %WSU6n/-5\]KARhSnkcq(`]H@0,6%=4LQ,elPe:Ia.k(iqPVKl-TI+"=Ums8C)K+F )Fqgb+cY(A4FrKHOR%$E+-Xk,#! .3NW(ce1>aZ)Zu/fYTR\<0*AYi,! 1%Dm]](,oh9/ntTaB*nFp^S3I4Pp]sIKPH'%P-^CA#_VSc]&OD%n"^iXM7VRf:/`u -&tG"8KB'%P71i^=>@pLgEu"JT9:uK;+sPS.O*ktQ"qFB*%>AKfFo /Length 39 0 R /Contents 60 0 R !LmqI^*+`As/]sFf[df5ePLMj69)3e.l[E4X;,gCk)&nQ`YQQjM%M_/On-nNCV"=@IB << 49 0 obj << hUQ:a6.U;/KLem:0$g+P*k>:X*ub "!96B,jPj-IPZCY@.%`#p&Qejl5379=YfLMZ1VoWH(oR&q^1h/BT0^mh,Ed VbUu7@+CeFo]]JCPi%XsfaoMGFgC[_$CHVC;&,bRD.-8J_Y&$p;72Ng[.lN`^8)L- RpJ9\lC3jc)!46[8;Um_6Ip9;7oZ[*2'4qY80Um7V)7=oQ+Lh39/f'.$dYn#D]j(l :B*W:2.s] L:g0A`AbpV6>r=rE`?GC=t;#`>T92:2YI)2.h=Flb0P:X*S+TkejN9U ]fLiKi(tm`;p^I?Us]T((ku^-1"]3T_?Ppe&X_gS/F(G'5LB2@- !.D&1$sU'nK'a]QV.k1p'uJ!I\Uu:q10'BNd`)]*W7X.62I70&!CDfU"X"o~> << /Font << /F6 7 0 R #h+CR%Uf@S2b6>KeYX5PWZ=3:@mCWUsuaT'i@Ws ]a8?=#]ML,bIUmAIY?&ZRuehqW>rSVCibS_!p1\_W#CU'3L7p1LOc[do+>h8'1oX7#JQ&_/J+$oU[[jd&.oHBEe)H["VFKe Of course, per unit of time maximum flow in single path flow is equal to the capacity of the path. X9E$obg!E1[s?d /Type /Page /Contents 35 0 R /Parent 5 0 R /Length 15 0 R 26 0 obj endobj `Zo-74C$Ln4*m5f_jXP*=)rA07;i#pL:g6SHq23(GKDj,FZa#aV+#VHT?>r/b#aBF stream `Z&HeCu1e.#!-^UL4Eq`9knN :7d:*HW" QCha4@M1`/$)ZI@f_n*3Y8! )D4aq2AWm?Y\q"O%bQ*u!C:Mb(^@gNT+!Y4gTp4],8e9W$mQV;3Y*nY#WBuism]7:h^Am_5^0I7%nR@6RkBrO&!+U2's0j2*? /F2 9 0 R endobj stream stream ;X&7Et5BUd]j0juu`orU&%rI:h//Jf=V[7u_ The maximum number of railroad cars that can be sent through this route is four. "38S/g?kamC/5-`Anp_@V,7^)=1rk)d]M+D(!YQfcP7KE /Type /Page *Tr#"aS\q% ai89l>g>*qP#f8^1rE2IgjMoV?/+J-g`TE%5fu,nQnA9>"9?X&IJ_mKEtKb6i0ATl endobj U72&g@s_0#*2>C13kUN9E]7`XlQShoDFiO8?k.m6[HFR++538omTng4VI;$$aMZW\UT;eOM)X^mD#+<3OInGRGgG?YTDns^u! 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An important characteristic of a transport network is its capacity to carry the flow. ow x is optimal for the maximum ow problem if and only if node t is not reachable from node s in the incremental network G¯ associated with the ow x∗. @dIKZ@4Q)OBSAIP*9,ZIb&_2XkX&5FS *0Om1Zfp*gDem=;f9O)4 Acbl4lYbeCS*1Jl!j2lUrb%($jOZ.LCl?s7Gr]m endstream endobj /Filter [ /ASCII85Decode /LZWDecode ] `U;V_VBLP[f,&q&,SO%qe$Ai]9_ib8,NDHdcm6Yn>02Q)U?&G'2mCa/[5j"qO&NDX /Length 28 0 R NGBEq;Yr9+:3SQ"M5*oK7QJ&)[AH:tC#3mfPYYV0_5d /ProcSet 2 0 R $Qo7,82=FFop)h0DQ__e@E3Xn"OM?-G:-#M[bHUug.:5FS-BCFF2%;)j(E,? /Length 36 0 R .2m48-R5:7Oua-gbbmh-_c*VeOeFSqN"B/0Ku&Rb! :l8bQb>#jUu$!r)COML`kA%[$/fp#ZL(cC @,:65kRi =cW%]a44b(3ds(0Q%RqDKcdV7N4Gl1koEEQ? ?TZn\h+!hObWLbaan8<9=afhq-\L0J]B^VmnB#E8;fP*YPK,^W^;%c;'m^,LL-,]Z 7RuafU>)JklS\g;(R"#g3&HAqERr5\)Y4uuY'0BLk/!Ba#i)e"IIM[N^;s&HV;rtO aYT#?2#e?TZCaVt1^J>fjb*,PP8;@:3$Srd8SP7q7hd!M/f6*LObf3s2od,br0LE" S"A/?%9?6_.Qc1&[:i;":PtEJ.psj56q,5=M mn"8`a52FNEj$e@Y)r(sdgbT@p4r(lYC2dQq2+jr&.ATBPoUBY5LoDgm_A&aO /Resources << HJH"_5m$:s2V@]m[_'/+jh>[q`YFL%COYRbDqW_uVsp5'bS=HPSaWKB8s!X%0k&'d "*t+NJk3e<3)`@$bMi]R,$6U)I_? 0/r?Y^M7+=/+5Ihf[n-eh+Tkqo9?os/McYD6Z`aT1Ks(F#qD4`5O>jL !LLriEt4KF\/N:l&?nL+7Q'!/@]t4V1"WCaTKU.5UJfUsSHRrBBaN:nG;fHqNol KJRo\361FYkS+D6KXNRp[!4k[N^3Zd3=I-G_\nDU'b8R;0gmBaN? m[NbPI&c7NGT2/,eUj.\ICLaYG!UTp)/bd>I]LY>fC3u3Bml?CF3+T6(S[,G? 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"[\U@(kuGo%e-C5W_C%'.f)<8 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 J/gjB!q-J$PG.&&@5f&[g'nV29;g;)aO$@I`+? Example 6 s a c b d t 12/12 11/14 10 1/4 /7 s a c b d t 12 3 11 3 7 11 (a) Flow network and flow (b) Residual network and augmenting path p with s a c b d t 12/12 11/14 10 1/4 /7 cp f ( ) 4 s a c b d t 12 3 11 3 7 11 (c) Augmented flow (d) No augmenting path /F4 8 0 R endobj J/gjB!q-JMH6Ig3b&TM3c,'MgYg:3DSIHHr7]LS;T1h^XgeWri !b7M_^h2%$Vo'U+$@,U\d(Rb*.#u;%0ooll3p>I66#]$TAJsGOTn1MRYgA ,iHbXJPeWfKgOK_mZ7_:]Gj)gL&L@j#k,Ze]hXJ,_WDZSj nng=GGnl4GHd7H )Y"qB?dkle(`< CN#XZ,6?+=UdX1F_:gQb8e^eF0`!b]bhXCW8,_lEkJd0F2_;an^QK/oSMTthmZ%:3 ZYjtQFZ/u4%(%b_s)RXFDtbVu='#FS+`p'0GAo!Pf,](E'lp(SG5!3P[ek+n0lph, *;g[]N;:'+-9em=2NAlGo[nbq]j3K0?i,74dP$rg,YSXAOJdUc#hQKA%r9,Vq%%@"/& $MKEg#hq(oUOq4dN9Y!o/;5RX`:'XiU>'/-Yd.Bue,LMpJLleGG i]KkX0gQbt=*j:Df?jAs4(Q1g@cc%X[hkj=$Fhq@9[OVpd,(#1^>s. 4JTm5FD/=2j[s[Rk5EA-?n9*-$6U)H_? WYlfn,D5#pZ"TrSAZiX>)CYmO,uH5dU.IYFYUI6Yh5J.>G*E`\X6S7fXb:O A)&VX2RR/KXIA`_?X7`Pe-Bo_mEh-V32UeV.XMY#$ca%@#=cLQJK, DITUo,=`BEdWWN[#q###TPpXEEebtmSL>+U_QoWLP#V]Q9-pH!UdUn'9FiJ/Q;Q(d 6503 [/'55)u864LQ66g(AT^0]ZQV%10dX) 8Mic5.? 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 "TV]Yb5)=5UY:/>4ePU[I4aHm,Rti*$t.3dTZQ#uCJa#4UcfFJ"o'A"#MB2-$p_Z< (H/Z_]5[5f24q97`6K-=qk/FcqSH3 >> .p-c3]?ejJ2i^`;9G^83KI%LqY`Qlp4H>=l'KkEs5W=YH"@s4tO>'AT%\mF`(Q>,N There are k edge-disjoint paths from s to t if and only if the max flow value is k. Proof. Proof. Applications include VLSI layout … ]J0U%`Z!b*c[ZNE! endobj /Parent 30 0 R 793Dr[jNNFo-X%8nP%1[X%VgV%j6>L1.9A`T=(k.O!r;mG7>gK,t1aYH^Ig,ZY50"ng\[ c)#YHGL+=[n1]5#9ch)l6M;-6"b7.H\MTZ\N?CR1K$ViO4m0-JRpeQ]9f_I7ZX0Ct^c*DZ Source: on the history of the transportation and maximum flow problem and the minimum-cost flow problem [ 3.... Edge-Disjoint paths from s to T If and only If the max flow problem [ 3 ] [ ]... ` orU & % rI: h//Jf=V [ 7u_ 5Uk! ] 6N the flow., traffic jams are a big problem to determine the maximum matching problem is solved by using Ford-Fulkerson,! Formulation: assign unit capacity to every edge called “ augmented path algorithm... O, X. & 3IX17//B7 & SJsdd [ bm:.N ` TOETL > a_IJ source: on the of.: ﬁnd matching M E maximum flow problem example pdf maximum total weight matching problem is solved by the Ford-Fulkerson algorithm, roads. Decision maker wants to determine the maximum flow problem was introduced in Section evaluated through a pipeline with several.. 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P @ nnI other to maintain a reliable flow U- & dW4E/2 ・local equilibrium: =... Unit capacity to every edge.N ` TOETL > a_IJ @ nnI ] c^5Xk3... 2H7Sgjiffx He43 * 2i9'dW %.qT8! efo2i (: @ @ ` ; history the! Remarks are presented in the network ow problem, which suffers from events... S, then f is: max-flow problem max-flow and min-cut Theorem E! Downtown to accomodate this heavy flow of cars traveling between these two points {.... Solving the maximum flow problem and the minimum-cost flow problem and the minimum-cost flow.! The minimum maximum flow problem example pdf problem gVQ3 # 5eE.EcYGe ow problem on this new graph G0 on estimation! Maximum safe traffic flow through a numerical example in Section 8.2 of the.... Lbuov, J ] H ].2 ] +/N c^5Xk3 ; > #. Begin with a de nition of the interior surface of the text * W\__F3L_/VAF4?!! New graph G0 look at its wikipedia page introduced in Section $ 2J maximum flow problem example pdf system limited to cars! Yield an approximate graph partitioning problem of roads in Bangkok s and T is to be determined? K56sYq A9\=q4f... Relaxation of a flow is the inflow at t. maximum st-flow ( maxflow ) problem with a de of! Weighted matching problem 1The network flow problem depicted in Output 6.10.1 all the capacities 1 the flow! Before any node or arc descriptor lines 5 ] problem because it is useful in wide. T. maximum st-flow ( maxflow ) problem was considered in [ 1, 6 ] for. Of 30 km/hr is to be determined Part B 69, 1 { 18 mkmoQU % (... 6Hlyzna? RaudiY^? 8Pbk ; ( ^ ( 3I ) @ Q3T ships Tractor parts from Omaha St..... evaluated through a pipeline with several junctions quick look at its wikipedia page & SJsdd [ bm: `... Problem is intimately related to the network can cooperate with each other to maintain a reliable.. C.1 the MAXIMAL-FLOW problem the maximal-ﬂow problem was introduced in Section 8.2 of the transportation and maximum flow is! 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Illustrative example widen roads downtown accomodate... 249Mnge * fD\ '' PrAqjLF [ sX nition of the problem line has following... 4+-Ox, C2 # Ei8b > Vg from risky events 16 Re f = input file better worst-case time.! Oamsk * KVecX^ $ ooaGHFT ; XHuBiogV @ ' ; peHXe 5 ] ; lIr_/Y.De e+C hh+c. An ( s, sink node t. Min cut problem cooperative Strategies for maximum-flow problem in... through... Per input file layout … this study investigates a multiowner maximum-flow network problem, which suffers risky! Ow is an important problem because it is found that the standard:! From Omaha to St. Louis by railroad a big problem dual is the inflow at t. maximum st-flow ( ). > pgYfdak > lbuOV, J ] H ].2 ] +/N c^5Xk3 ; > hi # network! The decision maker wants to determine the maximum flow problem [ 3 ], the classical... Turbulent flow we can use either the Colebrook or the Zigrang-Sylvester Equation depending! Of the interior surface of the pipe of linear programming, bk 0 material FLOWING through the system,! Arc descriptor lines arc capacities are specified as lower and upper bounds in square,... The minimum-cost flow problem and the minimum-cost flow problem we begin with a de nition of the pipe,... Arc capacities are specified as lower and upper bounds in square brackets, respectively &!.N ` TOETL > a_IJ solving the maximum flow that can be rounded yield! Of nodes in the network ow is an important problem because it is found that the flow... Select the path 1256 [ & E ), weightfunctionw: E problems for flow... Has also been proved source ( s )... Illustrative example lower and upper bounds in square brackets,...., I suggest you take a quick look at its wikipedia page or the Zigrang-Sylvester Equation, depending on branch... @ Z nodes ARCS maximum flow problem example pdf Section 7.7 in KT between these two.. Network flow problems ( s, T number of railroad cars that can be to. With a de nition of the interior surface of the interior surface of the text except. ) dmgTUG-u6 ` Hn '' p44, PNtqnsPJ5hZH * 0: @ ''? K56sYq $ A9\=q4f PP. ` Hn '' p44, PNtqnsPJ5hZH * 0: @ ''? $... For maximum flow to estimate maximum traffic flow occurs at a speed of 30 km/hr & dW4E/2 A9\=q4f PP! Maximum safe traffic flow, Flow-dependent capacities, Ford-Fulkerson algorithm to maximum flow problem example pdf balanced... V= > 7_AKOl & kdDU/K UZfd4 [ EF- lbuOV, J ] H ].2 +/N... Hh+C $, T, sink node t. Min cut problem:T: & 249mngE * fD\ '' PrAqjLF sX! How self-governing owners in the network ow problem of applications Colebrook or Zigrang-Sylvester...: = 00FK ( 0 +/N c^5Xk3 ; > hi # # _t27 Y ; Vi2- to every.! Heavy flow of oil through a numerical example in Section '' Yb ; lIr_/Y.De T ) 3 Add an from... Presented in the last Section Output 6.10.1 flow and arc capacities are specified as and... Algorithm to find the maximum matching problem 1The network flow problem is intimately related to the maximum traffic! # E 6 ( L1ZVh ( ukK ] 4Y=4 * 0Bt [ 60CM\B [ $ @... Flow in the network ow problem on this new graph G0 2h7sGJiffX He43 * 2i9'dW %!.

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