4941:"Determining a maximal steady state flow from one point to another in a network subject to capacity limitations on arcs ... was posed to the authors in the spring of 1955 by T.E. Harris, who, in conjunction with General F. S. Ross (Ret.) had formulated a simplified model of railway traffic flow, and pinpointed this particular problem as the central one suggested by the model. It was not long after this until the main result, Theorem 5.1, which we call the max-flow min-cut theorem, was conjectured and established. A number of proofs have since appeared."
3968:
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1488:
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2822:
655:
A flow can be visualized as a physical flow of a fluid through the network, following the direction of each edge. The capacity constraint then says that the volume flowing through each edge per unit time is less than or equal to the maximum capacity of the edge, and the conservation constraint says
2833:: the variables and sign constraints of the dual correspond to the constraints of the primal, and the constraints of the dual correspond to the variables and sign constraints of the primal. The resulting LP requires some explanation. The interpretation of the variables in the min-cut LP is:
2593:{\displaystyle {\begin{aligned}d_{uv}-z_{u}+z_{v}&\geq 0&&\forall (u,v)\in E,u\neq s,v\neq t\\d_{sv}+z_{v}&\geq 1&&\forall (s,v)\in E,v\neq t\\d_{ut}-z_{u}&\geq 0&&\forall (u,t)\in E,u\neq s\\d_{st}&\geq 1&&{\text{if }}(s,t)\in E\end{aligned}}}
4543:
The idea here is to 'flow' each project's profits through the 'pipes' of its machines. If we cannot fill the pipe from a machine, the machine's return is less than its cost, and the min cut algorithm will find it cheaper to cut the project's profit edge instead of the machine's cost edge.
3880:
4769:
1312:
809:
4025:
to purchase. Each project requires a number of machines and each machine can be shared by several projects. The problem is to determine which projects and machines should be selected and purchased respectively, so that the profit is maximized.
3038:
4196:
3681:
650:
2074:
2839:
854:
is the sink of the network. In the fluid analogy, it represents the amount of fluid entering the network at the source. Because of the conservation axiom for flows, this is the same as the amount of flow leaving the network at the sink.
2687:
2693:
4337:
4880:
1145:
5160:
2051:
3514:
The maximum flow problem can be formulated as the maximization of the electrical current through a network composed of nonlinear resistive elements. In this formulation, the limit of the current
1972:
2698:
2622:
2269:
2079:
1895:
78:
The theorem equates two quantities: the maximum flow through a network, and the minimum capacity of a cut of the network. To state the theorem, each of these notions must first be defined.
5258:
3757:
3110:
1471:
cut, and that furthermore a flow with maximal value and a cut with minimal capacity exist. The main theorem links the maximum flow value with the minimum cut capacity of the network.
5526:
5383:
4616:
3730:
296:
205:
3166:
3403:
3331:
3282:
1819:
1744:
455:
5594:
Both of the above statements prove that the capacity of cut obtained in the above described manner is equal to the flow obtained in the network. Also, the flow was obtained by
4885:
The above minimization problem can be formulated as a minimum-cut problem by constructing a network where the source (orange node) is connected to all the pixels with capacity
3446:
1171:
669:
4599:
are adjacent and have different assignments. The problem is to assign pixels to foreground or background such that the sum of their values minus the penalties is maximum.
3230:
2959:
2242:{\displaystyle {\begin{aligned}f_{uv}&\leq c_{uv}&&\forall (u,v)\in E\\\sum _{u}f_{uv}-\sum _{w}f_{vw}&=0&&v\in V\setminus \{s,t\}\end{aligned}}}
406:
4047:
3559:
2946:{\displaystyle d_{uv}={\begin{cases}1,&{\text{if }}u\in S{\text{ and }}v\in T{\text{ (the edge }}uv{\text{ is in the cut) }}\\0,&{\text{otherwise}}\end{cases}}}
526:
1351:
4342:
The above minimization problem can then be formulated as a minimum-cut problem by constructing a network, where the source is connected to the projects with capacity
3551:
1779:
1704:
1403:
1377:
361:
326:
1849:
1024:
2617:
895:
2817:{\displaystyle {\begin{aligned}d_{uv}&\geq 0&&\forall (u,v)\in E\\z_{v}&\in \mathbb {R} &&\forall v\in V\setminus \{s,t\}\end{aligned}}}
1423:
935:
915:
852:
832:
520:
500:
480:
5616:
A corollary from this proof is that the maximum flow through any set of edges in a cut of a graph is equal to the minimum capacity of all previous cuts.
1563:) of the arrow. The flows emanating from the source total five (2+3=5), as do the flows into the sink (2+3=5), establishing that the flow's value is 5.
5815:
P. Elias, A. Feinstein, and C. E. Shannon (1956) "A note on the maximum flow through a network", IRE. Transactions on
Information Theory, 2(4): 117–119
4215:
5605:
Also, since any flow in the network is always less than or equal to capacity of every cut possible in a network, the above described cut is also the
5840:
L. R. Ford & D. R. Fulkerson (1957) "A simple algorithm for finding the maximum network flows and an application to the
Hitchcock problem",
1036:
4780:
3959:
states that the maximum number of edge-disjoint s-t paths in an undirected graph is equal to the minimum number of edges in an s-t cut-set.
5856:(2001). "4.5. Combinatorial Implications of Max-Flow Min-Cut Theorem, 4.6. Linear Programming Interpretation of Max-Flow Min-Cut Theorem".
5080:
656:
that the amount that flows into each vertex equals the amount flowing out of each vertex, apart from the source and sink vertices.
1625:
The value of the flow is equal to the capacity of the cut, showing that the flow is a maximal flow and the cut is a minimal cut.
5686:
1467:
In the above situation, one can prove that the value of any flow through a network is less than or equal to the capacity of any
1985:
1154:
are removed, then no positive flow is possible, because there is no path in the resulting graph from the source to the sink.
996:
cut is a division of the vertices of the network into two parts, with the source in one part and the sink in the other. The
5625:
4518:
The minimum capacity of an s-t cut is 250 and the sum of the revenue of each project is 450; therefore the maximum profit
1913:
67:
5932:
4914:
capacity are added between two adjacent pixels. The s-t cut-set then represents the pixels assigned to the foreground in
3509:
3751:
has to satisfy not only the capacity constraint and the conservation of flows, but also the vertex capacity constraint
3476:
in the cut - we only have to guarantee that each edge that should be in the cut, is summed in the objective function.
5937:
5913:
5891:
5865:
1853:
946:
The other half of the max-flow min-cut theorem refers to a different aspect of a network: the collection of cuts. An
5169:
3875:{\displaystyle \forall v\in V\setminus \{s,t\}:\qquad \sum \nolimits _{\{u\in V\mid (u,v)\in E\}}f_{uv}\leq c(v).}
5803:
4764:{\displaystyle \max\{g\}=\sum _{i\in P}f_{i}+\sum _{i\in Q}b_{i}-\sum _{i\in P,j\in Q\lor j\in P,i\in Q}p_{ij}.}
1606:}. The capacities of the edges that cross this cut are 3 and 2, giving a cut capacity of 3+2=5. (The arrow from
3912:
states that the maximum value of an s-t flow is equal to the minimum capacity of an s-t cut in the new sense.
3053:
5942:
5468:
5325:
3694:
260:
169:
42:
36:
3115:
5640:
5011:
4992:
1428:
There are typically many cuts in a graph, but cuts with smaller weights are often more difficult to find.
51:, i.e., the smallest total weight of the edges which if removed would disconnect the source from the sink.
3359:
3287:
3238:
1783:
1708:
411:
5785:
5630:
1547:
The figure on the right shows a flow in a network. The numerical annotation on each arrow, in the form
1307:{\displaystyle c(S,T)=\sum \nolimits _{(u,v)\in X_{C}}c_{uv}=\sum \nolimits _{(i,j)\in E}c_{ij}d_{ij},}
24:
3408:
804:{\displaystyle |f|=\sum \nolimits _{\{v:(s,v)\in E\}}f_{sv}=\sum \nolimits _{\{v:(v,t)\in E\}}f_{vt},}
5875:
1636:
is at full capacity; this is always the case: a minimal cut represents a 'bottleneck' of the system.
3033:{\displaystyle z_{v}={\begin{cases}1,&{\text{if }}v\in S\\0,&{\text{otherwise}}\end{cases}}}
2981:
2864:
5595:
4191:{\displaystyle \max\{g\}=\sum _{i}r(p_{i})-\sum _{p_{i}\in P}r(p_{i})-\sum _{q_{j}\in Q}c(q_{j}).}
3676:{\displaystyle \lim _{V_{\text{in}}\to \infty }(I_{in})=\min _{X_{C}}\sum _{(u,v)\in X_{C}}c_{uv}}
3199:
645:{\displaystyle \sum \nolimits _{\{u:(u,v)\in E\}}f_{uv}=\sum \nolimits _{\{w:(v,w)\in E\}}f_{vw}.}
4421:
The figure on the right gives a network formulation of the following project selection problem:
373:
3043:
The minimization objective sums the capacity over all the edges that are contained in the cut.
1320:
3536:
3472:
Note that, since this is a minimization problem, we do not have to guarantee that an edge is
2829:
The max-flow LP is straightforward. The dual LP is obtained using the algorithm described in
2682:{\displaystyle {\begin{aligned}f_{uv}&\geq 0&&\forall (u,v)\in E\\\end{aligned}}}
3691:
In addition to edge capacity, consider there is capacity at each vertex, that is, a mapping
1754:
1679:
1382:
1356:
331:
301:
5791:
5715:
4553:
4415:
1827:
1002:
863:
4209:, this maximization problem can be formulated as a minimization problem instead, that is,
8:
5660:
3921:
3484:
2830:
1645:
870:
63:
4774:
This maximization problem can be formulated as a minimization problem instead, that is,
5650:
1648:
1408:
920:
900:
837:
817:
505:
485:
465:
5909:
5887:
5879:
5861:
5760:
5697:
3967:
4610:
be the set of points assigned to background, then the problem can be formulated as,
1478:
The maximum value of an s-t flow is equal to the minimum capacity over all s-t cuts.
5756:
20:
5788:
5777:
5645:
4935:
4332:{\displaystyle \min\{g'\}=\sum _{p_{i}\in P}r(p_{i})+\sum _{q_{j}\in Q}c(q_{j}).}
3971:
A network formulation of the project selection problem with the optimal solution
3926:
In the undirected edge-disjoint paths problem, we are given an undirected graph
5901:
5824:
5773:
5747:
Cederbaum, I. (August 1962). "On the optimal operation of communication nets".
4931:
95:
59:
5733:
4414:
respectively. By the max-flow min-cut theorem, one can solve the problem as a
1030:
is the set of edges that connect the source part of the cut to the sink part:
5926:
5853:
5874:
5827:
and D. R. Fulkerson (1956) "On the Max-Flow MinCut
Theorem of Networks", in
5802:
L. R. Ford Jr. and D. R. Fulkerson (1956) "Maximal flow through a network",
5655:
5635:
3524:
between the input terminals of the electrical network as the input voltage
3046:
The constraints guarantee that the variables indeed represent a legal cut:
55:
32:
4892:, and the sink (purple node) is connected by all the pixels with capacity
1140:{\displaystyle X_{C}:=\{(u,v)\in E\ :\ u\in S,v\in T\}=(S\times T)\cap E.}
5665:
4875:{\displaystyle \min\{g'\}=\sum _{i\in P,j\in Q\lor j\in P,i\in Q}p_{ij}.}
4041:
be the set of machines purchased, then the problem can be formulated as,
244:. It represents the maximum amount of flow that can pass through an edge.
48:
858:
The maximum flow problem asks for the largest flow on a given network.
3949:, and we have to find the maximum number of edge-disjoint s-t paths in
3495:*, such that the optimal values formed by the two solutions are equal.
1503:
is the edge's capacity. The flow value is 5. There are several minimal
522:(i.e. the source and sink, respectively), the following equality holds:
4558:
1653:
5610:
5599:
3487:, which states that if the primal program has an optimal solution,
1644:
The max-flow problem and min-cut problem can be formulated as two
5606:
5155:{\displaystyle c(S,T)=\sum \nolimits _{(u,v)\in S\times T}c_{uv}}
3893:
to be the set of vertices and edges such that for any path from
3889:
passing through a vertex cannot exceed its capacity. Define an
1628:
Note that the flow through each of the two arrows that connect
5578:, which is again a contradiction. Hence, any incoming edge
3901:, the path contains a member of the cut. In this case, the
3026:
2939:
1487:
4406:. The s-t cut-set represents the projects and machines in
4356:, and the sink is connected by the machines with capacity
3905:
is the sum of the capacity of each edge and vertex in it.
1165:
is the sum of the capacities of the edges in its cut-set,
3553:, is equal to the weight of the minimum-weight cut set.
4930:
An account of the discovery of the theorem was given by
2046:{\displaystyle \sum \nolimits _{(u,v)\in E}c_{uv}d_{uv}}
3686:
3491:*, then the dual program also has an optimal solution,
1491:
A maximal flow in a network. Each edge is labeled with
4201:
Since the first term does not depend on the choice of
5884:
5471:
5441:, which is a contradiction. Hence, any outgoing edge
5328:
5172:
5083:
4783:
4619:
4218:
4050:
3760:
3697:
3562:
3539:
3411:
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3290:
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1988:
1916:
1856:
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1711:
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1385:
1359:
1323:
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1005:
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903:
873:
840:
820:
672:
529:
508:
488:
468:
414:
376:
334:
304:
263:
172:
1967:{\displaystyle \sum \nolimits _{v:(s,v)\in E}f_{sv}}
5831:, Ann. Math. Studies, no. 38, Princeton, New Jersey
5685:Dantzig, G.B.; Fulkerson, D.R. (9 September 1964).
3503:
5520:
5377:
5252:
5154:
4874:
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4190:
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3724:
3675:
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3325:
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3224:
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2045:
1966:
1889:
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1813:
1773:
1738:
1698:
1417:
1397:
1371:
1345:
1306:
1139:
1018:
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909:
897:, that is, to route as much flow as possible from
889:
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826:
803:
644:
514:
494:
474:
449:
400:
355:
320:
290:
199:
5900:
5858:Combinatorial Optimization: Networks and Matroids
1458:such that the capacity of the s-t cut is minimal.
5924:
5684:
5311:To prove the above claim we consider two cases:
4784:
4620:
4606:be the set of pixels assigned to foreground and
4219:
4051:
3609:
3564:
5528:such that it carries some non-zero flow, i.e.,
1890:{\displaystyle \forall v\in V\setminus \{s,t\}}
47:is equal to the total weight of the edges in a
5014:), define two subsets of vertices as follows:
35:, the maximum amount of flow passing from the
5882:(1998). "6.1 The Max-Flow, Min-Cut Theorem".
5852:
5687:"On the max-flow min-cut theorem of networks"
5253:{\displaystyle value(f)=f_{out}(A)-f_{in}(A)}
4566:In the image segmentation problem, there are
4547:
1639:
4798:
4787:
4629:
4623:
4233:
4222:
4060:
4054:
3975:In the project selection problem, there are
3962:
3836:
3800:
3788:
3776:
3485:strong duality theorem in linear programming
2807:
2795:
2232:
2220:
1884:
1872:
1107:
1053:
780:
750:
724:
694:
621:
591:
565:
535:
363:, subject to the following two constraints:
73:
5904:(2004). "12. Introduction to LP-Duality".
4522:is 450 − 250 = 200, by selecting projects
5746:
3712:
3168:) guarantee that, for non-terminal nodes
2773:
1150:Thus, if all the edges in the cut-set of
278:
187:
4557:
3966:
3105:{\displaystyle d_{uv}-z_{u}+z_{v}\geq 0}
1486:
5521:{\displaystyle (y,x),x\in A,y\in A^{c}}
5378:{\displaystyle (x,y),x\in A,y\in A^{c}}
3725:{\displaystyle c:V\to \mathbb {R} ^{+}}
1980:
1908:
1750:
1675:
291:{\displaystyle f:E\to \mathbb {R} ^{+}}
200:{\displaystyle c:E\to \mathbb {R} ^{+}}
116:denotes the finite set of vertices and
5925:
3161:{\displaystyle d_{uv}\geq z_{u}-z_{v}}
5716:"Lecture 15 from CS261: Optimization"
5563:, therefore there exists a path from
5426:, therefore there exists a path from
5385:such that it is not saturated, i.e.,
5037:: the set of remaining vertices i.e.
5021:: the set of vertices reachable from
4918:and pixels assigned to background in
4479:Project 1 requires machines 1 and 2.
1614:is not considered, as it points from
5713:
5626:Approximate max-flow min-cut theorem
5544:. This implies, that there exists a
5407:. This implies, that there exists a
5010:(after the final flow assignment by
3910:generalized max-flow min-cut theorem
3687:Generalized max-flow min-cut theorem
5106:
4964:be a network (directed graph) with
4574:can be assigned a foreground value
3915:
3796:
1990:
1918:
1248:
1197:
746:
690:
587:
531:
13:
5734:"LP min-cut max-flow presentation"
5731:
3761:
3581:
3540:
3398:{\displaystyle d_{ut}-z_{u}\geq 0}
3326:{\displaystyle d_{sv}\geq 1-z_{v}}
3277:{\displaystyle d_{sv}+z_{v}\geq 1}
2780:
2727:
2651:
2494:
2415:
2324:
2118:
1857:
1814:{\displaystyle \forall (u,v)\in E}
1787:
1739:{\displaystyle \forall (u,v)\in E}
1712:
450:{\displaystyle f_{uv}\leq c_{uv}.}
14:
5954:
5749:Journal of the Franklin Institute
4972:being the source and the sink of
3773:
2792:
2217:
1869:
4562:Each black node denotes a pixel.
3510:Cederbaum's maximum flow theorem
3504:Cederbaum's maximum flow theorem
3441:{\displaystyle d_{ut}\geq z_{u}}
5842:Canadian Journal of Mathematics
5804:Canadian Journal of Mathematics
3794:
1462:
257:through a network is a mapping
5834:
5818:
5809:
5796:
5767:
5740:
5725:
5707:
5678:
5484:
5472:
5341:
5329:
5247:
5241:
5222:
5216:
5194:
5188:
5122:
5110:
5099:
5087:
4511:Project 3 requires machine 3.
4495:Project 2 requires machine 2.
4323:
4310:
4278:
4265:
4182:
4169:
4137:
4124:
4092:
4079:
3885:In other words, the amount of
3866:
3860:
3827:
3815:
3707:
3642:
3630:
3602:
3586:
3578:
3498:
2742:
2730:
2666:
2654:
2577:
2565:
2509:
2497:
2430:
2418:
2339:
2327:
2133:
2121:
2006:
1994:
1940:
1928:
1802:
1790:
1727:
1715:
1499:is the flow over the edge and
1264:
1252:
1213:
1201:
1190:
1178:
1125:
1113:
1068:
1056:
883:
875:
771:
759:
715:
703:
682:
674:
612:
600:
556:
544:
389:
377:
350:
338:
273:
182:
54:This is a special case of the
16:Concept in optimization theory
1:
5908:. Springer. pp. 93–100.
5671:
4392:capacity is added if project
3460:is counted in the cut (since
3345:is counted in the cut (since
1574:cut with value 5 is given by
1511:cuts with capacity 5; one is
166:function, which is a mapping
130:is the set of directed edges;
5761:10.1016/0016-0032(62)90401-5
5260:for any subset of vertices,
3908:In this new definition, the
3225:{\displaystyle d_{uv}\geq 1}
7:
5886:. Dover. pp. 120–128.
5860:. Dover. pp. 117–120.
5619:
4446:
4428:
4423:
10:
5961:
5933:Combinatorial optimization
5786:Princeton University Press
4925:
4551:
4548:Image segmentation problem
4509:
4493:
4477:
4426:
3919:
3507:
2916: is in the cut)
2256:
2066:
1640:Linear program formulation
1482:
401:{\displaystyle (u,v)\in E}
81:
62:and can be used to derive
5876:Christos H. Papadimitriou
3963:Project selection problem
3196:) is counted in the cut (
1476:Max-flow min-cut theorem.
663:of a flow is defined by
74:Definitions and statement
5938:Theorems in graph theory
5906:Approximation Algorithms
5641:Ford–Fulkerson algorithm
5602:of the network as well.
5596:Ford-Fulkerson algorithm
5012:Ford–Fulkerson algorithm
4995:. In the residual graph
4993:Ford–Fulkerson algorithm
4944:
4588:. There is a penalty of
3481:max-flow min-cut theorem
1433:Minimum s-t Cut Problem.
1346:{\displaystyle d_{ij}=1}
248:
29:max-flow min-cut theorem
4033:be the set of projects
3983:machines. Each project
3546:{\displaystyle \infty }
941:
5631:Edmonds–Karp algorithm
5522:
5379:
5254:
5156:
4876:
4765:
4581:or a background value
4563:
4333:
4192:
3972:
3876:
3726:
3677:
3547:
3442:
3399:
3327:
3278:
3226:
3162:
3106:
3034:
2947:
2818:
2683:
2594:
2243:
2047:
1968:
1891:
1845:
1815:
1775:
1774:{\displaystyle d_{uv}}
1740:
1700:
1699:{\displaystyle f_{uv}}
1555:, indicates the flow (
1544:
1419:
1399:
1398:{\displaystyle j\in T}
1373:
1372:{\displaystyle i\in S}
1347:
1308:
1141:
1020:
939:
931:
911:
891:
848:
828:
805:
646:
516:
496:
476:
451:
402:
357:
356:{\displaystyle f(u,v)}
322:
321:{\displaystyle f_{uv}}
292:
201:
68:Kőnig–Egerváry theorem
5523:
5380:
5303:to the cut must have
5292:from the cut must be
5255:
5157:
4877:
4766:
4561:
4334:
4193:
3970:
3877:
3743:, such that the flow
3727:
3678:
3548:
3448:) guarantee that, if
3443:
3400:
3333:) guarantee that, if
3328:
3279:
3227:
3163:
3107:
3035:
2948:
2905: (the edge
2819:
2684:
2595:
2244:
2048:
1969:
1892:
1846:
1844:{\displaystyle z_{v}}
1816:
1776:
1741:
1701:
1490:
1450:, that is, determine
1420:
1400:
1374:
1348:
1309:
1142:
1021:
1019:{\displaystyle X_{C}}
932:
912:
892:
864:Maximum Flow Problem.
860:
849:
829:
806:
647:
517:
497:
477:
460:Conservation of Flows
452:
403:
358:
323:
293:
202:
5943:Network flow problem
5696:: 13. Archived from
5590:must have zero flow.
5469:
5326:
5170:
5081:
4781:
4617:
4554:Maximum flow problem
4416:maximum flow problem
4216:
4048:
3758:
3695:
3560:
3537:
3479:The equality in the
3464:is by definition in
3409:
3360:
3349:is by definition in
3288:
3239:
3200:
3116:
3054:
2960:
2840:
2694:
2618:
2265:
2075:
1986:
1914:
1854:
1828:
1784:
1755:
1709:
1680:
1559:) and the capacity (
1409:
1383:
1357:
1321:
1172:
1037:
1003:
921:
901:
871:
838:
818:
670:
527:
506:
486:
466:
412:
374:
332:
302:
261:
170:
5829:Linear Inequalities
5453:is fully saturated.
4570:pixels. Each pixel
3903:capacity of the cut
2831:dual linear program
890:{\displaystyle |f|}
368:Capacity Constraint
25:optimization theory
5651:Linear programming
5609:which obtains the
5518:
5462:, there exists an
5375:
5319:, there exists an
5250:
5152:
4979:Consider the flow
4872:
4855:
4761:
4744:
4679:
4650:
4564:
4329:
4306:
4261:
4188:
4165:
4120:
4075:
3973:
3872:
3722:
3673:
3659:
3624:
3585:
3543:
3438:
3395:
3323:
3274:
3222:
3158:
3102:
3030:
3025:
2943:
2938:
2814:
2812:
2679:
2677:
2590:
2588:
2239:
2237:
2181:
2155:
2043:
1964:
1887:
1841:
1811:
1771:
1736:
1696:
1660:Max-flow (Primal)
1545:
1415:
1395:
1369:
1343:
1304:
1137:
1016:
964:is a partition of
927:
907:
887:
844:
834:is the source and
824:
801:
642:
512:
492:
472:
462:: For each vertex
447:
398:
353:
318:
288:
197:
5902:Vijay V. Vazirani
5880:Kenneth Steiglitz
5782:Flows in Networks
5264:. Therefore, for
4804:
4693:
4664:
4635:
4516:
4515:
4399:requires machine
4284:
4239:
4143:
4098:
4066:
4004:and each machine
3941:and two vertices
3625:
3608:
3575:
3563:
3483:follows from the
3188:, then the edge (
3021:
2995:
2934:
2917:
2906:
2892:
2878:
2827:
2826:
2563:
2172:
2146:
1418:{\displaystyle 0}
1085:
1079:
930:{\displaystyle t}
910:{\displaystyle s}
847:{\displaystyle t}
827:{\displaystyle s}
515:{\displaystyle t}
495:{\displaystyle s}
475:{\displaystyle v}
370:: For every edge
31:states that in a
5950:
5919:
5897:
5871:
5845:
5838:
5832:
5822:
5816:
5813:
5807:
5800:
5794:
5771:
5765:
5764:
5744:
5738:
5737:
5729:
5723:
5722:
5720:
5714:Trevisan, Luca.
5711:
5705:
5704:
5702:
5694:RAND Corporation
5691:
5682:
5661:Menger's theorem
5589:
5577:
5570:
5566:
5562:
5555:
5551:
5543:
5527:
5525:
5524:
5519:
5517:
5516:
5461:
5452:
5440:
5433:
5429:
5425:
5418:
5414:
5406:
5384:
5382:
5381:
5376:
5374:
5373:
5318:
5283:
5263:
5259:
5257:
5256:
5251:
5240:
5239:
5215:
5214:
5161:
5159:
5158:
5153:
5151:
5150:
5138:
5137:
5065:
5040:
5036:
5031:
5024:
5020:
5009:
5005:
4990:
4986:
4975:
4971:
4967:
4963:
4921:
4917:
4913:
4906:
4902:
4898:
4891:
4881:
4879:
4878:
4873:
4868:
4867:
4854:
4797:
4770:
4768:
4767:
4762:
4757:
4756:
4743:
4689:
4688:
4678:
4660:
4659:
4649:
4609:
4605:
4598:
4594:
4587:
4580:
4573:
4569:
4539:
4530:
4461:
4443:
4424:
4413:
4409:
4405:
4398:
4387:
4369:
4355:
4338:
4336:
4335:
4330:
4322:
4321:
4305:
4298:
4297:
4277:
4276:
4260:
4253:
4252:
4232:
4208:
4204:
4197:
4195:
4194:
4189:
4181:
4180:
4164:
4157:
4156:
4136:
4135:
4119:
4112:
4111:
4091:
4090:
4074:
4040:
4032:
4024:
4010:
4003:
3989:
3982:
3978:
3957:Menger's theorem
3952:
3948:
3944:
3940:
3922:Menger's Theorem
3916:Menger's theorem
3881:
3879:
3878:
3873:
3853:
3852:
3840:
3839:
3750:
3742:
3731:
3729:
3728:
3723:
3721:
3720:
3715:
3682:
3680:
3679:
3674:
3672:
3671:
3658:
3657:
3656:
3623:
3622:
3621:
3601:
3600:
3584:
3577:
3576:
3573:
3552:
3550:
3549:
3544:
3532:
3523:
3456:, then the edge
3447:
3445:
3444:
3439:
3437:
3436:
3424:
3423:
3404:
3402:
3401:
3396:
3388:
3387:
3375:
3374:
3356:The constraints
3341:, then the edge
3332:
3330:
3329:
3324:
3322:
3321:
3303:
3302:
3283:
3281:
3280:
3275:
3267:
3266:
3254:
3253:
3235:The constraints
3231:
3229:
3228:
3223:
3215:
3214:
3167:
3165:
3164:
3159:
3157:
3156:
3144:
3143:
3131:
3130:
3111:
3109:
3108:
3103:
3095:
3094:
3082:
3081:
3069:
3068:
3050:The constraints
3039:
3037:
3036:
3031:
3029:
3028:
3022:
3019:
2996:
2993:
2972:
2971:
2952:
2950:
2949:
2944:
2942:
2941:
2935:
2932:
2918:
2915:
2907:
2904:
2893:
2890:
2879:
2876:
2855:
2854:
2823:
2821:
2820:
2815:
2813:
2778:
2776:
2764:
2763:
2725:
2713:
2712:
2688:
2686:
2685:
2680:
2678:
2649:
2637:
2636:
2611:sign constraints
2599:
2597:
2596:
2591:
2589:
2564:
2561:
2558:
2546:
2545:
2492:
2480:
2479:
2467:
2466:
2413:
2401:
2400:
2388:
2387:
2322:
2310:
2309:
2297:
2296:
2284:
2283:
2248:
2246:
2245:
2240:
2238:
2206:
2194:
2193:
2180:
2168:
2167:
2154:
2116:
2114:
2113:
2094:
2093:
2052:
2050:
2049:
2044:
2042:
2041:
2029:
2028:
2016:
2015:
1973:
1971:
1970:
1965:
1963:
1962:
1950:
1949:
1896:
1894:
1893:
1888:
1850:
1848:
1847:
1842:
1840:
1839:
1820:
1818:
1817:
1812:
1780:
1778:
1777:
1772:
1770:
1769:
1745:
1743:
1742:
1737:
1705:
1703:
1702:
1697:
1695:
1694:
1654:
1457:
1453:
1449:
1424:
1422:
1421:
1416:
1404:
1402:
1401:
1396:
1378:
1376:
1375:
1370:
1352:
1350:
1349:
1344:
1336:
1335:
1313:
1311:
1310:
1305:
1300:
1299:
1287:
1286:
1274:
1273:
1243:
1242:
1230:
1229:
1228:
1227:
1153:
1146:
1144:
1143:
1138:
1083:
1077:
1049:
1048:
1029:
1025:
1023:
1022:
1017:
1015:
1014:
987:
977:
967:
963:
936:
934:
933:
928:
916:
914:
913:
908:
896:
894:
893:
888:
886:
878:
853:
851:
850:
845:
833:
831:
830:
825:
810:
808:
807:
802:
797:
796:
784:
783:
741:
740:
728:
727:
685:
677:
651:
649:
648:
643:
638:
637:
625:
624:
582:
581:
569:
568:
521:
519:
518:
513:
501:
499:
498:
493:
481:
479:
478:
473:
456:
454:
453:
448:
443:
442:
427:
426:
407:
405:
404:
399:
362:
360:
359:
354:
327:
325:
324:
319:
317:
316:
297:
295:
294:
289:
287:
286:
281:
243:
228:
213:
206:
204:
203:
198:
196:
195:
190:
158:
145:
129:
111:
64:Menger's theorem
21:computer science
5960:
5959:
5953:
5952:
5951:
5949:
5948:
5947:
5923:
5922:
5916:
5894:
5868:
5849:
5848:
5839:
5835:
5823:
5819:
5814:
5810:
5801:
5797:
5778:D. R. Fulkerson
5772:
5768:
5745:
5741:
5732:Keller, Orgad.
5730:
5726:
5718:
5712:
5708:
5700:
5689:
5683:
5679:
5674:
5646:GNRS conjecture
5622:
5598:, so it is the
5579:
5576:
5572:
5568:
5564:
5561:
5557:
5553:
5549:
5529:
5512:
5508:
5470:
5467:
5466:
5459:
5442:
5439:
5435:
5431:
5427:
5424:
5420:
5416:
5412:
5404:
5386:
5369:
5365:
5327:
5324:
5323:
5316:
5294:fully saturated
5265:
5261:
5232:
5228:
5204:
5200:
5171:
5168:
5167:
5143:
5139:
5109:
5105:
5082:
5079:
5078:
5074:is defined by
5047:
5038:
5034:
5030:
5026:
5022:
5018:
5007:
5002:
4996:
4988:
4980:
4973:
4969:
4965:
4950:
4947:
4928:
4919:
4915:
4912:
4908:
4904:
4900:
4897:
4893:
4890:
4886:
4860:
4856:
4808:
4790:
4782:
4779:
4778:
4749:
4745:
4697:
4684:
4680:
4668:
4655:
4651:
4639:
4618:
4615:
4614:
4607:
4603:
4596:
4593:
4589:
4586:
4582:
4579:
4575:
4571:
4567:
4556:
4550:
4538:
4532:
4529:
4523:
4458:
4449:
4440:
4431:
4411:
4407:
4404:
4400:
4397:
4393:
4384:
4377:
4371:
4366:
4357:
4352:
4343:
4317:
4313:
4293:
4289:
4288:
4272:
4268:
4248:
4244:
4243:
4225:
4217:
4214:
4213:
4206:
4202:
4176:
4172:
4152:
4148:
4147:
4131:
4127:
4107:
4103:
4102:
4086:
4082:
4070:
4049:
4046:
4045:
4038:
4030:
4021:
4012:
4009:
4005:
4000:
3991:
3990:yields revenue
3988:
3984:
3980:
3976:
3965:
3950:
3946:
3942:
3927:
3924:
3918:
3845:
3841:
3799:
3795:
3759:
3756:
3755:
3744:
3733:
3716:
3711:
3710:
3696:
3693:
3692:
3689:
3664:
3660:
3652:
3648:
3629:
3617:
3613:
3612:
3593:
3589:
3572:
3568:
3567:
3561:
3558:
3557:
3538:
3535:
3534:
3531:
3525:
3522:
3515:
3512:
3506:
3501:
3432:
3428:
3416:
3412:
3410:
3407:
3406:
3405:(equivalent to
3383:
3379:
3367:
3363:
3361:
3358:
3357:
3317:
3313:
3295:
3291:
3289:
3286:
3285:
3284:(equivalent to
3262:
3258:
3246:
3242:
3240:
3237:
3236:
3207:
3203:
3201:
3198:
3197:
3152:
3148:
3139:
3135:
3123:
3119:
3117:
3114:
3113:
3112:(equivalent to
3090:
3086:
3077:
3073:
3061:
3057:
3055:
3052:
3051:
3024:
3023:
3018:
3016:
3007:
3006:
2992:
2990:
2977:
2976:
2967:
2963:
2961:
2958:
2957:
2937:
2936:
2931:
2929:
2920:
2919:
2914:
2903:
2891: and
2889:
2875:
2873:
2860:
2859:
2847:
2843:
2841:
2838:
2837:
2811:
2810:
2777:
2772:
2765:
2759:
2755:
2752:
2751:
2724:
2714:
2705:
2701:
2697:
2695:
2692:
2691:
2676:
2675:
2648:
2638:
2629:
2625:
2621:
2619:
2616:
2615:
2587:
2586:
2560:
2557:
2547:
2538:
2534:
2531:
2530:
2491:
2481:
2475:
2471:
2459:
2455:
2452:
2451:
2412:
2402:
2396:
2392:
2380:
2376:
2373:
2372:
2321:
2311:
2305:
2301:
2292:
2288:
2276:
2272:
2268:
2266:
2263:
2262:
2236:
2235:
2205:
2195:
2186:
2182:
2176:
2160:
2156:
2150:
2143:
2142:
2115:
2106:
2102:
2095:
2086:
2082:
2078:
2076:
2073:
2072:
2034:
2030:
2021:
2017:
1993:
1989:
1987:
1984:
1983:
1955:
1951:
1921:
1917:
1915:
1912:
1911:
1855:
1852:
1851:
1835:
1831:
1829:
1826:
1825:
1785:
1782:
1781:
1762:
1758:
1756:
1753:
1752:
1710:
1707:
1706:
1687:
1683:
1681:
1678:
1677:
1665:Min-cut (Dual)
1649:linear programs
1642:
1485:
1465:
1455:
1451:
1436:
1410:
1407:
1406:
1384:
1381:
1380:
1358:
1355:
1354:
1328:
1324:
1322:
1319:
1318:
1292:
1288:
1279:
1275:
1251:
1247:
1235:
1231:
1223:
1219:
1200:
1196:
1173:
1170:
1169:
1151:
1044:
1040:
1038:
1035:
1034:
1027:
1010:
1006:
1004:
1001:
1000:
979:
969:
965:
950:
944:
922:
919:
918:
902:
899:
898:
882:
874:
872:
869:
868:
839:
836:
835:
819:
816:
815:
814:where as above
789:
785:
749:
745:
733:
729:
693:
689:
681:
673:
671:
668:
667:
630:
626:
590:
586:
574:
570:
534:
530:
528:
525:
524:
523:
507:
504:
503:
487:
484:
483:
467:
464:
463:
435:
431:
419:
415:
413:
410:
409:
375:
372:
371:
333:
330:
329:
309:
305:
303:
300:
299:
282:
277:
276:
262:
259:
258:
251:
230:
215:
212:
208:
191:
186:
185:
171:
168:
167:
150:
137:
117:
98:
84:
76:
60:linear programs
56:duality theorem
17:
12:
11:
5:
5958:
5957:
5946:
5945:
5940:
5935:
5921:
5920:
5914:
5898:
5892:
5872:
5866:
5847:
5846:
5833:
5825:George Dantzig
5817:
5808:
5795:
5774:L. R. Ford Jr.
5766:
5755:(2): 130–141.
5739:
5724:
5706:
5703:on 5 May 2018.
5676:
5675:
5673:
5670:
5669:
5668:
5663:
5658:
5653:
5648:
5643:
5638:
5633:
5628:
5621:
5618:
5592:
5591:
5574:
5559:
5515:
5511:
5507:
5504:
5501:
5498:
5495:
5492:
5489:
5486:
5483:
5480:
5477:
5474:
5455:
5454:
5437:
5422:
5402:
5372:
5368:
5364:
5361:
5358:
5355:
5352:
5349:
5346:
5343:
5340:
5337:
5334:
5331:
5309:
5308:
5301:incoming edges
5297:
5290:outgoing edges
5249:
5246:
5243:
5238:
5235:
5231:
5227:
5224:
5221:
5218:
5213:
5210:
5207:
5203:
5199:
5196:
5193:
5190:
5187:
5184:
5181:
5178:
5175:
5166:Now, we know,
5164:
5163:
5149:
5146:
5142:
5136:
5133:
5130:
5127:
5124:
5121:
5118:
5115:
5112:
5108:
5104:
5101:
5098:
5095:
5092:
5089:
5086:
5042:
5041:
5032:
5028:
5000:
4976:respectively.
4946:
4943:
4927:
4924:
4910:
4895:
4888:
4883:
4882:
4871:
4866:
4863:
4859:
4853:
4850:
4847:
4844:
4841:
4838:
4835:
4832:
4829:
4826:
4823:
4820:
4817:
4814:
4811:
4807:
4803:
4800:
4796:
4793:
4789:
4786:
4772:
4771:
4760:
4755:
4752:
4748:
4742:
4739:
4736:
4733:
4730:
4727:
4724:
4721:
4718:
4715:
4712:
4709:
4706:
4703:
4700:
4696:
4692:
4687:
4683:
4677:
4674:
4671:
4667:
4663:
4658:
4654:
4648:
4645:
4642:
4638:
4634:
4631:
4628:
4625:
4622:
4591:
4584:
4577:
4549:
4546:
4536:
4527:
4514:
4513:
4508:
4505:
4502:
4498:
4497:
4492:
4489:
4486:
4482:
4481:
4476:
4473:
4470:
4466:
4465:
4463:
4456:
4445:
4438:
4427:
4402:
4395:
4382:
4375:
4364:
4350:
4340:
4339:
4328:
4325:
4320:
4316:
4312:
4309:
4304:
4301:
4296:
4292:
4287:
4283:
4280:
4275:
4271:
4267:
4264:
4259:
4256:
4251:
4247:
4242:
4238:
4235:
4231:
4228:
4224:
4221:
4199:
4198:
4187:
4184:
4179:
4175:
4171:
4168:
4163:
4160:
4155:
4151:
4146:
4142:
4139:
4134:
4130:
4126:
4123:
4118:
4115:
4110:
4106:
4101:
4097:
4094:
4089:
4085:
4081:
4078:
4073:
4069:
4065:
4062:
4059:
4056:
4053:
4019:
4007:
3998:
3986:
3964:
3961:
3917:
3914:
3883:
3882:
3871:
3868:
3865:
3862:
3859:
3856:
3851:
3848:
3844:
3838:
3835:
3832:
3829:
3826:
3823:
3820:
3817:
3814:
3811:
3808:
3805:
3802:
3798:
3793:
3790:
3787:
3784:
3781:
3778:
3775:
3772:
3769:
3766:
3763:
3719:
3714:
3709:
3706:
3703:
3700:
3688:
3685:
3684:
3683:
3670:
3667:
3663:
3655:
3651:
3647:
3644:
3641:
3638:
3635:
3632:
3628:
3620:
3616:
3611:
3607:
3604:
3599:
3596:
3592:
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3580:
3571:
3566:
3542:
3529:
3520:
3505:
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3500:
3497:
3470:
3469:
3435:
3431:
3427:
3422:
3419:
3415:
3394:
3391:
3386:
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3366:
3354:
3320:
3316:
3312:
3309:
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3270:
3265:
3261:
3257:
3252:
3249:
3245:
3233:
3221:
3218:
3213:
3210:
3206:
3155:
3151:
3147:
3142:
3138:
3134:
3129:
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3122:
3101:
3098:
3093:
3089:
3085:
3080:
3076:
3072:
3067:
3064:
3060:
3041:
3040:
3027:
3017:
3015:
3012:
3009:
3008:
3005:
3002:
2999:
2991:
2989:
2986:
2983:
2982:
2980:
2975:
2970:
2966:
2954:
2953:
2940:
2930:
2928:
2925:
2922:
2921:
2913:
2910:
2902:
2899:
2896:
2888:
2885:
2882:
2874:
2872:
2869:
2866:
2865:
2863:
2858:
2853:
2850:
2846:
2825:
2824:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2775:
2771:
2768:
2766:
2762:
2758:
2754:
2753:
2750:
2747:
2744:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2720:
2717:
2715:
2711:
2708:
2704:
2700:
2699:
2689:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2639:
2635:
2632:
2628:
2624:
2623:
2613:
2607:
2606:
2601:
2600:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2559:
2556:
2553:
2550:
2548:
2544:
2541:
2537:
2533:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2482:
2478:
2474:
2470:
2465:
2462:
2458:
2454:
2453:
2450:
2447:
2444:
2441:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2405:
2403:
2399:
2395:
2391:
2386:
2383:
2379:
2375:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2312:
2308:
2304:
2300:
2295:
2291:
2287:
2282:
2279:
2275:
2271:
2270:
2255:
2250:
2249:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2196:
2192:
2189:
2185:
2179:
2175:
2171:
2166:
2163:
2159:
2153:
2149:
2145:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2112:
2109:
2105:
2101:
2098:
2096:
2092:
2089:
2085:
2081:
2080:
2065:
2059:
2058:
2040:
2037:
2033:
2027:
2024:
2020:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1992:
1979:
1961:
1958:
1954:
1948:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1920:
1907:
1901:
1900:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1838:
1834:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1768:
1765:
1761:
1749:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1714:
1693:
1690:
1686:
1674:
1668:
1667:
1662:
1657:
1641:
1638:
1484:
1481:
1480:
1479:
1464:
1461:
1460:
1459:
1414:
1394:
1391:
1388:
1368:
1365:
1362:
1342:
1339:
1334:
1331:
1327:
1315:
1314:
1303:
1298:
1295:
1291:
1285:
1282:
1278:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1250:
1246:
1241:
1238:
1234:
1226:
1222:
1218:
1215:
1212:
1209:
1206:
1203:
1199:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1148:
1147:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1082:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1047:
1043:
1013:
1009:
988:. That is, an
943:
940:
926:
906:
885:
881:
877:
843:
823:
812:
811:
800:
795:
792:
788:
782:
779:
776:
773:
770:
767:
764:
761:
758:
755:
752:
748:
744:
739:
736:
732:
726:
723:
720:
717:
714:
711:
708:
705:
702:
699:
696:
692:
688:
684:
680:
676:
653:
652:
641:
636:
633:
629:
623:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
589:
585:
580:
577:
573:
567:
564:
561:
558:
555:
552:
549:
546:
543:
540:
537:
533:
511:
491:
471:
457:
446:
441:
438:
434:
430:
425:
422:
418:
397:
394:
391:
388:
385:
382:
379:
352:
349:
346:
343:
340:
337:
315:
312:
308:
285:
280:
275:
272:
269:
266:
250:
247:
246:
245:
210:
194:
189:
184:
181:
178:
175:
160:
131:
96:directed graph
83:
80:
75:
72:
15:
9:
6:
4:
3:
2:
5956:
5955:
5944:
5941:
5939:
5936:
5934:
5931:
5930:
5928:
5917:
5915:3-540-65367-8
5911:
5907:
5903:
5899:
5895:
5893:0-486-40258-4
5889:
5885:
5881:
5877:
5873:
5869:
5867:0-486-41453-1
5863:
5859:
5855:
5854:Eugene Lawler
5851:
5850:
5843:
5837:
5830:
5826:
5821:
5812:
5805:
5799:
5793:
5790:
5787:
5783:
5779:
5775:
5770:
5762:
5758:
5754:
5750:
5743:
5735:
5728:
5717:
5710:
5699:
5695:
5688:
5681:
5677:
5667:
5664:
5662:
5659:
5657:
5654:
5652:
5649:
5647:
5644:
5642:
5639:
5637:
5634:
5632:
5629:
5627:
5624:
5623:
5617:
5614:
5612:
5608:
5603:
5601:
5597:
5587:
5583:
5547:
5546:backward edge
5541:
5537:
5533:
5513:
5509:
5505:
5502:
5499:
5496:
5493:
5490:
5487:
5481:
5478:
5475:
5465:
5464:incoming edge
5457:
5456:
5450:
5446:
5410:
5405:
5398:
5394:
5390:
5370:
5366:
5362:
5359:
5356:
5353:
5350:
5347:
5344:
5338:
5335:
5332:
5322:
5321:outgoing edge
5314:
5313:
5312:
5306:
5302:
5298:
5295:
5291:
5287:
5286:
5285:
5281:
5277:
5273:
5269:
5266:value(
5244:
5236:
5233:
5229:
5225:
5219:
5211:
5208:
5205:
5201:
5197:
5191:
5185:
5182:
5179:
5176:
5173:
5147:
5144:
5140:
5134:
5131:
5128:
5125:
5119:
5116:
5113:
5102:
5096:
5093:
5090:
5084:
5077:
5076:
5075:
5073:
5069:
5063:
5059:
5055:
5051:
5048:value(
5046:
5033:
5017:
5016:
5015:
5013:
5006:obtained for
5003:
4994:
4987:computed for
4984:
4977:
4961:
4957:
4953:
4942:
4939:
4937:
4933:
4923:
4899:. Two edges (
4869:
4864:
4861:
4857:
4851:
4848:
4845:
4842:
4839:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4809:
4805:
4801:
4794:
4791:
4777:
4776:
4775:
4758:
4753:
4750:
4746:
4740:
4737:
4734:
4731:
4728:
4725:
4722:
4719:
4716:
4713:
4710:
4707:
4704:
4701:
4698:
4694:
4690:
4685:
4681:
4675:
4672:
4669:
4665:
4661:
4656:
4652:
4646:
4643:
4640:
4636:
4632:
4626:
4613:
4612:
4611:
4600:
4560:
4555:
4545:
4541:
4535:
4526:
4521:
4512:
4506:
4503:
4500:
4499:
4496:
4490:
4487:
4484:
4483:
4480:
4474:
4471:
4468:
4467:
4464:
4462:
4459:
4452:
4444:
4441:
4434:
4425:
4422:
4419:
4417:
4391:
4385:
4378:
4367:
4360:
4353:
4346:
4326:
4318:
4314:
4307:
4302:
4299:
4294:
4290:
4285:
4281:
4273:
4269:
4262:
4257:
4254:
4249:
4245:
4240:
4236:
4229:
4226:
4212:
4211:
4210:
4185:
4177:
4173:
4166:
4161:
4158:
4153:
4149:
4144:
4140:
4132:
4128:
4121:
4116:
4113:
4108:
4104:
4099:
4095:
4087:
4083:
4076:
4071:
4067:
4063:
4057:
4044:
4043:
4042:
4037:selected and
4036:
4027:
4022:
4015:
4001:
3994:
3979:projects and
3969:
3960:
3958:
3954:
3938:
3934:
3930:
3923:
3913:
3911:
3906:
3904:
3900:
3896:
3892:
3888:
3869:
3863:
3857:
3854:
3849:
3846:
3842:
3833:
3830:
3824:
3821:
3818:
3812:
3809:
3806:
3803:
3791:
3785:
3782:
3779:
3770:
3767:
3764:
3754:
3753:
3752:
3748:
3740:
3736:
3717:
3704:
3701:
3698:
3668:
3665:
3661:
3653:
3649:
3645:
3639:
3636:
3633:
3626:
3618:
3614:
3605:
3597:
3594:
3590:
3569:
3556:
3555:
3554:
3528:
3519:
3511:
3496:
3494:
3490:
3486:
3482:
3477:
3475:
3467:
3463:
3459:
3455:
3451:
3433:
3429:
3425:
3420:
3417:
3413:
3392:
3389:
3384:
3380:
3376:
3371:
3368:
3364:
3355:
3352:
3348:
3344:
3340:
3336:
3318:
3314:
3310:
3307:
3304:
3299:
3296:
3292:
3271:
3268:
3263:
3259:
3255:
3250:
3247:
3243:
3234:
3219:
3216:
3211:
3208:
3204:
3195:
3191:
3187:
3183:
3179:
3175:
3171:
3153:
3149:
3145:
3140:
3136:
3132:
3127:
3124:
3120:
3099:
3096:
3091:
3087:
3083:
3078:
3074:
3070:
3065:
3062:
3058:
3049:
3048:
3047:
3044:
3013:
3010:
3003:
3000:
2997:
2987:
2984:
2978:
2973:
2968:
2964:
2956:
2955:
2926:
2923:
2911:
2908:
2900:
2897:
2894:
2886:
2883:
2880:
2870:
2867:
2861:
2856:
2851:
2848:
2844:
2836:
2835:
2834:
2832:
2804:
2801:
2798:
2789:
2786:
2783:
2769:
2767:
2760:
2756:
2748:
2745:
2739:
2736:
2733:
2721:
2718:
2716:
2709:
2706:
2702:
2690:
2672:
2669:
2663:
2660:
2657:
2645:
2642:
2640:
2633:
2630:
2626:
2614:
2612:
2609:
2608:
2605:
2604:
2583:
2580:
2574:
2571:
2568:
2554:
2551:
2549:
2542:
2539:
2535:
2527:
2524:
2521:
2518:
2515:
2512:
2506:
2503:
2500:
2488:
2485:
2483:
2476:
2472:
2468:
2463:
2460:
2456:
2448:
2445:
2442:
2439:
2436:
2433:
2427:
2424:
2421:
2409:
2406:
2404:
2397:
2393:
2389:
2384:
2381:
2377:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2336:
2333:
2330:
2318:
2315:
2313:
2306:
2302:
2298:
2293:
2289:
2285:
2280:
2277:
2273:
2261:
2260:
2259:
2254:
2253:
2229:
2226:
2223:
2214:
2211:
2208:
2202:
2199:
2197:
2190:
2187:
2183:
2177:
2173:
2169:
2164:
2161:
2157:
2151:
2147:
2139:
2136:
2130:
2127:
2124:
2110:
2107:
2103:
2099:
2097:
2090:
2087:
2083:
2071:
2070:
2069:
2064:
2061:
2060:
2057:
2056:
2053:
2038:
2035:
2031:
2025:
2022:
2018:
2012:
2009:
2003:
2000:
1997:
1978:
1977:
1974:
1959:
1956:
1952:
1946:
1943:
1937:
1934:
1931:
1925:
1922:
1906:
1903:
1902:
1899:
1898:
1881:
1878:
1875:
1866:
1863:
1860:
1836:
1832:
1823:
1822:
1808:
1805:
1799:
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1793:
1766:
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1759:
1748:
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1733:
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1724:
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1718:
1691:
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1673:
1670:
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1573:
1569:
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1558:
1554:
1550:
1542:
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1530:
1526:
1522:
1518:
1514:
1510:
1506:
1502:
1498:
1494:
1489:
1477:
1474:
1473:
1472:
1470:
1447:
1443:
1439:
1434:
1431:
1430:
1429:
1426:
1412:
1392:
1389:
1386:
1366:
1363:
1360:
1340:
1337:
1332:
1329:
1325:
1301:
1296:
1293:
1289:
1283:
1280:
1276:
1270:
1267:
1261:
1258:
1255:
1244:
1239:
1236:
1232:
1224:
1220:
1216:
1210:
1207:
1204:
1193:
1187:
1184:
1181:
1175:
1168:
1167:
1166:
1164:
1160:
1155:
1134:
1131:
1128:
1122:
1119:
1116:
1110:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1080:
1074:
1071:
1065:
1062:
1059:
1050:
1045:
1041:
1033:
1032:
1031:
1011:
1007:
999:
995:
991:
986:
982:
976:
972:
961:
957:
953:
949:
938:
924:
904:
879:
866:
865:
859:
856:
841:
821:
798:
793:
790:
786:
777:
774:
768:
765:
762:
756:
753:
742:
737:
734:
730:
721:
718:
712:
709:
706:
700:
697:
686:
678:
666:
665:
664:
662:
657:
639:
634:
631:
627:
618:
615:
609:
606:
603:
597:
594:
583:
578:
575:
571:
562:
559:
553:
550:
547:
541:
538:
509:
489:
469:
461:
458:
444:
439:
436:
432:
428:
423:
420:
416:
395:
392:
386:
383:
380:
369:
366:
365:
364:
347:
344:
341:
335:
313:
310:
306:
283:
270:
267:
264:
256:
242:
238:
234:
226:
222:
218:
192:
179:
176:
173:
165:
161:
157:
153:
149:
144:
140:
136:
132:
128:
124:
120:
115:
109:
105:
101:
97:
93:
92:
91:
90:consists of
89:
79:
71:
69:
65:
61:
57:
52:
50:
46:
45:
40:
39:
34:
30:
26:
22:
5905:
5883:
5857:
5841:
5836:
5828:
5820:
5811:
5798:
5781:
5769:
5752:
5748:
5742:
5727:
5709:
5698:the original
5693:
5680:
5656:Maximum flow
5636:Flow network
5615:
5604:
5593:
5585:
5581:
5545:
5539:
5535:
5531:
5463:
5448:
5444:
5409:forward edge
5408:
5400:
5396:
5392:
5388:
5320:
5310:
5304:
5300:
5293:
5289:
5279:
5275:
5271:
5267:
5165:
5071:
5067:
5066:, where the
5061:
5057:
5053:
5049:
5044:
5043:
4998:
4982:
4978:
4959:
4955:
4951:
4948:
4940:
4929:
4884:
4773:
4601:
4565:
4542:
4533:
4524:
4519:
4517:
4510:
4494:
4478:
4454:
4450:
4447:
4436:
4432:
4429:
4420:
4389:
4380:
4373:
4362:
4358:
4348:
4344:
4341:
4200:
4034:
4028:
4017:
4013:
3996:
3992:
3974:
3956:
3955:
3936:
3932:
3928:
3925:
3909:
3907:
3902:
3898:
3894:
3890:
3886:
3884:
3746:
3738:
3734:
3690:
3526:
3517:
3513:
3492:
3488:
3480:
3478:
3473:
3471:
3465:
3461:
3457:
3453:
3449:
3350:
3346:
3342:
3338:
3334:
3193:
3189:
3185:
3181:
3177:
3173:
3169:
3045:
3042:
2828:
2610:
2603:
2602:
2257:
2252:
2251:
2068:subject to
2067:
2062:
2055:
2054:
1981:
1976:
1975:
1909:
1904:
1897:
1824:
1821:
1751:
1746:
1676:
1671:
1664:
1659:
1643:
1633:
1629:
1627:
1624:
1619:
1615:
1611:
1607:
1603:
1599:
1595:
1591:
1587:
1583:
1579:
1575:
1571:
1567:
1565:
1560:
1556:
1552:
1548:
1546:
1540:
1536:
1532:
1528:
1524:
1520:
1516:
1512:
1508:
1504:
1500:
1496:
1492:
1475:
1468:
1466:
1463:Main theorem
1445:
1441:
1437:
1432:
1427:
1316:
1162:
1158:
1156:
1149:
997:
993:
989:
984:
980:
974:
970:
959:
955:
951:
947:
945:
862:
861:
857:
813:
660:
658:
654:
459:
367:
254:
252:
240:
236:
232:
224:
220:
216:
163:
155:
151:
147:
142:
138:
134:
126:
122:
118:
113:
107:
103:
99:
87:
85:
77:
53:
43:
37:
33:flow network
28:
18:
5666:Minimum cut
5270: ) =
5052: ) =
3732:denoted by
3533:approaches
3499:Application
2258:subject to
2063:constraints
1646:primal-dual
1425:otherwise.
482:apart from
298:denoted by
207:denoted by
49:minimum cut
5927:Categories
5806:8: 399–404
5784:, page 1,
5672:References
4595:if pixels
4552:See also:
4370:. An edge
3920:See also:
3508:See also:
968:such that
5844:9: 210–18
5506:∈
5494:∈
5363:∈
5351:∈
5305:zero flow
5284:we need:
5226:−
5132:×
5126:∈
5107:∑
4938:in 1962:
4936:Fulkerson
4849:∈
4837:∈
4831:∨
4825:∈
4813:∈
4806:∑
4738:∈
4726:∈
4720:∨
4714:∈
4702:∈
4695:∑
4691:−
4673:∈
4666:∑
4644:∈
4637:∑
4300:∈
4286:∑
4255:∈
4241:∑
4159:∈
4145:∑
4141:−
4114:∈
4100:∑
4096:−
4068:∑
3855:≤
3831:∈
3813:∣
3807:∈
3797:∑
3774:∖
3768:∈
3762:∀
3708:→
3646:∈
3627:∑
3582:∞
3579:→
3541:∞
3426:≥
3390:≥
3377:−
3311:−
3305:≥
3269:≥
3217:≥
3146:−
3133:≥
3097:≥
3071:−
3020:otherwise
3001:∈
2933:otherwise
2898:∈
2884:∈
2793:∖
2787:∈
2781:∀
2770:∈
2746:∈
2728:∀
2719:≥
2670:∈
2652:∀
2643:≥
2581:∈
2552:≥
2525:≠
2513:∈
2495:∀
2486:≥
2469:−
2446:≠
2434:∈
2416:∀
2407:≥
2367:≠
2355:≠
2343:∈
2325:∀
2316:≥
2286:−
2218:∖
2212:∈
2174:∑
2170:−
2148:∑
2137:∈
2119:∀
2100:≤
2010:∈
1991:∑
1982:minimize
1944:∈
1919:∑
1910:maximize
1905:objective
1870:∖
1864:∈
1858:∀
1806:∈
1788:∀
1731:∈
1713:∀
1672:variables
1435:Minimize
1390:∈
1364:∈
1268:∈
1249:∑
1217:∈
1198:∑
1129:∩
1120:×
1102:∈
1090:∈
1072:∈
1026:of a cut
867:Maximize
775:∈
747:∑
719:∈
691:∑
616:∈
588:∑
560:∈
532:∑
429:≤
393:∈
274:→
183:→
94:a finite
5620:See also
5611:max-flow
5600:max-flow
5542:) > 0
5534: (
5391: (
5068:capacity
5004: )
4887: f
4795:′
4576: f
4448:Machine
4430:Project
4390:infinite
4230:′
2994:if
2877:if
2562:if
1618:back to
1495:, where
1159:capacity
164:capacity
112:, where
66:and the
5792:0159700
5780:(1962)
5607:min-cut
5530:
5399:) <
5387:
5072:s-t cut
4985:
4981:
4926:History
4907:) with
4903:) and (
3891:s-t cut
3749:
3745:
3516:
1483:Example
1163:s-t cut
998:cut-set
948:s-t cut
88:network
82:Network
41:to the
5912:
5890:
5864:
5776:&
5070:of an
5045:Claim.
4011:costs
3452:is in
3337:is in
3184:is in
3176:is in
1586:} and
1523:} and
1317:where
1161:of an
1084:
1078:
146:and a
135:source
38:source
27:, the
5719:(PDF)
5701:(PDF)
5690:(PDF)
5548:from
5411:from
5039:V − A
4945:Proof
4388:with
3458:(u,t)
3343:(s,v)
3172:, if
661:value
249:Flows
5910:ISBN
5888:ISBN
5862:ISBN
5299:All
5288:All
4968:and
4949:Let
4934:and
4932:Ford
4905:j, i
4901:i, j
4602:Let
4597:i, j
4531:and
4491:100
4475:200
4410:and
4205:and
4029:Let
3945:and
3887:flow
3180:and
1622:.)
1566:One
1454:and
1379:and
1157:The
978:and
942:Cuts
659:The
502:and
255:flow
239:) ∈
229:for
148:sink
58:for
44:sink
23:and
5757:doi
5753:274
5571:in
5567:to
5556:in
5552:to
5458:In
5434:in
5430:to
5419:in
5415:to
5315:In
5025:in
4991:by
4954:= (
4785:min
4621:max
4507:50
4504:150
4488:200
4472:100
4220:min
4052:max
4035:not
3931:= (
3897:to
3610:min
3565:lim
3474:not
3170:u,v
1632:to
1610:to
1493:f/c
1469:s-t
1353:if
954:= (
917:to
328:or
214:or
102:= (
19:In
5929::
5878:,
5789:MR
5751:.
5692:.
5613:.
5584:,
5538:,
5447:,
5403:xy
5395:,
5278:,
5060:,
4958:,
4922:.
4911:ij
4592:ij
4540:.
4501:3
4485:2
4469:1
4418:.
4379:,
3953:.
3935:,
3574:in
3530:in
3521:in
3468:).
3353:).
3232:).
1651:.
1602:,
1598:,
1594:,
1590:={
1578:={
1543:}.
1539:,
1535:,
1531:,
1527:={
1515:={
1444:,
1405:,
1051::=
983:∈
973:∈
958:,
937:.
408:,
253:A
223:,
211:uv
162:a
154:∈
141:∈
133:a
121:⊆
106:,
86:A
70:.
5918:.
5896:.
5870:.
5763:.
5759::
5736:.
5721:.
5588:)
5586:x
5582:y
5580:(
5575:f
5573:G
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5558:G
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5540:x
5536:y
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5510:A
5503:y
5500:,
5497:A
5491:x
5488:,
5485:)
5482:x
5479:,
5476:y
5473:(
5460:G
5451:)
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5445:x
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5438:f
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5432:y
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5371:c
5367:A
5360:y
5357:,
5354:A
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5339:y
5336:,
5333:x
5330:(
5317:G
5307:.
5296:.
5282:)
5280:A
5276:A
5274:(
5272:c
5268:f
5262:A
5248:)
5245:A
5242:(
5237:n
5234:i
5230:f
5223:)
5220:A
5217:(
5212:t
5209:u
5206:o
5202:f
5198:=
5195:)
5192:f
5189:(
5186:e
5183:u
5180:l
5177:a
5174:v
5162:.
5148:v
5145:u
5141:c
5135:T
5129:S
5123:)
5120:v
5117:,
5114:u
5111:(
5103:=
5100:)
5097:T
5094:,
5091:S
5088:(
5085:c
5064:)
5062:A
5058:A
5056:(
5054:c
5050:f
5035:A
5029:f
5027:G
5023:s
5019:A
5008:G
5001:f
4999:G
4997:(
4989:G
4983:f
4974:G
4970:t
4966:s
4962:)
4960:E
4956:V
4952:G
4920:Q
4916:P
4909:p
4896:i
4894:b
4889:i
4870:.
4865:j
4862:i
4858:p
4852:Q
4846:i
4843:,
4840:P
4834:j
4828:Q
4822:j
4819:,
4816:P
4810:i
4802:=
4799:}
4792:g
4788:{
4759:.
4754:j
4751:i
4747:p
4741:Q
4735:i
4732:,
4729:P
4723:j
4717:Q
4711:j
4708:,
4705:P
4699:i
4686:i
4682:b
4676:Q
4670:i
4662:+
4657:i
4653:f
4647:P
4641:i
4633:=
4630:}
4627:g
4624:{
4608:Q
4604:P
4590:p
4585:i
4583:b
4578:i
4572:i
4568:n
4537:3
4534:p
4528:2
4525:p
4520:g
4460:)
4457:j
4455:q
4453:(
4451:c
4442:)
4439:i
4437:p
4435:(
4433:r
4412:Q
4408:P
4403:j
4401:q
4396:i
4394:p
4386:)
4383:j
4381:q
4376:i
4374:p
4372:(
4368:)
4365:j
4363:q
4361:(
4359:c
4354:)
4351:i
4349:p
4347:(
4345:r
4327:.
4324:)
4319:j
4315:q
4311:(
4308:c
4303:Q
4295:j
4291:q
4282:+
4279:)
4274:i
4270:p
4266:(
4263:r
4258:P
4250:i
4246:p
4237:=
4234:}
4227:g
4223:{
4207:Q
4203:P
4186:.
4183:)
4178:j
4174:q
4170:(
4167:c
4162:Q
4154:j
4150:q
4138:)
4133:i
4129:p
4125:(
4122:r
4117:P
4109:i
4105:p
4093:)
4088:i
4084:p
4080:(
4077:r
4072:i
4064:=
4061:}
4058:g
4055:{
4039:Q
4031:P
4023:)
4020:j
4018:q
4016:(
4014:c
4008:j
4006:q
4002:)
3999:i
3997:p
3995:(
3993:r
3987:i
3985:p
3981:m
3977:n
3951:G
3947:t
3943:s
3939:)
3937:E
3933:V
3929:G
3899:t
3895:s
3870:.
3867:)
3864:v
3861:(
3858:c
3850:v
3847:u
3843:f
3837:}
3834:E
3828:)
3825:v
3822:,
3819:u
3816:(
3810:V
3804:u
3801:{
3792::
3789:}
3786:t
3783:,
3780:s
3777:{
3771:V
3765:v
3747:f
3741:)
3739:v
3737:(
3735:c
3718:+
3713:R
3705:V
3702::
3699:c
3669:v
3666:u
3662:c
3654:C
3650:X
3643:)
3640:v
3637:,
3634:u
3631:(
3619:C
3615:X
3606:=
3603:)
3598:n
3595:i
3591:I
3587:(
3570:V
3527:V
3518:I
3493:y
3489:x
3466:T
3462:t
3454:S
3450:u
3434:u
3430:z
3421:t
3418:u
3414:d
3393:0
3385:u
3381:z
3372:t
3369:u
3365:d
3351:S
3347:s
3339:T
3335:v
3319:v
3315:z
3308:1
3300:v
3297:s
3293:d
3272:1
3264:v
3260:z
3256:+
3251:v
3248:s
3244:d
3220:1
3212:v
3209:u
3205:d
3194:v
3192:,
3190:u
3186:T
3182:v
3178:S
3174:u
3154:v
3150:z
3141:u
3137:z
3128:v
3125:u
3121:d
3100:0
3092:v
3088:z
3084:+
3079:u
3075:z
3066:v
3063:u
3059:d
3014:,
3011:0
3004:S
2998:v
2988:,
2985:1
2979:{
2974:=
2969:v
2965:z
2927:,
2924:0
2912:v
2909:u
2901:T
2895:v
2887:S
2881:u
2871:,
2868:1
2862:{
2857:=
2852:v
2849:u
2845:d
2808:}
2805:t
2802:,
2799:s
2796:{
2790:V
2784:v
2774:R
2761:v
2757:z
2749:E
2743:)
2740:v
2737:,
2734:u
2731:(
2722:0
2710:v
2707:u
2703:d
2673:E
2667:)
2664:v
2661:,
2658:u
2655:(
2646:0
2634:v
2631:u
2627:f
2584:E
2578:)
2575:t
2572:,
2569:s
2566:(
2555:1
2543:t
2540:s
2536:d
2528:s
2522:u
2519:,
2516:E
2510:)
2507:t
2504:,
2501:u
2498:(
2489:0
2477:u
2473:z
2464:t
2461:u
2457:d
2449:t
2443:v
2440:,
2437:E
2431:)
2428:v
2425:,
2422:s
2419:(
2410:1
2398:v
2394:z
2390:+
2385:v
2382:s
2378:d
2370:t
2364:v
2361:,
2358:s
2352:u
2349:,
2346:E
2340:)
2337:v
2334:,
2331:u
2328:(
2319:0
2307:v
2303:z
2299:+
2294:u
2290:z
2281:v
2278:u
2274:d
2233:}
2230:t
2227:,
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