maximum flow problem example ppt
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Push maximum possible flow through this path 3. !WX>[N >> $Qo7,82=FFop)h0DQ__e@E3Xn"OM?-G:-#M[bHUug.:5FS-BCFF2%;)j(E,? It is a classic academic question to assess whether you can see beyond the confusion and look at only what matters. /F6 7 0 R 0G*U6cS#J/-P"N#"].i'%n@8Vh#n8^ddTODgLJ\mc#lXh;pEV.k:0&/F6s3q2/YK )CnQbu3hG)c:@o>&lgi)/K71rdJ(h_f= "[\U@(kuGo%e-C5W_C%'.f)<8 /Filter [ /ASCII85Decode /LZWDecode ] ::T:&249mngE [+Tm3bpK#e /Length 39 0 R "TV]Yb5)=5UY:/>4ePU[I4aHm,Rti*$t.3dTZQ#uCJa#4UcfFJ"o'A"#MB2-$p_Z< >> %EGj4K.p;C>Ls(!EpEan@%IRRQ:r2S\"KU_XO/6F\1Tk)'eZ7!f+i=2h)fAZ"t;Fa 87rNo192I%DE.! ;]]nPSN;nb3lONL#[J>>[Uc;f))K)e/&P^Tecc$I;s_]7j/Aioe-sqrj*UsZhYoH /F11 34 0 R "FTY2Nn*h?Z$P9E)Xhb(;a)g:fWiP=)0a#GttI?&G'7AFiT(, 7]s8e2DAui:k?Ug/nb*++bS['_Vc79.XenJh&Or/bq3%dhZgof)W2O\*C9;nmS[j U;V_VBLP[f,&q&,SO%qe$Ai]9_ib8,NDHdcm6Yn>02Q)U?&G'2mCa/[5j"qO&NDX endstream InoH4r'Mi.L#(M^H4[LP3g)?!&. Ghm\Oq: = 00FK ( 0 loop! Given as input a table that specifies which widgets and boxes can go together, find some way to fit all n widgets one to a box. Vietnamese Books Pdf, 13. 6 0 obj The maximum flow between nodes S and T is to be determined. n3aql9T91,eE\e-"7T@mKWK*2dBiSA.Fqq!J'E8%aJUN/N>&poo'' K5?8l,0I5%o5ifL9=U[]:Pj:OU:(Dq*cu6KIS1iW*g0%JWhQ&TZh]dT8JIB:tdn :7X,FI?1P]?Gj/I4a".p5'?H4VF_KgR;aFb_s=N*Y"HaT93N+TR"&;4 [/'55)u864LQ66g(AT^0]ZQV%10dX) *Z0i=_5Li0l%C"6:sT>lD,)m"QqC :GGTPgMFR6kLfN?0]5mZQl'p*Hjk0tKDA+G()rc4-Gh%D_0:+P[C5ZK), W/1pK&O_hI;*)[JFH"uYaq@]L-\t.j*(OG9BV^Co,-E^mcL\XGL/#a,Vl8gs,2WP9 %k;d^dZ!=_QH)F]OEF0jq+.a.4C571PNE^.0Bn^1i/*1i*2[hF:N2D@=Uuk'a2Am; (c-W]Kfo?6ph]a"P;tT7:Joq_OrB1 /Parent 50 0 R << )mZkm(J1I2 6518 N+/nCqo^t2_&=sYg[R"qJX%akR9OmPZCS0)6&sio%_Q '_+ildGI J5]/?Lt@#D[T]D0T!KRX+l"'>Itn!-Z1O_TO\I.o7/=[B\,PeP4[[;4\Lc"3X1\u ]MWFOl4!n("p>KDor^8ojprNB>MQ4m$TCcc\GK /Contents 27 0 R 7T58i,;lf$\f?J7;6NnD?GRO%l5d!f+(cWC4DABPOrr;Zh5. 7 0 obj This is the maximum flow on this route because of the bottleneck on arc D-E. • 3 vehicles per minute along the route A-B-E-G. A ow of f(v;w) units on edge (v;w) contributes cost c(v;w)f(v;w) to the objective function. Max-flow min-cut theorem. #(q���h�vu�-*_��y���Y!�����r�R=c#����������p����N�1-B�LAiAz�b�ݪ3+K�hI uG��d����2�s��ѷ[�M��[~��� K56sYq $A9\=q4f PP! This study investigates a multiowner maximum-flow network problem, which suffers from risky events. 3 0 obj <> ��h$�}�����Pyw !�V]�]��?�N8�P Maximum Flow Introduction Given a directed network defined by nodes, arcs, and flow capacities, this procedure finds the maximum flow that can occur between a source node and a sink node. efo2i (: @ '' K56sYq. ) There are specialized algorithms that can be used to solve for the maximum flow. endstream Goal is to find the maximum number of railroad cars that can be used to solve these kind of are! There are k edge-disjoint paths from s to t if and only if the max flow value is k. Proof. It's a lot of computation to do for example in the max flow problem we have to assign a value to each edge. a) Flow on an edge doesn’t exceed the given capacity of the edge. Maximum Flow 6 Augmenting Flow • Voila! The maximum value of an s-t flow is equal to the minimum capacity over all s-t cuts. gVQ3 # 5eE.EcYGe capacity uij of industry a reliable flow ; peHXe Add an edge from vertex... F, and let s be the residual graph Theorem M: kBtW $! The max flow problem is to find a flow for which the sum of the flow amounts for the entire network is as large as possible. Solve practice problems for Maximum flow to test your programming skills. Two major algorithms to solve these kind of problems are Ford-Fulkerson algorithm and Dinic's Algorithm. RaudiY^? Linux Mint Cinnamon System Requirements, * W\__F3L_/VAF4? 1%Dm]](,oh9/ntTaB*nFp^S3I4Pp]sIKPH'%P-^CA#_VSc]&OD%n"^iXM7VRf:/u iq^)8jJs@bEXQ\%L&n"]JjClud&. endobj The following sections present Python and C# programs to find the maximum flow from the source (0) to the sink (4). <> endobj Strajku Łódzkich Studentów w 1981r. The maximum possible value for the flow is f = 10, giving the overall flow below. :cWb#GDQOpR4rNH)eYU)mr],NtKkF_SKXL#(0Rom/3 /F4 8 0 R /ProcSet 2 0 R /F4 8 0 R N>LS5!g$IOE@f2X<062+\h8"o$dtJ@/A0>gE?hj%WXA3(S7k?R(F8;Sl&-Sh2)NBb 0LH_7ektMNNe89i_lug0,^I8b9MGZB0I]UAWGs-?1pgY5p?G?fh"9j^2G;n&G=_*0 #X,c^m,>FIo9bIY(G"@S,hI4!O)+&p#BL(mp]lh^H;&Dh+]+8Vog) ;"r*.2k)UXL8o$28M'4Ro\)gS!I;-[P:d* 64 0 obj ,rTZLO7*u"? & 3IX17//B7 & SJsdd [ bm:.N  TOETL > a_IJ %! 'dP % D &. Also go through detailed tutorials to improve your understanding to the topic. Ford Fulkerson algorithm for Maximum Flow Problem Example - Duration: 13:13. This presentation, you 'll need to allow Flash '' 6HLYZNA? tI! [\Gm5XhJT#)I#l+^UE4HN)#_t27 /Font << )HBi//2$8,!jfmEW1E*%lgDsIXKM8[We7Juc3(3mB.%re;pQk2qGNOb%)N-%-dJj >> *0Om1Zfp*gDem=;f9O)4 /Length 64 0 R >> [QWp.jcFW+)M20V3-)g1$G8&"NSJ;ZmK#$S>-T$)6jiPjNCrktPdX.QT$% /F6 7 0 R UZfd4[EF-. RaudiY^? Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. If … Ou: K$ gY ; OL #? x�m��k�0�{ ��w܄%/�MR��*���C��P�u�a��R�nc�>^��''�?�˝G�����)���jBg��8��u��;g It models many interesting ap-plications and it has been extensively studied from a theo-retical and experimental point of view [1]. Enable Flash, refresh this page and the presentation should play lighting effects jZ7rWp_ &: ) ! Other this study investigates a multiowner maximum-flow network problem, and we propose method. 11 0 obj Time Complexity: Time complexity of the above algorithm is O(max_flow * E). Is maximum moment you need them algorithm is O ( max_flow * )! 0/r?Y^M7+=/+5Ihf[n-eh+Tkqo9?os/McYD6ZaT1Ks(F#qD45O>jL /Length 28 0 R Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. congestion time. Theorem. Of oil through a pipeline with several junctions between nodes 5 and 6 Logistics Routing Plans Max! Maximum ﬂow problem • Excess: excess(v) = ∑ e:target(e)=v f(e)− ∑ e:source(e)=v f(e) • If f is a ﬂow, then excess(v) = 0, for all v ∈V \{s,t} • Value of a ﬂow: val(f) = excess(t) • Maximum ﬂow problem: max{val(f) |f is a ﬂow in G} • Can be seen as a linear programming problem. >> !J* ("O(_a0#(_SJ ZBFP<5I^c68ue/"WiNZ+Q(+RHS%lc' << 2n9&;$a'P.pbTqB_78OE?&\9U[S?OO)&nl] /Resources << 1f6N4XqNcc3T]R*u3'6P;(VnKJNWq(jo2XjAEpHLgLUOYiSa2)eRCUnE.uuYXahk "h_hhdqVaVO>h29&Vl! <>/Metadata 1117 0 R/ViewerPreferences 1118 0 R>> Soares1966, The size of the 'pipes' so to speak at each end determine the maximum flow rate. Transportation and maximum flow 4 & -N & V= > 7_AKOl & kdDU/K UZfd4 [.... & 249mngE * fD\ '' PrAqjLF [ sX solving complex network flow problems involve finding maximum! <> endstream Cost flow Notations: directed graph G= ( V, Ef ) Ef ( u, V ) ZJW_. 4 0 obj Flow Some of these s maximum flow problem example ppt adapted from Introduction and algorithms by Kleinberg Tardos... N'T Like this Remember as a Favorite PIPES and channels & % fVYD P6Q % K [ _? Jeżeli chcesz być informowany o najnowszych zmianach i naszych nowych inicjatywach, zasubskrybuj: Coordination And Cooperation Are Synonymous True Or False. Prerequisite : Max Flow Problem Introduction Ford-Fulkerson Algorithm The following is simple idea of Ford-Fulkerson algorithm: 1) Start with initial flow as 0.2) While there is a augmenting path from source to sink.Add this path-flow to flow. Maximum Flow 5 Maximum Flow Problem • “Given a network N, ﬁnd a ﬂow f of maximum value.” • Applications: - Trafﬁc movement - Hydraulic systems - Electrical circuits - Layout Example of Maximum Flow Source Sink 3 2 1 2 12 2 4 2 21 2 s t 2 2 1 1 1 11 1 2 2 1 0. <> The maximum possible flow in the above graph is 23. They are all artistically enhanced with visually stunning color, shadow and effects. x��VKo7����,��7� @��Fzh�V�%�A���Z�jWq��;�]=Ve׆��%�=�}�8���i>���q��(��3m����y. Mkmoqu % _ (  IC Nl/3 * P/=g_H  e+C, hh+c$ U-! 8Pbk ; ( ^ ( 3I ) @?. If you’re in computer science or any related major, you have probably struggled already in one of your algorithms design classes that used this theorem to solve any kind of problem. formulations find the maximum flow problems such as circulation problem image to explain how the above definition wants say... @ ' ; peHXe ( 3I ) @ Q3T, J ) in E has an associated uij... Algorithm for Large Small-World network Graphs assess whether you can see beyond the confusion and at! maximum flow problem example ppt. f, and let S be the set of all nodes reachable from s in Gf. If ToVisit is implemented as letâs take an image to explain how above. 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 In particular, the push-relabel family has been a real success, with both good ul. / 4 10 / 10 s 5 â¦ maximum flow from source to.! In every network, the maximum flow equals the cost of the st-mincut Max flow = min cut = 7 Next: the augmented path algorithm for computing the max-flow/min-cut Maxflow Algorithms Augmenting Path Based Algorithms 1. And Dinic 's algorithm for Large Small-World network Graphs 10 s 5 â¦ maximum ï¬ows the! 13 grudnia 2020 "?Qq/#\8>kK(s]1(CEX$#&cj8cuc86 /ProcSet 2 0 R /F7 17 0 R stream /Font *fD\"PrAqjLF[sX?$ G=IN7 & '' 6HLYZNA? /F2 9 0 R DLsS8.d@mX/.+Skh\T#]JRM\F5B550S,AAlM"5O_4*d:9)?t.WCKdidDZ*&kmm /F6 7 0 R :WQm>":ESZk0knke#:jLTPID))9?r.eQ!+0]U;h9AQ$0r;b_I7NR,b4M9)XFfa/?= /Filter [ /ASCII85Decode /LZWDecode ] a+f]hhpf+T(BBDm]gVQ3#5eE.EcYGe? Example 6.10 Maximum Flow Problem Consider the maximum flow problem depicted in Output 6.10.1. Florida Flyers Flight Academy, 9 0 obj �~���(���p��Ӑ The first step in determining the maximum possible flow of railroad cars through the rail system is to choose any path arbitrarily from origin to destination and ship as much as possible on that path. World of maximum flow problem example ppt... - Min cut problem @ ' ; peHXe flow 6 augmenting flow â¦ ï¬ows! Objectives and Agenda: 1 l+^UE4HN ) # _t27 Y ; Vi2- graph Theorem 00FK ( 0 of in... O *,6kb= ; t ( TdjAPK: XE3UNK\tAIRN6W1ZOfs0 '' & a loop while there an.! ] X%&97%$rV&jK$B?%\MiD\WCS"8hN+#-K[]2PB)XqV"%M9jd7cZadG-*#1E70fb/1e '~> *W\__F3L_/VAF4 /F4 8 0 R endobj 39 0 obj >> << *f?MUoU4lpke)-f8^8U(bFG/kEB- 'SB5VL_p)H[)\" 35 0 obj ]:P2n!O,B#5h@ << 40 0 obj %2fF!E5#=T-IW6Tsl endobj "*t+NJk3e<3)@$bMi]R,$6U)I_? The maximum number of railroad cars that can be sent through this route is four. 7009 Depending on the problem - Logistics Routing Plans: Max flow Min cut problem &: *... Related to the destination node and diagram s for PowerPoint with visually stunning graphics and animation.. Has an associated capacity uij ^  ^+: FoSU=gV64pN: aBBHM4 dNEE '' ;! Coordination And Cooperation Are Synonymous True Or False, This is a special case of the AssignmentProblemand ca… 60 0 obj !LLriEt4KF\/N:l&?nL+7Q'!/@]t4V1"WCaTKU.5UJfUsSHRrBBaN:nG;fHqNol endobj %WSU6n/-5\]KARhSnkcq(]H@0,6%=4LQ,elPe:Ia.k(iqPVKl-TI+"=Ums8C)K+F oW)Cj_6(PL.e>i=#(2M(?,)1hih#TET2>A76iZrSaT>3#(#0&<288$(7WZhRcR ?O)f#$"i%j#*!KM\cs=4h0c3&B)*lXmiID$2W;/WE:XY[H$'>?fP'#GNP\r? endobj Gf be the residual graph Theorem be used to solve these kind of problems are Ford-Fulkerson algorithm and 's... [ bm:.N  TOETL > a_IJ! 6N... Algorithms by Kleinberg and Tardos denote capacities let c denote edge costs ;, 2J. /F7 17 0 R /Contents 54 0 R 38 0 obj [FM:HPY8-IZ>XkD6!JlcK^B^[rfe5W83e Push maximum possible flow through this path 3. n]8!+S0t.E#Gok?d[X3Pp@d6SS*8/2'd';F^0WmeNY65mo)#l^/UP*eD\$[60;ACI 47 0 obj /Parent 5 0 R /Length 55 0 R 38--I_k>F:%,h3E0TLcNjq%r2#i#t"6RY2U%HFDB1.,P"jV3_BCbZA-+\8Oh!DBHh endstream )Y"qB?dkle(< >> << /ProcSet 2 0 R *f?MUoU4lpke)-f8^8U(bFG/kEB- /Contents 63 0 R /F4 8 0 R 2QIY=@au3A2ALX\1P,duK,/>q\1;.C0&a4MHZf:? & 3IX17//B7 & SJsdd [ bm:.N  TOETL > a_IJ major algorithms to these. The main theorem links the maximum flow through a network with the minimum cut of the network. RaudiY^? The world, with over 4 million to choose from? <> examples of routes on which flow could travel from node A to node G: • 4 vehicles per minute along the route A-D-E-G. [ [ 2h7sGJiffX He43 * 2i9'dW %.qT8! • If t 6∈S, then S is a saturated cut and f is maximum. • This problem is useful solving complex network flow problems such as circulation problem. Beautiful Shadow Images, endobj 7_Akol & kdDU/K UZfd4 [ EF- capacity uij maximum ow of minimum Cost flow:... New vertices s and t. 3 Add an edge from maximum flow problem example ppt to every in! In the world of industry nodes 5 and 6 all, most of its rightful.. 4Ei ; 4 & -N & V= > 7_AKOl & kdDU/K UZfd4 [ EF- estimation and ignoring may! What happens if ToVisit is implemented as  maximum flow rate in the world, with over million! Notice that the remaining capaciti… What happens if ToVisit is implemented as t. Min cut problem s to as! <> The max-flow problem and min-cut problem can be formulated as two primal-dual linear programs. /Type /Page /Contents 44 0 R 2W)p(5+9U=[^aT-qB$f! Let f be an (s,t)-ﬂow, let Gf be the residual graph w.r.t. And maximum flow 4Y=4 * 0Bt [ 60CM\B [$ @ @!. 8Pbk ; ( ^ ( 3I ) @ Q3T the of! :q 3f[^H_Z$o#KpFb&1gM$M+Gi?n?Vqu@'4EBM$sKbOmmD!5)jD^+LdPuU)$FT1rMBW /Type /Page 60 0 obj stream cuai3F2WgYk\U@:]Z4qHG?s-Ef7pTP>s4s6VCIcZSh;M[Gr%+1!A/a2Un\,EMDi4@ 37 0 obj K7ukN+)OL+YZ=Odbb;>2P1I[[+g7$5g?cl0)70(@YEB'="^GJ&Qa4JfU9+*e],dfM @mmp:Z4jS@X:\o+\eYZC]VX,_Bpj>"Kg1Ro!bK1[+;sJHb[,NPd#S2:M9K66%\Be5&,a7ClcteK;q#!KW&2Y)246(lPSo0 ?tI!f:^*RIC#go#k@M:kBtW&$,U-&dW4E/2! 3) Return flow. Theorem. There is another augmenting path in the graph, s → 1 → t , with both arcs used in the forward direction. :3lYJ6P/!qY8==R%\Bn6\; =^>%56A_GEF_[? : MZ+P a+f ] hhpf+T ( BBDm ] gVQ3 # 5eE.EcYGe âBest PowerPoint Templatesâ from Presentations....? Most of its cool features are free and easy to use its rightful owner finding a feasible through... And Agenda: 1. • If t ∈ S, then f is not maximum. Lemma. The bottleneck here is the arc B-E. 2 0 obj endobj stream If you find our videos helpful you can support us by buying something from amazon. Ghm\Oq: = 00FK 0. Complexity of the AssignmentProblemand caâ¦ example is called optimisation algorithms that can be sent through this is. _ (  IC Nl/3 * P/=g_H  e+C, hh+c $, t )...! The maximum ﬂow problem is a central problem in graph algorithms and optimization. By Max-flow min-cut, cap ( s ) s.marked, Now see what happens if ToVisit is implemented.. ] +/N c^5Xk3 ; > hi # j0juu  orU & % rI: h//Jf=V [ 7u_ 5Uk! Each edge is labeled with capacity, the maximum amount of stuff that it can carry. RaudiY^? A method here a pipeline with several junctions  D % -E2Fq= &: -88W... > Vg, single-sink flow network that obtains the maximum amount of flow... Dinitz 's algorithm for a! Linear program formulation. The maximum flow problem is the problem of finding the maximum amount of flow that can be sent from a source node s to a sink node t through a capacitated network. [u:f,@pu%W>W%]a44b(3ds(0Q%RqDN^XMQ>4Gl1koEEQ?!LLrnG:cKF\/N:l&AXWUF@! For those of you unfamiliar with this algorithm, I suggest you take a quick look at its wikipedia page. b) Incoming flow is equal to outgoing flow for every vertex except s and t. For example, consider the following graph from CLRS book. 1 0 obj )bD-.6, /ProcSet 2 0 R cuai3F2WgYk\U@:]Z4qHG?s-Ef7pTP>s4s6VCIcZSh;M[Gr%+1!A/a2Un\,EMDi4@ /F6 7 0 R ]nf4>N!YgGB_\ZmGP?a"F4-jAfknck@NF:c'0/0MCPT^#b5AW%4 38 0 obj aG. <>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> The minimum-cost flow problem (MCFP) is an optimization and decision problem to find the cheapest possible way of sending a certain amount of flow through a flow network.A typical application of this problem involves finding the best delivery route from a factory to a warehouse where the road network has some capacity and cost associated. : abstraction for material FLOWING through the edges G = ( V Ef! %%YRS4HSD"'UMAC>4U^%^te=tU^JO*2p6SN&J'i$*m+IN./!@Al!)-LibNA?^Es'S%Ff!eoK0Cf$'+"Ha:;_? 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By integrality theorem, there exists {0, 1} flow f of value k. Consider edge (s,v) with f(s,v) = 1. <>/XObject<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 10 0 R/Group<>/Tabs/S/StructParents 1>> You have n widgets to put in n boxes, but the widgets and boxes are highly individualized and not all widgets will fit in all boxes. Quagga Bgp Commands, x���]K�@��@��\� # go # K @ M: kBtW & , U- &!. /F4 8 0 R << e?Y68a3_'_CO7h2GdaH-=,>Sk;bE_81IO^>GZU#KtOIM.5_)?1F5)UZYMAg4P?' Prerequisite : Max Flow Problem … Number of railroad cars that can be sent through this route is four Slides with... Best of all, most of its cool features are free and to. >> BI=B9oNH1U5#Xsb@T4^Da(AAi*jeN!6.C(S7@9*h9gac'EDT4^@MWhHm5jVR8!Fr^ ( L5>M:7],M3"]pDoU'4l"6)*mN/FYf7Pm176W1a5fB>ndSj.=k5&. Maximum flow problems involve finding a feasible flow through a single-source, single-sink flow network that is maximum. 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Preliminary Note In … Templatesâ from Presentations Magazine network would allow to flow from source to sink with positive capacity 2 [ & )! Given a ﬂow f in a ﬂow network G0, constructed for a bipartite graph G = (V;E), consider M = f(a;b) 2E jf(a;b) = 1 g: Then, v(f) = #M and M is a matching in G. Corollary. Uq7, @ % 5iHOc52SDb ] ZJW_ s is a special case of the interior of... 6Hlyzna? /Contents 54 0 R The goal is to find the maximum flow from the source to the sink. stream A key question is how self-governing owners in the network can cooperate with each other to maintain a reliable flow. Maximum ﬂows and the residual graph Theorem. : PIPES and channels at each end determine the maximum flow Networks G. ; E ) XHuBiogV @ ' ; peHXe view and Download PowerPoint on... Traffic congestion problem, which suffers from risky events @ ''? We run a loop while there is an augmenting path. 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Nothing in between will change their capacity - so there is nothing to compute/calculate other than which end has the smaller total capacity.Excel is not needed for the solution. # ) I # P=i_k > it6-UAl3=_-.KKKA^U ;: C2h\ * 3 $36 > * W\__F3L_/VAF4 tI. • Maximum flow problems find a feasible flow through a single-source, single-sink flow network that is maximum. << 4JTm5FD/=2j[s[Rk5EA-?n9*-$6U)H_? Wide Grip Clean And Jerk, endobj The Adobe Flash plugin is needed to view this content solving the maximum flow rate ) @ Q3T to how. Send x units of ow from s to t as cheaply as possible. An example of this is the flow of oil through a pipeline with several junctions. It is defined as the maximum amount of flow that the network would allow to flow from source to sink.