22:
3676:
3430:
7669:
7661:
8287:
6845:
293:
5763:
6893:. A team is eliminated if it has no chance to finish the season in the first place. The task of the baseball elimination problem is to determine which teams are eliminated at each point during the season. Schwartz proposed a method which reduces this problem to maximum network flow. In this method a network is created to determine whether team
2138:
The push relabel algorithm maintains a preflow, i.e. a flow function with the possibility of excess in the vertices. The algorithm runs while there is a vertex with positive excess, i.e. an active vertex in the graph. The push operation increases the flow on a residual edge, and a height function on
1280:
Note that several maximum flows may exist, and if arbitrary real (or even arbitrary rational) values of flow are permitted (instead of just integers), there is either exactly one maximum flow, or infinitely many, since there are infinitely many linear combinations of the base maximum flows. In other
3140:
Gao, Liu, and Peng's algorithm revolves around dynamically maintaining the augmenting electrical flows at the core of the interior point method based algorithm from . This entails designing data structures that, in limited settings, return edges with large electric energy in a graph undergoing
8141:
5668:. The latter case is always applicable. In the former case, the total number of edges in the cover is increased by 1 and the number of paths stays the same; in the latter case the number of paths is increased and the number of edges stays the same. It is now clear that after covering all
7915:
946:
7485:
and obviously maximum bipartite matching in this graph produces an airline schedule with minimum number of crews. As it is mentioned in the
Application part of this article, the maximum cardinality bipartite matching is an application of maximum flow problem.
124:
It 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
111:
Consider a rail network connecting two cities by way of a number of intermediate cities, where each link of the network has a number assigned to it representing its capacity. Assuming a steady state condition, find a maximal flow from one given city to the
4621:
4420:
4494:
2583:
2139:
the vertices controls through which residual edges can flow be pushed. The height function is changed by the relabel operation. The proper definitions of these operations guarantee that the resulting flow function is a maximum flow.
8281:
5968:
7209:
In the airline industry a major problem is the scheduling of the flight crews. The airline scheduling problem can be considered as an application of extended maximum network flow. The input of this problem is a set of flights
136:
Over the years, various improved solutions to the maximum flow problem were discovered, notably the shortest augmenting path algorithm of
Edmonds and Karp and independently Dinitz; the blocking flow algorithm of Dinitz; the
1236:
149:; and the binary blocking flow algorithm of Goldberg and Rao. The algorithms of Sherman and Kelner, Lee, Orecchia and Sidford, respectively, find an approximately optimal maximum flow but only work in undirected graphs.
7167:
4324:
7964:
3136:
7743:
2450:
756:
2194:
Push-relabel algorithm variant which always selects the most recently active vertex, and performs push operations while the excess is positive and there are admissible residual edges from this vertex.
2715:
2366:
3293:
4962:
3819:
3050:
6831:
is the task of finding the maximum-weight or minimum-weight closure in a vertex-weighted directed graph. It may be solved in polynomial time using a reduction to the maximum flow problem.
2964:
1929:
280:
with negative weights another particular case of minimum-cost flow problem an algorithm in almost-linear time has also been reported. Both algorithms were deemed best papers at the 2022
3589:
3531:
2370:
The algorithm builds limited size trees on the residual graph regarding to the height function. These trees provide multilevel push operations, i.e. pushing along an entire saturating
5850:
603:
1094:
7498:
for maximum goods that can flow through it. The problem is to find if there is a circulation that satisfies the demand. This problem can be transformed into a maximum-flow problem.
9288:
Brand, J. vd; Lee, Y.T.; Liu, Y.P.; Saranurak, T.; Sidford, A; Song, Z.; Wang, D. (2021). "Minimum Cost Flows, MDPs, and â1-Regression in Nearly Linear Time for Dense
Instances".
7643:
2786:
7214:
which contains the information about where and when each flight departs and arrives. In one version of airline scheduling the goal is to produce a feasible schedule with at most
3209:
2864:
2252:
646:
5127:
1275:
699:
7494:
There are some factories that produce goods and some villages where the goods have to be delivered. They are connected by a networks of roads with each road having a capacity
5332:
4021:
3952:
270:
6268:
6165:
6091:
4352:
1003:
7649:
If there exists a circulation, looking at the max-flow solution would give the answer as to how much goods have to be sent on a particular road for satisfying the demands.
6241:
6118:
6064:
3728:
4426:
4115:
2466:
1457:
and route the flow on remaining edges accordingly, to obtain another maximum flow. If flow values can be any real or rational numbers, then there are infinitely many such
2082:
2035:
1409:
1044:
740:
9243:
8988:
7692:
an image. They present an algorithm to find the background and the foreground in an image. More precisely, the algorithm takes a bitmap as an input modelled as follows:
5646:
5588:
3338:
481:
4500:
2893:
2134:
1784:
1727:
1435:
206:
41:) has a preference for only a subset of the humans. Find any matching of pets to humans such that the maximum number of pets are adopted by one of its preferred humans.
3374:
2190:
1975:
380:
6656:
6435:
6314:
5806:
5357:
5152:
4889:
4841:
4650:
4228:
3847:
3473:
1475:
549:
348:
8382:, with the smallest cost. In most variants, the cost-coefficients may be either positive or negative. There are various polynomial-time algorithms for this problem.
6270:(see Fig. 4.4.1). In this expanded network, the vertex capacity constraint is removed and therefore the problem can be treated as the original maximum flow problem.
6037:
4084:
4047:
3978:
3298:
Chen, Kyng, Liu, Peng, Gutenberg and
Sachdeva's algorithm solves maximum flow and minimum-cost flow in almost linear time by building the flow through a sequence of
1345:
511:
5014:
2747:
2660:
2626:
1826:
1674:
6214:
5552:
to the cover, we can either add it to an existing path, or create a new path of length zero starting at that vertex. The former case is applicable whenever either
5752:
5530:
4776:
1852:
1501:
6799:
6779:
6759:
6736:
6716:
6696:
6676:
6618:
6598:
6575:
6555:
6535:
6515:
6495:
6475:
6455:
6394:
6374:
6354:
6334:
6185:
6138:
6011:
5991:
5870:
5726:
5706:
5686:
5666:
5608:
5550:
5504:
5481:
5461:
5437:
5417:
5397:
5377:
5300:
5280:
5260:
5240:
5216:
5196:
5176:
5094:
5074:
5054:
5034:
4982:
4909:
4864:
4816:
4796:
4750:
4730:
4710:
4690:
4670:
4344:
4252:
4179:
4159:
4139:
3907:
3887:
3867:
3752:
3657:
3633:
3609:
2294:
2274:
1593:
1569:
1549:
1529:
1455:
1365:
1319:
1299:
443:
423:
400:
211:
In 2022 Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst
Gutenberg, and Sushant Sachdeva published an almost-linear time algorithm running in
3754:, that is a matching that contains the largest possible number of edges. This problem can be transformed into a maximum flow problem by constructing a network
6741:
3. In addition to the paths being edge-disjoint and/or vertex disjoint, the paths also have a length constraint: we count only paths whose length is exactly
8718:
7422:
Another version of airline scheduling is finding the minimum needed crews to perform all the flights. To find an answer to this problem, a bipartite graph
5878:
3416:
applications (see below), where the flow across an edge may encode whether the item corresponding to that edge is to be included in the set sought or not.
8152:
9487:
9599:
8844:
Bernstein, Aaron; Nanongkai, Danupon; Wulff-Nilsen, Christian (30 October 2022). "Negative-Weight Single-Source
Shortest Paths in Near-linear Time".
8917:
1102:
9455:
9716:
8653:
8136:{\displaystyle q'(A,B)=\sum _{i\in A}b_{i}+\sum _{i\in B}a_{i}+\sum _{\begin{matrix}i,j{\text{ adjacent}}\\|A\cap \{i,j\}|=1\end{matrix}}p_{ij}}
7910:{\displaystyle q(A,B)=\sum _{i\in A}a_{i}+\sum _{i\in B}b_{i}-\sum _{\begin{matrix}i,j{\text{ adjacent}}\\|A\cap \{i,j\}|=1\end{matrix}}p_{ij}}
5766:
Fig. 4.4.1. Transformation of a maximum flow problem with vertex capacities constraint into the original maximum flow problem by node splitting
7060:
3067:
8472:
Gass, Saul I.; Assad, Arjang A. (2005). "Mathematical, algorithmic and professional developments of operations research from 1951 to 1956".
281:
8866:
8294:
We now construct the network whose nodes are the pixel, plus a source and a sink, see Figure on the right. We connect the source to pixel
8819:
7672:
Network built from the bitmap. The source is on the left, the sink on the right. The darker an edge is, the bigger is its capacity. a
4257:
941:{\displaystyle \forall v\in V\setminus \{s,t\}:\quad \sum _{u:(u,v)\in E,f_{uv}>0}f_{uv}=\sum _{u:(v,u)\in E,f_{vu}>0}f_{vu}.}
8331:. Now, it remains to compute a minimum cut in that network (or equivalently a maximum flow). The last figure shows a minimum cut.
3433:
Fig. 4.1.1. Transformation of a multi-source multi-sink maximum flow problem into a single-source single-sink maximum flow problem
8797:
Chen, L.; Kyng, R.; Liu, Y.P.; Gutenberg, M.P.; Sachdeva, S. (2022). "Maximum Flow and
Minimum-Cost Flow in Almost-Linear Time".
6969:â which represents the number of plays between these two teams. We also add a team node for each team and connect each game node
6932:â which represents the number of plays between these two teams. We also add a team node for each team and connect each game node
2387:
138:
9517:
8512:
6989:
to ensure one of them wins. One does not need to restrict the flow value on these edges. Finally, edges are made from team node
747:
The sum of the flows entering a node must equal the sum of the flows exiting that node, except for the source and the sink. Or:
1629:
9692:
9439:
9094:
8686:
8628:
8489:
8397:
says that, in a certain pair of edges, at least one must have a nonzero flow. With negative constraints, the problem becomes
2665:
2312:
107:. In their 1955 paper, Ford and Fulkerson wrote that the problem of Harris and Ross is formulated as follows (see p. 5):
3215:
8401:
even for simple networks. With positive constraints, the problem is polynomial if fractional flows are allowed, but may be
4914:
277:
9273:
Brand, J. vd; Lee, Y.T.; Nanongkai, D.; Peng, R.; Saranurak, T.; Sidford, A.; Song, Z.; Wang, D. (16â19 November 2020).
6399:
1. The paths must be edge-disjoint. This problem can be transformed to a maximum flow problem by constructing a network
5973:
In other words, the amount of flow passing through a vertex cannot exceed its capacity. To find the maximum flow across
8654:"An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations"
5872:
has to satisfy not only the capacity constraint and the conservation of flows, but also the vertex capacity constraint
1678:
As long as there is an open path through the residual graph, send the minimum of the residual capacities on that path.
9330:
Itai, A.; Perl, Y.; Shiloach, Y. (1982). "The complexity of finding maximum disjoint paths with length constraints".
8770:
2981:
2898:
8556:
1857:
9309:
Gao, Y.; Liu, Y.P.; Peng, R. (2021). "Fully
Dynamic Electrical Flows: Sparse Maxflow Faster Than Goldberg-Rao".
3757:
3536:
3478:
5811:
1571:
refers to the largest edge capacity after rescaling all capacities to integer values (if the network contains
564:
9711:
1058:
69:
65:
9412:
9542:
Schauer, Joachim; Pferschy, Ulrich (1 July 2013). "The maximum flow problem with disjunctive constraints".
9495:
3731:
1979:
A modification of Dinic's algorithm with a different approach to constructing blocking flows. Refer to the
1637:
104:
7602:
2752:
60:
The maximum flow problem can be seen as a special case of more complex network flow problems, such as the
3154:
2801:
2214:
617:
9077:
Cheriyan, J.; Maheshwari, S. N. (1988). "Analysis of preflow push algorithms for maximum network flow".
9429:
8508:
8476:. International Series in Operations Research & Management Science. Vol. 75. pp. 79â110.
5103:
4415:{\displaystyle V_{\textrm {out}}=\{v_{\textrm {out}}\mid v\in V\land v{\text{ has outgoing edge(s)}}\}}
2199:
2144:
2090:
1683:
1251:
658:
89:
46:
8700:
5808:
be a network. Suppose there is capacity at each node in addition to edge capacity, that is, a mapping
5308:
8341:
4489:{\displaystyle V_{\textrm {in}}=\{v_{\textrm {in}}\mid v\in V\land v{\text{ has incoming edge(s)}}\}}
2578:{\displaystyle O\left(E\cdot \min\{V^{2/3},E^{1/2}\}\cdot \log {\frac {V^{2}}{E}}\cdot \log U\right)}
273:
214:
9556:
9488:"Project imagesegmentationwithmaxflow, that contains the source code to produce these illustrations"
8753:
6246:
6143:
6069:
959:
9620:
8441:
7729:
are placed one in the foreground and the other in the background. The goal is to find a partition (
6219:
6096:
6042:
3689:
3405:
77:
9606:(1999). "An analysis of the highest-level selection rule in the preflow-push max-flow algorithm".
7589:) be this new network. There exists a circulation that satisfies the demand if and only if :
4616:{\displaystyle E'=\{(u_{\textrm {out}},v_{\textrm {in}})\in V_{out}\times V_{in}\mid (u,v)\in E\}}
3340:
approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized
3983:
3914:
3399:
If each edge in a flow network has integral capacity, then there exists an integral maximal flow.
2046:
1999:
1370:
1008:
704:
9204:
8947:
5993:, we can transform the problem into the maximum flow problem in the original sense by expanding
5613:
5555:
3404:
The claim is not only that the value of the flow is an integer, which follows directly from the
3301:
448:
9615:
9551:
9470:
8748:
8436:
4190:
3611:. We can transform the multi-source multi-sink problem into a maximum flow problem by adding a
3054:
Interior point methods and dynamic maintenance of electric flows with expander decompositions.
2871:
2301:
2100:
1750:
1693:
1414:
1248:
is to route as much flow as possible from the source to the sink, in other words find the flow
159:
6698:. Then the value of the maximum flow is equal to the maximum number of independent paths from
3679:
Fig. 4.3.1. Transformation of a maximum bipartite matching problem into a maximum flow problem
3343:
2159:
1944:
353:
6623:
6402:
6281:
5773:
4195:
4089:
3440:
1460:
561:
of an edge is the maximum amount of flow that can pass through an edge. Formally it is a map
516:
315:
7193:. In the mentioned article it is proved that this flow value is the maximum flow value from
6016:
4054:
4026:
3957:
1324:
486:
9645:
9417:
9376:
4987:
3413:
2720:
2633:
2599:
1988:
1799:
1789:
1740:
1732:
1647:
6190:
8:
7222:
5731:
5509:
4755:
4231:
3675:
1831:
1480:
96:
61:
9380:
8427:
Schrijver, A. (2002). "On the history of the transportation and maximum flow problems".
5337:
5132:
4869:
4821:
4630:
3827:
9577:
9449:
9392:
9310:
9289:
9181:
9162:
9153:
9135:
9118:
King, V.; Rao, S.; Tarjan, R. (1994). "A Faster
Deterministic Maximum Flow Algorithm".
9059:
9040:
9027:
8845:
8798:
8776:
8692:
8664:
8634:
8606:
8454:
7689:
6784:
6764:
6744:
6721:
6701:
6681:
6661:
6603:
6583:
6560:
6540:
6520:
6500:
6480:
6460:
6440:
6379:
6359:
6339:
6319:
6170:
6123:
5996:
5976:
5855:
5711:
5691:
5671:
5651:
5593:
5535:
5489:
5466:
5446:
5422:
5402:
5382:
5362:
5285:
5265:
5245:
5225:
5201:
5181:
5161:
5079:
5059:
5039:
5019:
4967:
4894:
4849:
4801:
4781:
4735:
4715:
4695:
4675:
4655:
4329:
4237:
4164:
4161:, and a maximum cardinality matching can be found by taking those edges that have flow
4144:
4124:
3892:
3872:
3852:
3737:
3642:
3618:
3594:
3429:
2279:
2259:
1615:
1578:
1554:
1534:
1514:
1440:
1350:
1304:
1284:
428:
408:
385:
142:
73:
9629:
7376:
is reachable with a reasonable amount of time and cost from the destination of flight
4818:. Therefore, the problem can be solved by finding the maximum cardinality matching in
9688:
9668:
9664:
9637:
9569:
9435:
9347:
9100:
9090:
9008:
8780:
8766:
8682:
8624:
8523:
8485:
5963:{\displaystyle \sum _{i\in V}f_{iv}\leq c(v)\qquad \forall v\in V\backslash \{s,t\}.}
1572:
9063:
8696:
8638:
8458:
7668:
9660:
9641:
9625:
9581:
9561:
9421:
9413:
9384:
9339:
9171:
9127:
9082:
9049:
9031:
9004:
8758:
8674:
8616:
8565:
8477:
8446:
8402:
8398:
8286:
8276:{\displaystyle q(A,B)=\sum _{i\in A\cup B}a_{i}+\sum _{i\in A\cup B}b_{i}-q'(A,B).}
7660:
9185:
9139:
8891:
8547:
8543:
7258:
as the source and the sink nodes. For the source and destination of every flight
6810:
3684:
2256:
Push-relabel algorithm variant which always selects the most distant vertex from
1511:
The following table lists algorithms for solving the maximum flow problem. Here,
100:
8943:
8661:
Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms
8603:
Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science
3591:
instead of only one source and one sink, we are to find the maximum flow across
9425:
9365:
Schwartz, B. L. (1966). "Possible Winners in Partially Completed Tournaments".
8719:"New algorithm can dramatically streamline solutions to the 'max flow' problem"
8678:
8393:
says that a certain pair of edges cannot simultaneously have a nonzero flow; a
7652:
The problem can be extended by adding a lower bound on the flow on some edges.
7395:
In the mentioned method, it is claimed and proved that finding a flow value of
6844:
6678:
with vertex capacities, where the capacities of all vertices and all edges are
1793:
1055:
is the amount of flow passing from the source to the sink. Formally for a flow
153:
9565:
2296:(i.e. the highest label vertex) but otherwise proceeds as the FIFO algorithm.
2043:
data structure speeds up the maximum flow computation in the layered graph to
1231:{\displaystyle |f|=\sum _{v:\ (s,v)\in E}f_{sv}=\sum _{u:\ (u,t)\in E}f_{ut}.}
9705:
9680:
9672:
9603:
9573:
9351:
9104:
9086:
2040:
146:
8762:
8481:
6781:. Most variants of this problem are NP-complete, except for small values of
276:
problem of which for the maximum flow problem is a particular case. For the
9343:
9131:
8570:
8551:
7226:
5532:, we start with an empty cover and build it incrementally. To add a vertex
3212:
The exact complexity is not stated clearly in the paper, but appears to be
1625:
54:
9176:
9157:
8745:
Proceedings of the forty-fifth annual ACM symposium on Theory of Computing
8450:
2895:-norm flows. Builds on earlier algorithm of Madry, which achieved runtime
9367:
8620:
9054:
9035:
7162:{\displaystyle r(S-\{k\})=\sum _{i,j\in \{S-\{k\}\} \atop i<j}r_{ij}}
9396:
6860:
teams competing in a league. At a specific stage of the league season,
3131:{\displaystyle {\tilde {O}}(E^{{\frac {3}{2}}-{\frac {1}{328}}}\log U)}
9636:
8601:
Sherman, Jonah (2013). "Nearly Maximum Flows in Nearly Linear Time".
292:
92:
and F. S. Ross as a simplified model of Soviet railway traffic flow.
9388:
9081:. Lecture Notes in Computer Science. Vol. 338. pp. 30â48.
7229:
problems, with the added constraint of a lower bound on edge flows.
6580:
2. The paths must be independent, i.e., vertex-disjoint (except for
1854:. In networks with unit capacities, Dinic's algorithm terminates in
9315:
9294:
9275:
Bipartite Matching in Nearly-linear Time on Moderately Dense Graphs
9079:
Foundations of Software Technology and Theoretical Computer Science
8850:
8803:
6853:
5688:
vertices, the sum of the number of paths and edges in the cover is
8669:
8611:
1731:
A specialization of FordâFulkerson, finding augmenting paths with
1551:
denote the number of vertices and edges of the network. The value
655:. The flow of an edge cannot exceed its capacity, in other words:
9598:
8867:"Finally, a Fast Algorithm for Shortest Paths on Negative Graphs"
5762:
8820:"Researchers Achieve 'Absurdly Fast' Algorithm for Network Flow"
6926:
being the source and the sink respectively. One adds a game node
4184:
3424:
76:(i.e., cut severing s from t) in the network, as stated in the
8918:"FOCS 2022 Best Paper Award for Prof. Aaron Bernstein's Paper"
8843:
8743:
Orlin, James B. (2013). "Max flows in O(nm) time, or better".
4319:{\displaystyle G'=(V_{\textrm {out}}\cup V_{\textrm {in}},E')}
8652:
Kelner, J. A.; Lee, Y. T.; Orecchia, L.; Sidford, A. (2014).
8513:"Fundamentals of a Method for Evaluating Rail Net Capacities"
8359:
in addition to its capacity. If the flow through the edge is
7688:
In their book, Kleinberg and Tardos present an algorithm for
7054:
be the set of all teams participating in the league and let
6848:
Construction of network flow for baseball elimination problem
9199:
Kathuria, T.; Liu, Y.P.; Sidford, A. (16â19 November 2020).
8942:
Malhotra, V.M.; Kumar, M. Pramodh; Maheshwari, S.N. (1978).
8290:
Minimum cut displayed on the network (triangles VS circles).
7737:) of the set of pixels that maximize the following quantity
21:
2445:{\displaystyle O\left(VE\log _{\frac {E}{V\log V}}V\right)}
1796:. The maximum flow in a layered graph can be calculated in
131:
Fundamentals of a Method for Evaluating Rail net Capacities
7225:
called bounded circulation which is the generalization of
5262:. In either case, no more than one edge leaves any vertex
3146:
Chen, Kyng, Liu, Peng, Gutenberg and Sachdeva's algorithm
5443:
Thus no vertex has two incoming or two outgoing edges in
5359:â if it is matched, there is a single incoming edge into
3670:
3667:) with infinite capacity on each edge (See Fig. 4.1.1.).
1788:
In each phase the algorithms builds a layered graph with
88:
The maximum flow problem was first formulated in 1954 by
8651:
9260:
Computing Maximum Flow with Augmenting Electrical Flows
8941:
7411:
is equal to finding a feasible schedule for flight set
2710:{\displaystyle E\leq O(V^{{\frac {16}{15}}-\epsilon })}
2361:{\displaystyle O\left(VE\log {\frac {V^{2}}{E}}\right)}
1980:
8057:
7831:
7177:
is not eliminated if and only if a flow value of size
6273:
5939:
5757:
3288:{\displaystyle E\exp O(\log ^{7/8}E\log \log E)\log U}
2591:
James B Orlin's + KRT (King, Rao, Tarjan)'s algorithm
9272:
9207:
8950:
8155:
7967:
7746:
7605:
7063:
6787:
6767:
6747:
6724:
6704:
6684:
6664:
6626:
6606:
6586:
6563:
6543:
6523:
6503:
6483:
6463:
6443:
6405:
6382:
6362:
6342:
6322:
6284:
6249:
6222:
6193:
6173:
6146:
6126:
6099:
6072:
6045:
6019:
5999:
5979:
5881:
5858:
5814:
5776:
5734:
5714:
5694:
5674:
5654:
5616:
5596:
5558:
5538:
5512:
5492:
5469:
5449:
5425:
5405:
5385:
5365:
5340:
5311:
5288:
5268:
5248:
5228:
5204:
5184:
5164:
5135:
5106:
5082:
5062:
5042:
5022:
4990:
4970:
4917:
4897:
4872:
4852:
4824:
4804:
4784:
4758:
4738:
4718:
4698:
4678:
4658:
4633:
4503:
4429:
4355:
4332:
4260:
4240:
4198:
4167:
4147:
4127:
4092:
4057:
4029:
3986:
3960:
3917:
3895:
3875:
3855:
3830:
3760:
3740:
3692:
3645:
3621:
3597:
3539:
3481:
3443:
3346:
3304:
3218:
3157:
3070:
2984:
2901:
2874:
2804:
2755:
2723:
2668:
2636:
2602:
2469:
2390:
2315:
2282:
2262:
2217:
2162:
2103:
2049:
2002:
1947:
1860:
1834:
1802:
1753:
1696:
1650:
1581:
1557:
1537:
1517:
1483:
1463:
1443:
1417:
1373:
1353:
1327:
1307:
1287:
1254:
1105:
1061:
1011:
962:
759:
707:
661:
620:
567:
519:
489:
451:
431:
411:
388:
356:
318:
217:
162:
9287:
9262:. New Brunswick, New Jersey: IEEE. pp. 593â602.
8796:
6517:
respectively, and assigning each edge a capacity of
5708:. Therefore, if the number of edges in the cover is
4957:{\displaystyle u_{\mathrm {out} },v_{\mathrm {in} }}
64:. The maximum value of an s-t flow (i.e., flow from
304:. The numbers next to the edges are the capacities.
9237:
9198:
9076:
8982:
8275:
8135:
7909:
7637:
7221:To solve this problem one uses a variation of the
7161:
6793:
6773:
6753:
6730:
6710:
6690:
6670:
6650:
6612:
6592:
6569:
6549:
6529:
6509:
6489:
6469:
6449:
6429:
6388:
6368:
6356:, we are to find the maximum number of paths from
6348:
6328:
6308:
6262:
6235:
6208:
6179:
6159:
6132:
6112:
6085:
6058:
6031:
6005:
5985:
5962:
5864:
5844:
5800:
5746:
5720:
5700:
5680:
5660:
5640:
5602:
5582:
5544:
5524:
5498:
5475:
5455:
5431:
5411:
5391:
5371:
5351:
5326:
5294:
5274:
5254:
5234:
5222:, in which case there is exactly one edge leaving
5210:
5190:
5170:
5146:
5121:
5088:
5068:
5048:
5028:
5008:
4976:
4956:
4903:
4883:
4858:
4835:
4810:
4790:
4770:
4744:
4724:
4704:
4684:
4664:
4644:
4615:
4488:
4414:
4338:
4318:
4246:
4222:
4173:
4153:
4133:
4109:
4078:
4041:
4015:
3972:
3946:
3901:
3881:
3861:
3841:
3813:
3746:
3722:
3651:
3627:
3603:
3583:
3525:
3467:
3368:
3332:
3287:
3203:
3130:
3044:
2958:
2887:
2858:
2780:
2741:
2709:
2654:
2620:
2577:
2444:
2360:
2288:
2268:
2246:
2184:
2128:
2076:
2029:
1969:
1923:
1846:
1820:
1778:
1721:
1668:
1587:
1563:
1543:
1523:
1495:
1469:
1449:
1429:
1403:
1359:
1339:
1313:
1293:
1269:
1230:
1088:
1038:
997:
940:
734:
693:
640:
597:
543:
505:
475:
437:
417:
394:
374:
342:
264:
200:
9685:Combinatorial Optimization: Networks and Matroids
9703:
9471:"Max-flow extensions: circulations with demands"
9329:
8990:algorithm for finding maximum flows in networks"
8792:
8790:
8385:2. The maximum-flow problem can be augmented by
4141:is equal to the size of the maximum matching in
2484:
1867:
9541:
9434:. MIT Press and McGraw-Hill. pp. 643â668.
9022:
9020:
9018:
7441:is created where each flight has a copy in set
3045:{\displaystyle {\tilde {O}}((E+V^{3/2})\log U)}
2868:Interior point methods and edge boosting using
9026:
8378:It is required to find a flow of a given size
3383:
9679:
9117:
8787:
8542:
7943:. On the border, between two adjacent pixels
7489:
6878:is the number of games left to play for team
2959:{\displaystyle {\tilde {O}}(E^{10/7}U^{1/7})}
57:that obtains the maximum possible flow rate.
9454:: CS1 maint: multiple names: authors list (
9070:
9036:"A new approach to the maximum-flow problem"
9015:
8474:An Annotated Timeline of Operations Research
8101:
8089:
7958:. It is equivalent to minimize the quantity
7875:
7863:
7506:and add edges from it to every factory node
7328:An edge with capacity between each pair of
7128:
7125:
7119:
7110:
7082:
7076:
5954:
5942:
5096:is vertex-disjoint, consider the following:
4610:
4515:
4483:
4445:
4409:
4371:
4185:Minimum path cover in directed acyclic graph
3794:
3782:
3578:
3546:
3520:
3488:
3425:Multi-source multi-sink maximum flow problem
2529:
2487:
1936:MKM (Malhotra, Kumar, Maheshwari) algorithm
1912:
1870:
787:
775:
282:Symposium on Foundations of Computer Science
9152:
5178:, in which case there are no edges leaving
1924:{\displaystyle O(\min\{V^{2/3},E^{1/2}\}E)}
25:Flow network for the problem: Each human (r
9468:
9431:Introduction to Algorithms, Second Edition
9308:
9277:. Durham, NC, USA: IEEE. pp. 919â930.
9247:. Durham, NC, USA: IEEE. pp. 119â130.
8507:
7680:when the pixel is not green. The penalty p
6834:
3814:{\displaystyle N=(X\cup Y\cup \{s,t\},E')}
1828:time, and the maximum number of phases is
278:single-source shortest path (SSSP) problem
72:t) is equal to the minimum capacity of an
53:involve finding a feasible flow through a
9619:
9555:
9314:
9293:
9175:
9053:
8849:
8817:
8802:
8752:
8668:
8610:
8569:
8471:
8440:
8426:
6889:is the number of games left against team
6819:of a directed graph is a set of vertices
5829:
3584:{\displaystyle T=\{t_{1},\ldots ,t_{m}\}}
3526:{\displaystyle S=\{s_{1},\ldots ,s_{n}\}}
1624:Constraints given by the definition of a
1076:
634:
582:
9364:
8864:
8285:
7936:(considered as the background), we gain
7925:(considered as the foreground), we gain
7667:
7659:
6843:
5845:{\displaystyle c:V\to \mathbb {R} ^{+},}
5761:
3674:
3428:
3389:
598:{\displaystyle c:E\to \mathbb {R} ^{+}.}
291:
129:where refers to the 1955 secret report
20:
9653:Journal of Computer and System Sciences
9258:Madry, Aleksander (9â11 October 2016).
9158:"Beyond the flow decomposition barrier"
8738:
8736:
8600:
7449:. If the same plane can perform flight
7288:. One also adds the following edges to
6839:
6537:. In this network, the maximum flow is
4230:, we are to find the minimum number of
1367:in another maximum flow, then for each
1089:{\displaystyle f:E\to \mathbb {R} ^{+}}
103:created the first known algorithm, the
9717:Computational problems in graph theory
9704:
9408:
9406:
8716:
7721:is the penalty if two adjacent pixels
6167:is connected to edges coming out from
4911:from it. Intuitively, if two vertices
4121:Then the value of the maximum flow in
3671:Maximum cardinality bipartite matching
3412:is integral. This is crucial for many
3394:The integral flow theorem states that
9544:Journal of Combinatorial Optimization
9257:
8742:
7655:
7204:
6396:. This problem has several variants:
5590:and some path in the cover starts at
4254:. We can construct a bipartite graph
3376:time using a dynamic data structure.
2630:Orlin's algorithm solves max-flow in
16:Computational problem in graph theory
9646:"A data structure for dynamic trees"
8733:
8591:, Princeton University Press (1962).
8583:
8581:
8538:
8536:
8503:
8501:
8422:
8420:
8418:
7638:{\displaystyle =\sum _{i\in v}d_{i}}
7349:An edge with capacity between each
7312:An edge with capacity between each
2781:{\displaystyle E>V^{1+\epsilon }}
2379:KRT (King, Rao, Tarjan)'s algorithm
133:by Harris and Ross (see p. 5).
120:, in 1962, Ford and Fulkerson wrote:
9403:
8915:
6274:Maximum number of paths from s to t
5758:Maximum flow with vertex capacities
3204:{\displaystyle O(E^{1+o(1)}\log U)}
2859:{\displaystyle E^{4/3+o(1)}U^{1/3}}
2247:{\displaystyle O(V^{2}{\sqrt {E}})}
641:{\displaystyle f:E\to \mathbb {R} }
13:
9592:
8587:Ford, L.R., Jr.; Fulkerson, D.R.,
7676:is high when the pixel is green, b
7527:is the production rate of factory
7284:as the destination node of flight
7173:In this method it is claimed team
7096:
6804:
5927:
4948:
4945:
4930:
4927:
4924:
1464:
1424:
1374:
760:
308:First we establish some notation:
14:
9728:
8865:Brubaker, Ben (18 January 2023).
8578:
8533:
8498:
8415:
8405:when the flows must be integral.
8309:to the sink by an edge of weight
7710:â„ 0 in the likelihood that pixel
7699:â„ 0 is the likelihood that pixel
6956:, and connects each of them from
6497:being the source and the sink of
6120:is connected by edges going into
5122:{\displaystyle v_{\textrm {out}}}
5036:. Clearly the number of edges in
4712:has a vertex-disjoint path cover
1270:{\displaystyle f_{\textrm {max}}}
772:
694:{\displaystyle f_{uv}\leq c_{uv}}
382:being the source and the sink of
9201:Unit Capacity Maxflow in Almost
8818:Klarreich, Erica (8 June 2022).
8717:Knight, Helen (7 January 2014).
8552:"Maximal flow through a network"
8395:positive disjunctive constraints
7541:and add edges from all villages
5327:{\displaystyle v_{\textrm {in}}}
4846:Assume we have found a matching
156:published a paper describing an
9535:
9510:
9480:
9462:
9358:
9323:
9302:
9281:
9266:
9251:
9192:
9146:
9111:
8935:
8909:
8884:
8858:
8837:
8811:
8557:Canadian Journal of Mathematics
8391:negative disjunctive constraint
7714:belongs to the background, and
7383:An edge with capacity between
7296:An edge with capacity between
5926:
3382:For additional algorithms, see
2793:Kathuria-Liu-Sidford algorithm
2458:Binary blocking flow algorithm
793:
265:{\displaystyle O(|E|^{1+o(1)})}
9608:Information Processing Letters
9232:
9211:
8997:Information Processing Letters
8977:
8967:
8958:
8954:
8710:
8645:
8594:
8465:
8267:
8255:
8171:
8159:
8105:
8079:
7988:
7976:
7879:
7853:
7762:
7750:
7566:is the demand rate of village
7085:
7067:
6856:elimination problem there are
6645:
6633:
6620:). We can construct a network
6424:
6412:
6303:
6291:
6263:{\displaystyle v_{\text{out}}}
6203:
6197:
6160:{\displaystyle v_{\text{out}}}
6086:{\displaystyle v_{\text{out}}}
5923:
5917:
5824:
5795:
5783:
5629:
5617:
5571:
5559:
5003:
4991:
4601:
4589:
4548:
4518:
4313:
4272:
4217:
4205:
4067:
4061:
3999:
3987:
3930:
3918:
3808:
3767:
3717:
3699:
3462:
3450:
3419:
3361:
3355:
3325:
3319:
3273:
3231:
3198:
3184:
3178:
3161:
3125:
3083:
3077:
3039:
3027:
3000:
2997:
2991:
2953:
2914:
2908:
2833:
2827:
2736:
2727:
2704:
2678:
2649:
2640:
2615:
2606:
2241:
2221:
2179:
2166:
2123:
2107:
2071:
2053:
2024:
2006:
1964:
1951:
1918:
1864:
1815:
1806:
1773:
1757:
1716:
1700:
1663:
1654:
1398:
1380:
1201:
1189:
1148:
1136:
1115:
1107:
1071:
1024:
1012:
998:{\displaystyle f_{uv}=-f_{vu}}
889:
877:
817:
805:
720:
708:
648:that satisfies the following:
630:
577:
535:
523:
464:
452:
425:is a function on the edges of
337:
325:
259:
254:
248:
234:
225:
221:
195:
191:
183:
178:
170:
166:
29:) is willing to adopt a cat (w
1:
9630:10.1016/S0020-0190(99)00019-8
8408:
8334:
6236:{\displaystyle v_{\text{in}}}
6113:{\displaystyle v_{\text{in}}}
6059:{\displaystyle v_{\text{in}}}
4798:is the number of vertices in
3723:{\displaystyle G=(X\cup Y,E)}
3615:connecting to each vertex in
2971:BLNPSSSW / BLLSSSW algorithm
1506:
287:
9683:(2001). "4. Network Flows".
9665:10.1016/0022-0000(83)90006-5
9428:(2001). "26. Maximum Flow".
9009:10.1016/0020-0190(78)90016-9
4891:, and constructed the cover
3732:maximum cardinality matching
3639:connected by each vertex in
3384:Goldberg & Tarjan (1988)
956:. Flows are skew symmetric:
296:A flow network, with source
7:
9687:. Dover. pp. 109â177.
7703:belongs to the foreground,
6823:, such that no edges leave
5463:, which means all paths in
4016:{\displaystyle (y,t)\in E'}
3947:{\displaystyle (s,x)\in E'}
2077:{\displaystyle O(VE\log V)}
2030:{\displaystyle O(VE\log V)}
1404:{\displaystyle \Delta \in }
1039:{\displaystyle (u,v)\in E.}
735:{\displaystyle (u,v)\in E.}
10:
9733:
9238:{\displaystyle O(m^{4/3})}
8983:{\displaystyle O(|V|^{3})}
8896:focs2022.eecs.berkeley.edu
8679:10.1137/1.9781611973402.16
7490:Circulationâdemand problem
6869:is the number of wins and
6808:
5641:{\displaystyle (v,u)\in E}
5583:{\displaystyle (u,v)\in E}
5305:Similarly for each vertex
4627:Then it can be shown that
4480: has incoming edge(s)
4406: has outgoing edge(s)
3333:{\displaystyle E^{1+o(1)}}
2374:instead of a single edge.
476:{\displaystyle (u,v)\in E}
83:
9566:10.1007/s10878-011-9438-7
8366:, then the total cost is
8342:minimum-cost flow problem
7664:Source image of size 8x8.
6960:by an edge with capacity
5728:, the number of paths is
5419:has no incoming edges in
4181:in an integral max-flow.
2888:{\displaystyle \ell _{p}}
2129:{\displaystyle O(V^{2}E)}
1779:{\displaystyle O(V^{2}E)}
1722:{\displaystyle O(VE^{2})}
1430:{\displaystyle x+\Delta }
1321:in one maximum flow, and
201:{\displaystyle O(|V||E|)}
9087:10.1007/3-540-50517-2_69
8429:Mathematical Programming
8305:. We connect the pixel
7262:, one adds two nodes to
4234:to cover each vertex in
3406:max-flow min-cut theorem
3369:{\displaystyle E^{o(1)}}
2185:{\displaystyle O(V^{3})}
1970:{\displaystyle O(V^{3})}
1638:FordâFulkerson algorithm
375:{\displaystyle s,t\in V}
105:FordâFulkerson algorithm
78:max-flow min-cut theorem
8763:10.1145/2488608.2488705
8482:10.1007/0-387-25837-X_5
8387:disjunctive constraints
7481:induces a schedule for
7275:as the source and node
7030:from winning more than
7026:is set to prevent team
6835:Real world applications
6651:{\displaystyle N=(V,E)}
6430:{\displaystyle N=(V,E)}
6309:{\displaystyle G=(V,E)}
6278:Given a directed graph
6216:to the edge connecting
6187:, then assign capacity
5801:{\displaystyle N=(V,E)}
5486:To show that the cover
4223:{\displaystyle G=(V,E)}
4110:{\displaystyle e\in E'}
3468:{\displaystyle N=(V,E)}
3408:, but that the flow on
3059:Gao-Liu-Peng algorithm
2717:while KRT solves it in
1470:{\displaystyle \Delta }
544:{\displaystyle g(u,v).}
343:{\displaystyle N=(V,E)}
37:2). However each pet (p
9344:10.1002/net.3230120306
9239:
9132:10.1006/jagm.1994.1044
8984:
8916:Santosh, Nagarakatte.
8571:10.4153/CJM-1956-045-5
8511:; Ross, F. S. (1955).
8291:
8277:
8137:
7921:Indeed, for pixels in
7911:
7685:
7665:
7639:
7163:
6849:
6795:
6775:
6755:
6732:
6712:
6692:
6672:
6652:
6614:
6594:
6571:
6551:
6531:
6511:
6491:
6471:
6451:
6431:
6390:
6370:
6350:
6330:
6310:
6264:
6237:
6210:
6181:
6161:
6134:
6114:
6087:
6060:
6033:
6032:{\displaystyle v\in V}
6007:
5987:
5964:
5866:
5846:
5802:
5767:
5748:
5722:
5702:
5682:
5662:
5648:and some path ends at
5642:
5604:
5584:
5546:
5526:
5500:
5477:
5457:
5433:
5413:
5393:
5373:
5353:
5328:
5296:
5276:
5256:
5236:
5212:
5192:
5172:
5148:
5123:
5090:
5070:
5050:
5030:
5010:
4978:
4958:
4905:
4885:
4860:
4837:
4812:
4792:
4772:
4746:
4726:
4706:
4686:
4666:
4646:
4617:
4490:
4416:
4340:
4320:
4248:
4224:
4191:directed acyclic graph
4175:
4155:
4135:
4111:
4080:
4079:{\displaystyle c(e)=1}
4043:
4042:{\displaystyle y\in Y}
4017:
3974:
3973:{\displaystyle x\in X}
3948:
3903:
3883:
3863:
3849:contains the edges in
3843:
3815:
3748:
3724:
3680:
3653:
3629:
3605:
3585:
3527:
3475:with a set of sources
3469:
3434:
3370:
3334:
3289:
3205:
3132:
3046:
2960:
2889:
2860:
2782:
2743:
2711:
2656:
2622:
2579:
2446:
2362:
2302:Push-relabel algorithm
2290:
2270:
2248:
2206:vertex selection rule
2200:Pushârelabel algorithm
2186:
2151:vertex selection rule
2145:Pushârelabel algorithm
2130:
2091:pushârelabel algorithm
2078:
2031:
1971:
1925:
1848:
1822:
1780:
1723:
1684:EdmondsâKarp algorithm
1670:
1589:
1565:
1545:
1525:
1497:
1471:
1451:
1431:
1405:
1361:
1341:
1340:{\displaystyle y>x}
1315:
1301:units of flow on edge
1295:
1271:
1232:
1090:
1040:
999:
942:
745:Conservation of flows.
736:
695:
642:
599:
545:
507:
506:{\displaystyle g_{uv}}
477:
439:
419:
396:
376:
344:
305:
266:
202:
139:push-relabel algorithm
127:
114:
42:
9240:
9177:10.1145/290179.290181
9120:Journal of Algorithms
8985:
8451:10.1007/s101070100259
8298:by an edge of weight
8289:
8278:
8138:
7912:
7671:
7663:
7640:
7189:}) exists in network
7164:
6847:
6796:
6776:
6756:
6733:
6713:
6693:
6673:
6653:
6615:
6595:
6577:edge-disjoint paths.
6572:
6552:
6532:
6512:
6492:
6472:
6452:
6432:
6391:
6371:
6351:
6331:
6311:
6265:
6238:
6211:
6182:
6162:
6135:
6115:
6088:
6061:
6034:
6008:
5988:
5965:
5867:
5847:
5803:
5765:
5749:
5723:
5703:
5683:
5663:
5643:
5605:
5585:
5547:
5527:
5501:
5483:are vertex-disjoint.
5478:
5458:
5434:
5414:
5394:
5374:
5354:
5329:
5297:
5277:
5257:
5237:
5213:
5193:
5173:
5149:
5124:
5091:
5071:
5051:
5031:
5011:
5009:{\displaystyle (u,v)}
4979:
4959:
4906:
4886:
4861:
4838:
4813:
4793:
4773:
4747:
4727:
4707:
4687:
4667:
4647:
4618:
4491:
4417:
4341:
4321:
4249:
4232:vertex-disjoint paths
4225:
4176:
4156:
4136:
4112:
4081:
4044:
4018:
3975:
3949:
3904:
3884:
3864:
3844:
3816:
3749:
3725:
3678:
3654:
3630:
3606:
3586:
3528:
3470:
3432:
3390:Integral flow theorem
3371:
3335:
3290:
3206:
3133:
3047:
2961:
2890:
2861:
2783:
2744:
2742:{\displaystyle O(VE)}
2712:
2657:
2655:{\displaystyle O(VE)}
2623:
2621:{\displaystyle O(VE)}
2580:
2447:
2363:
2291:
2271:
2249:
2187:
2131:
2079:
2032:
1972:
1926:
1849:
1823:
1821:{\displaystyle O(VE)}
1781:
1724:
1671:
1669:{\displaystyle O(EU)}
1590:
1566:
1546:
1526:
1498:
1477:values for each pair
1472:
1452:
1432:
1406:
1362:
1342:
1316:
1296:
1272:
1233:
1091:
1041:
1000:
943:
737:
696:
643:
600:
546:
508:
478:
440:
420:
397:
377:
345:
295:
267:
203:
122:
109:
51:maximum flow problems
24:
9712:Network flow problem
9418:Charles E. Leiserson
9205:
8948:
8747:. pp. 765â774.
8621:10.1109/FOCS.2013.36
8605:. pp. 263â269.
8153:
7965:
7932:; for all pixels in
7744:
7603:
7244:) be a network with
7061:
6997:and the capacity of
6981:with two team nodes
6912:) be a network with
6840:Baseball elimination
6785:
6765:
6745:
6722:
6702:
6682:
6662:
6624:
6604:
6584:
6561:
6541:
6521:
6501:
6481:
6461:
6441:
6403:
6380:
6360:
6340:
6320:
6282:
6247:
6220:
6209:{\displaystyle c(v)}
6191:
6171:
6144:
6124:
6097:
6070:
6043:
6017:
5997:
5977:
5879:
5856:
5812:
5774:
5732:
5712:
5692:
5672:
5652:
5614:
5594:
5556:
5536:
5510:
5490:
5467:
5447:
5423:
5403:
5383:
5363:
5338:
5309:
5286:
5266:
5246:
5226:
5202:
5182:
5162:
5133:
5104:
5080:
5060:
5040:
5020:
4988:
4968:
4915:
4895:
4870:
4850:
4822:
4802:
4782:
4756:
4736:
4716:
4696:
4676:
4656:
4631:
4501:
4427:
4353:
4330:
4258:
4238:
4196:
4165:
4145:
4125:
4090:
4055:
4027:
3984:
3958:
3915:
3893:
3873:
3853:
3828:
3758:
3738:
3690:
3643:
3619:
3595:
3537:
3479:
3441:
3344:
3302:
3216:
3155:
3141:resistance updates.
3068:
2982:
2899:
2872:
2802:
2753:
2721:
2666:
2634:
2600:
2467:
2388:
2313:
2280:
2260:
2215:
2160:
2101:
2047:
2000:
1945:
1858:
1832:
1800:
1790:breadth-first search
1751:
1733:breadth-first search
1694:
1648:
1579:
1555:
1535:
1515:
1481:
1461:
1441:
1415:
1371:
1351:
1325:
1305:
1285:
1277:with maximum value.
1252:
1246:maximum flow problem
1103:
1059:
1009:
960:
757:
705:
659:
618:
565:
517:
487:
449:
429:
409:
386:
354:
316:
215:
160:
101:Delbert R. Fulkerson
9498:on 22 December 2019
9381:1966SIAMR...8..302S
9055:10.1145/48014.61051
8520:Research Memorandum
8316:. We connect pixel
7594:Maximum flow value(
7223:circulation problem
5852:such that the flow
5747:{\displaystyle n-m}
5525:{\displaystyle n-m}
4771:{\displaystyle n-m}
3730:, we are to find a
3613:consolidated source
3533:and a set of sinks
2304:with dynamic trees
1991:with dynamic trees
1847:{\displaystyle V-1}
1496:{\displaystyle x,y}
653:Capacity constraint
97:Lester R. Ford, Jr.
62:circulation problem
47:optimization theory
9518:"Algorithm Design"
9235:
9163:Journal of the ACM
9156:; Rao, S. (1998).
9041:Journal of the ACM
8980:
8922:www.cs.rutgers.edu
8529:on 8 January 2014.
8292:
8273:
8233:
8198:
8133:
8119:
8117:
8038:
8009:
7907:
7893:
7891:
7812:
7783:
7686:
7666:
7656:Image segmentation
7635:
7624:
7502:Add a source node
7205:Airline scheduling
7159:
7145:
6850:
6791:
6771:
6751:
6728:
6708:
6688:
6668:
6648:
6610:
6590:
6567:
6547:
6527:
6507:
6487:
6467:
6447:
6427:
6386:
6366:
6346:
6326:
6306:
6260:
6233:
6206:
6177:
6157:
6130:
6110:
6083:
6056:
6029:
6003:
5983:
5960:
5897:
5862:
5842:
5798:
5768:
5744:
5718:
5698:
5678:
5658:
5638:
5600:
5580:
5542:
5522:
5496:
5473:
5453:
5429:
5409:
5389:
5369:
5352:{\displaystyle G'}
5349:
5324:
5292:
5272:
5252:
5232:
5208:
5188:
5168:
5147:{\displaystyle G'}
5144:
5119:
5086:
5066:
5046:
5026:
5006:
4974:
4954:
4901:
4884:{\displaystyle G'}
4881:
4856:
4836:{\displaystyle G'}
4833:
4808:
4788:
4768:
4742:
4722:
4702:
4682:
4662:
4645:{\displaystyle G'}
4642:
4613:
4486:
4412:
4336:
4316:
4244:
4220:
4171:
4151:
4131:
4107:
4076:
4039:
4013:
3970:
3944:
3899:
3879:
3859:
3842:{\displaystyle E'}
3839:
3811:
3744:
3720:
3681:
3649:
3637:consolidated sink
3625:
3601:
3581:
3523:
3465:
3435:
3366:
3330:
3285:
3201:
3128:
3042:
2956:
2885:
2856:
2778:
2739:
2707:
2652:
2618:
2575:
2442:
2358:
2286:
2266:
2244:
2182:
2126:
2074:
2027:
1967:
1921:
1844:
1818:
1776:
1719:
1666:
1616:Linear programming
1595:may be infinite).
1585:
1561:
1541:
1521:
1493:
1467:
1447:
1427:
1401:
1357:
1337:
1311:
1291:
1281:words, if we send
1267:
1228:
1211:
1158:
1086:
1036:
995:
938:
921:
849:
732:
691:
638:
595:
541:
503:
473:
445:then its value on
435:
415:
392:
372:
350:be a network with
340:
306:
262:
198:
43:
33:1) and/or a dog (w
9694:978-0-486-41453-9
9638:Daniel D. Sleator
9441:978-0-262-03293-3
9096:978-3-540-50517-4
8688:978-1-61197-338-9
8630:978-0-7695-5135-7
8589:Flows in Networks
8491:978-1-4020-8116-3
8212:
8177:
8072:
8052:
8023:
7994:
7846:
7826:
7797:
7768:
7609:
7143:
7091:
6993:to the sink node
6794:{\displaystyle k}
6774:{\displaystyle k}
6754:{\displaystyle k}
6731:{\displaystyle t}
6711:{\displaystyle s}
6691:{\displaystyle 1}
6671:{\displaystyle G}
6613:{\displaystyle t}
6593:{\displaystyle s}
6570:{\displaystyle k}
6550:{\displaystyle k}
6530:{\displaystyle 1}
6510:{\displaystyle N}
6490:{\displaystyle t}
6470:{\displaystyle s}
6450:{\displaystyle G}
6389:{\displaystyle t}
6369:{\displaystyle s}
6349:{\displaystyle t}
6329:{\displaystyle s}
6316:and two vertices
6257:
6230:
6180:{\displaystyle v}
6154:
6133:{\displaystyle v}
6107:
6080:
6053:
6006:{\displaystyle N}
5986:{\displaystyle N}
5882:
5865:{\displaystyle f}
5721:{\displaystyle m}
5701:{\displaystyle n}
5681:{\displaystyle n}
5661:{\displaystyle v}
5603:{\displaystyle v}
5545:{\displaystyle u}
5499:{\displaystyle C}
5476:{\displaystyle C}
5456:{\displaystyle C}
5432:{\displaystyle C}
5412:{\displaystyle v}
5392:{\displaystyle C}
5372:{\displaystyle v}
5320:
5295:{\displaystyle C}
5275:{\displaystyle v}
5255:{\displaystyle C}
5235:{\displaystyle v}
5211:{\displaystyle C}
5191:{\displaystyle v}
5171:{\displaystyle M}
5115:
5089:{\displaystyle C}
5069:{\displaystyle m}
5049:{\displaystyle C}
5029:{\displaystyle C}
4977:{\displaystyle M}
4904:{\displaystyle C}
4859:{\displaystyle M}
4811:{\displaystyle G}
4791:{\displaystyle n}
4745:{\displaystyle m}
4725:{\displaystyle C}
4705:{\displaystyle G}
4685:{\displaystyle m}
4665:{\displaystyle M}
4544:
4529:
4481:
4456:
4438:
4407:
4382:
4364:
4339:{\displaystyle G}
4298:
4283:
4247:{\displaystyle V}
4174:{\displaystyle 1}
4154:{\displaystyle G}
4134:{\displaystyle N}
4117:(See Fig. 4.3.1).
3902:{\displaystyle Y}
3882:{\displaystyle X}
3862:{\displaystyle G}
3747:{\displaystyle G}
3652:{\displaystyle T}
3628:{\displaystyle S}
3604:{\displaystyle N}
3380:
3379:
3112:
3099:
3080:
2994:
2911:
2694:
2556:
2428:
2351:
2289:{\displaystyle t}
2269:{\displaystyle s}
2239:
1989:Dinic's algorithm
1741:Dinic's algorithm
1588:{\displaystyle U}
1564:{\displaystyle U}
1544:{\displaystyle E}
1524:{\displaystyle V}
1450:{\displaystyle u}
1360:{\displaystyle u}
1347:units of flow on
1314:{\displaystyle u}
1294:{\displaystyle x}
1263:
1188:
1175:
1135:
1122:
866:
794:
438:{\displaystyle N}
418:{\displaystyle g}
395:{\displaystyle N}
274:minimum-cost flow
9724:
9698:
9676:
9650:
9642:Robert E. Tarjan
9633:
9623:
9586:
9585:
9559:
9539:
9533:
9532:
9530:
9528:
9514:
9508:
9507:
9505:
9503:
9494:. Archived from
9484:
9478:
9477:
9475:
9469:Carl Kingsford.
9466:
9460:
9459:
9453:
9445:
9422:Ronald L. Rivest
9414:Thomas H. Cormen
9410:
9401:
9400:
9362:
9356:
9355:
9327:
9321:
9320:
9318:
9306:
9300:
9299:
9297:
9285:
9279:
9278:
9270:
9264:
9263:
9255:
9249:
9248:
9244:
9242:
9241:
9236:
9231:
9230:
9226:
9196:
9190:
9189:
9179:
9150:
9144:
9143:
9115:
9109:
9108:
9074:
9068:
9067:
9057:
9024:
9013:
9012:
8994:
8989:
8987:
8986:
8981:
8976:
8975:
8970:
8961:
8939:
8933:
8932:
8930:
8928:
8913:
8907:
8906:
8904:
8902:
8888:
8882:
8881:
8879:
8877:
8862:
8856:
8855:
8853:
8841:
8835:
8834:
8832:
8830:
8815:
8809:
8808:
8806:
8794:
8785:
8784:
8756:
8740:
8731:
8730:
8728:
8726:
8714:
8708:
8707:
8706:on 3 March 2016.
8705:
8699:. Archived from
8672:
8658:
8649:
8643:
8642:
8614:
8598:
8592:
8585:
8576:
8575:
8573:
8548:Fulkerson, D. R.
8540:
8531:
8530:
8528:
8522:. Archived from
8517:
8505:
8496:
8495:
8469:
8463:
8462:
8444:
8424:
8403:strongly NP-hard
8399:strongly NP-hard
8351:cost-coefficient
8282:
8280:
8279:
8274:
8254:
8243:
8242:
8232:
8208:
8207:
8197:
8142:
8140:
8139:
8134:
8132:
8131:
8118:
8108:
8082:
8073:
8070:
8048:
8047:
8037:
8019:
8018:
8008:
7975:
7916:
7914:
7913:
7908:
7906:
7905:
7892:
7882:
7856:
7847:
7844:
7822:
7821:
7811:
7793:
7792:
7782:
7644:
7642:
7641:
7636:
7634:
7633:
7623:
7599:
7572:
7565:
7558:
7551:
7547:
7540:
7537:Add a sink node
7533:
7526:
7519:
7512:
7505:
7497:
7480:
7477:. A matching in
7476:
7467:is connected to
7466:
7440:
7257:
7168:
7166:
7165:
7160:
7158:
7157:
7144:
7142:
7131:
7049:
7025:
6980:
6943:
6925:
6800:
6798:
6797:
6792:
6780:
6778:
6777:
6772:
6760:
6758:
6757:
6752:
6737:
6735:
6734:
6729:
6717:
6715:
6714:
6709:
6697:
6695:
6694:
6689:
6677:
6675:
6674:
6669:
6657:
6655:
6654:
6649:
6619:
6617:
6616:
6611:
6599:
6597:
6596:
6591:
6576:
6574:
6573:
6568:
6556:
6554:
6553:
6548:
6536:
6534:
6533:
6528:
6516:
6514:
6513:
6508:
6496:
6494:
6493:
6488:
6476:
6474:
6473:
6468:
6456:
6454:
6453:
6448:
6436:
6434:
6433:
6428:
6395:
6393:
6392:
6387:
6375:
6373:
6372:
6367:
6355:
6353:
6352:
6347:
6335:
6333:
6332:
6327:
6315:
6313:
6312:
6307:
6269:
6267:
6266:
6261:
6259:
6258:
6255:
6242:
6240:
6239:
6234:
6232:
6231:
6228:
6215:
6213:
6212:
6207:
6186:
6184:
6183:
6178:
6166:
6164:
6163:
6158:
6156:
6155:
6152:
6139:
6137:
6136:
6131:
6119:
6117:
6116:
6111:
6109:
6108:
6105:
6092:
6090:
6089:
6084:
6082:
6081:
6078:
6065:
6063:
6062:
6057:
6055:
6054:
6051:
6038:
6036:
6035:
6030:
6012:
6010:
6009:
6004:
5992:
5990:
5989:
5984:
5969:
5967:
5966:
5961:
5910:
5909:
5896:
5871:
5869:
5868:
5863:
5851:
5849:
5848:
5843:
5838:
5837:
5832:
5807:
5805:
5804:
5799:
5753:
5751:
5750:
5745:
5727:
5725:
5724:
5719:
5707:
5705:
5704:
5699:
5687:
5685:
5684:
5679:
5667:
5665:
5664:
5659:
5647:
5645:
5644:
5639:
5609:
5607:
5606:
5601:
5589:
5587:
5586:
5581:
5551:
5549:
5548:
5543:
5531:
5529:
5528:
5523:
5505:
5503:
5502:
5497:
5482:
5480:
5479:
5474:
5462:
5460:
5459:
5454:
5438:
5436:
5435:
5430:
5418:
5416:
5415:
5410:
5398:
5396:
5395:
5390:
5378:
5376:
5375:
5370:
5358:
5356:
5355:
5350:
5348:
5333:
5331:
5330:
5325:
5323:
5322:
5321:
5318:
5301:
5299:
5298:
5293:
5281:
5279:
5278:
5273:
5261:
5259:
5258:
5253:
5241:
5239:
5238:
5233:
5217:
5215:
5214:
5209:
5197:
5195:
5194:
5189:
5177:
5175:
5174:
5169:
5153:
5151:
5150:
5145:
5143:
5128:
5126:
5125:
5120:
5118:
5117:
5116:
5113:
5095:
5093:
5092:
5087:
5075:
5073:
5072:
5067:
5055:
5053:
5052:
5047:
5035:
5033:
5032:
5027:
5016:is contained in
5015:
5013:
5012:
5007:
4984:, then the edge
4983:
4981:
4980:
4975:
4963:
4961:
4960:
4955:
4953:
4952:
4951:
4935:
4934:
4933:
4910:
4908:
4907:
4902:
4890:
4888:
4887:
4882:
4880:
4865:
4863:
4862:
4857:
4842:
4840:
4839:
4834:
4832:
4817:
4815:
4814:
4809:
4797:
4795:
4794:
4789:
4777:
4775:
4774:
4769:
4751:
4749:
4748:
4743:
4731:
4729:
4728:
4723:
4711:
4709:
4708:
4703:
4691:
4689:
4688:
4683:
4671:
4669:
4668:
4663:
4651:
4649:
4648:
4643:
4641:
4622:
4620:
4619:
4614:
4585:
4584:
4569:
4568:
4547:
4546:
4545:
4542:
4532:
4531:
4530:
4527:
4511:
4495:
4493:
4492:
4487:
4482:
4479:
4459:
4458:
4457:
4454:
4441:
4440:
4439:
4436:
4421:
4419:
4418:
4413:
4408:
4405:
4385:
4384:
4383:
4380:
4367:
4366:
4365:
4362:
4345:
4343:
4342:
4337:
4325:
4323:
4322:
4317:
4312:
4301:
4300:
4299:
4296:
4286:
4285:
4284:
4281:
4268:
4253:
4251:
4250:
4245:
4229:
4227:
4226:
4221:
4180:
4178:
4177:
4172:
4160:
4158:
4157:
4152:
4140:
4138:
4137:
4132:
4116:
4114:
4113:
4108:
4106:
4085:
4083:
4082:
4077:
4048:
4046:
4045:
4040:
4022:
4020:
4019:
4014:
4012:
3979:
3977:
3976:
3971:
3953:
3951:
3950:
3945:
3943:
3908:
3906:
3905:
3900:
3888:
3886:
3885:
3880:
3868:
3866:
3865:
3860:
3848:
3846:
3845:
3840:
3838:
3820:
3818:
3817:
3812:
3807:
3753:
3751:
3750:
3745:
3729:
3727:
3726:
3721:
3658:
3656:
3655:
3650:
3634:
3632:
3631:
3626:
3610:
3608:
3607:
3602:
3590:
3588:
3587:
3582:
3577:
3576:
3558:
3557:
3532:
3530:
3529:
3524:
3519:
3518:
3500:
3499:
3474:
3472:
3471:
3466:
3437:Given a network
3375:
3373:
3372:
3367:
3365:
3364:
3339:
3337:
3336:
3331:
3329:
3328:
3294:
3292:
3291:
3286:
3251:
3250:
3246:
3210:
3208:
3207:
3202:
3188:
3187:
3137:
3135:
3134:
3129:
3115:
3114:
3113:
3105:
3100:
3092:
3082:
3081:
3073:
3051:
3049:
3048:
3043:
3026:
3025:
3021:
2996:
2995:
2987:
2965:
2963:
2962:
2957:
2952:
2951:
2947:
2934:
2933:
2929:
2913:
2912:
2904:
2894:
2892:
2891:
2886:
2884:
2883:
2865:
2863:
2862:
2857:
2855:
2854:
2850:
2837:
2836:
2817:
2787:
2785:
2784:
2779:
2777:
2776:
2748:
2746:
2745:
2740:
2716:
2714:
2713:
2708:
2703:
2702:
2695:
2687:
2661:
2659:
2658:
2653:
2627:
2625:
2624:
2619:
2584:
2582:
2581:
2576:
2574:
2570:
2557:
2552:
2551:
2542:
2528:
2527:
2523:
2507:
2506:
2502:
2451:
2449:
2448:
2443:
2441:
2437:
2430:
2429:
2427:
2410:
2367:
2365:
2364:
2359:
2357:
2353:
2352:
2347:
2346:
2337:
2295:
2293:
2292:
2287:
2275:
2273:
2272:
2267:
2253:
2251:
2250:
2245:
2240:
2235:
2233:
2232:
2204:maximum distance
2191:
2189:
2188:
2183:
2178:
2177:
2135:
2133:
2132:
2127:
2119:
2118:
2083:
2081:
2080:
2075:
2036:
2034:
2033:
2028:
1976:
1974:
1973:
1968:
1963:
1962:
1930:
1928:
1927:
1922:
1911:
1910:
1906:
1890:
1889:
1885:
1853:
1851:
1850:
1845:
1827:
1825:
1824:
1819:
1785:
1783:
1782:
1777:
1769:
1768:
1728:
1726:
1725:
1720:
1715:
1714:
1675:
1673:
1672:
1667:
1598:
1597:
1594:
1592:
1591:
1586:
1570:
1568:
1567:
1562:
1550:
1548:
1547:
1542:
1530:
1528:
1527:
1522:
1502:
1500:
1499:
1494:
1476:
1474:
1473:
1468:
1456:
1454:
1453:
1448:
1436:
1434:
1433:
1428:
1410:
1408:
1407:
1402:
1366:
1364:
1363:
1358:
1346:
1344:
1343:
1338:
1320:
1318:
1317:
1312:
1300:
1298:
1297:
1292:
1276:
1274:
1273:
1268:
1266:
1265:
1264:
1261:
1237:
1235:
1234:
1229:
1224:
1223:
1210:
1186:
1171:
1170:
1157:
1133:
1118:
1110:
1096:it is given by:
1095:
1093:
1092:
1087:
1085:
1084:
1079:
1045:
1043:
1042:
1037:
1004:
1002:
1001:
996:
994:
993:
975:
974:
947:
945:
944:
939:
934:
933:
920:
913:
912:
862:
861:
848:
841:
840:
741:
739:
738:
733:
700:
698:
697:
692:
690:
689:
674:
673:
647:
645:
644:
639:
637:
604:
602:
601:
596:
591:
590:
585:
550:
548:
547:
542:
512:
510:
509:
504:
502:
501:
482:
480:
479:
474:
444:
442:
441:
436:
424:
422:
421:
416:
401:
399:
398:
393:
381:
379:
378:
373:
349:
347:
346:
341:
271:
269:
268:
263:
258:
257:
237:
228:
207:
205:
204:
199:
194:
186:
181:
173:
118:Flows in Network
9732:
9731:
9727:
9726:
9725:
9723:
9722:
9721:
9702:
9701:
9695:
9648:
9600:Joseph Cheriyan
9595:
9593:Further reading
9590:
9589:
9557:10.1.1.414.4496
9540:
9536:
9526:
9524:
9516:
9515:
9511:
9501:
9499:
9486:
9485:
9481:
9473:
9467:
9463:
9447:
9446:
9442:
9411:
9404:
9389:10.1137/1008062
9363:
9359:
9328:
9324:
9307:
9303:
9286:
9282:
9271:
9267:
9256:
9252:
9222:
9218:
9214:
9206:
9203:
9202:
9197:
9193:
9154:Goldberg, A. V.
9151:
9147:
9116:
9112:
9097:
9075:
9071:
9028:Goldberg, A. V.
9025:
9016:
8992:
8971:
8966:
8965:
8957:
8949:
8946:
8945:
8940:
8936:
8926:
8924:
8914:
8910:
8900:
8898:
8890:
8889:
8885:
8875:
8873:
8871:Quanta Magazine
8863:
8859:
8842:
8838:
8828:
8826:
8824:Quanta Magazine
8816:
8812:
8795:
8788:
8773:
8754:10.1.1.259.5759
8741:
8734:
8724:
8722:
8715:
8711:
8703:
8689:
8663:. p. 217.
8656:
8650:
8646:
8631:
8599:
8595:
8586:
8579:
8541:
8534:
8526:
8515:
8506:
8499:
8492:
8470:
8466:
8425:
8416:
8411:
8375:
8371:
8364:
8357:
8349:,v) also has a
8337:
8329:
8314:
8303:
8247:
8238:
8234:
8216:
8203:
8199:
8181:
8154:
8151:
8150:
8124:
8120:
8116:
8115:
8104:
8078:
8075:
8074:
8069:
8056:
8043:
8039:
8027:
8014:
8010:
7998:
7968:
7966:
7963:
7962:
7956:
7941:
7930:
7898:
7894:
7890:
7889:
7878:
7852:
7849:
7848:
7843:
7830:
7817:
7813:
7801:
7788:
7784:
7772:
7745:
7742:
7741:
7719:
7708:
7697:
7683:
7679:
7675:
7658:
7629:
7625:
7613:
7604:
7601:
7600:
7593:
7571:
7567:
7564:
7560:
7557:
7553:
7549:
7546:
7542:
7538:
7532:
7528:
7525:
7521:
7518:
7514:
7511:
7507:
7503:
7495:
7492:
7478:
7468:
7458:
7426:
7423:
7375:
7366:
7357:
7345:
7336:
7320:
7308:
7283:
7274:
7245:
7207:
7150:
7146:
7132:
7097:
7095:
7062:
7059:
7058:
7048:
7039:
7031:
7024:
7015:
7006:
6998:
6970:
6968:
6933:
6931:
6913:
6897:is eliminated.
6888:
6877:
6868:
6842:
6837:
6829:closure problem
6813:
6811:Closure problem
6807:
6805:Closure problem
6786:
6783:
6782:
6766:
6763:
6762:
6746:
6743:
6742:
6723:
6720:
6719:
6703:
6700:
6699:
6683:
6680:
6679:
6663:
6660:
6659:
6625:
6622:
6621:
6605:
6602:
6601:
6585:
6582:
6581:
6562:
6559:
6558:
6542:
6539:
6538:
6522:
6519:
6518:
6502:
6499:
6498:
6482:
6479:
6478:
6462:
6459:
6458:
6442:
6439:
6438:
6404:
6401:
6400:
6381:
6378:
6377:
6361:
6358:
6357:
6341:
6338:
6337:
6321:
6318:
6317:
6283:
6280:
6279:
6276:
6254:
6250:
6248:
6245:
6244:
6227:
6223:
6221:
6218:
6217:
6192:
6189:
6188:
6172:
6169:
6168:
6151:
6147:
6145:
6142:
6141:
6125:
6122:
6121:
6104:
6100:
6098:
6095:
6094:
6077:
6073:
6071:
6068:
6067:
6050:
6046:
6044:
6041:
6040:
6039:is replaced by
6018:
6015:
6014:
5998:
5995:
5994:
5978:
5975:
5974:
5902:
5898:
5886:
5880:
5877:
5876:
5857:
5854:
5853:
5833:
5828:
5827:
5813:
5810:
5809:
5775:
5772:
5771:
5760:
5733:
5730:
5729:
5713:
5710:
5709:
5693:
5690:
5689:
5673:
5670:
5669:
5653:
5650:
5649:
5615:
5612:
5611:
5595:
5592:
5591:
5557:
5554:
5553:
5537:
5534:
5533:
5511:
5508:
5507:
5491:
5488:
5487:
5468:
5465:
5464:
5448:
5445:
5444:
5424:
5421:
5420:
5404:
5401:
5400:
5384:
5381:
5380:
5364:
5361:
5360:
5341:
5339:
5336:
5335:
5317:
5316:
5312:
5310:
5307:
5306:
5287:
5284:
5283:
5267:
5264:
5263:
5247:
5244:
5243:
5227:
5224:
5223:
5218:; or it can be
5203:
5200:
5199:
5183:
5180:
5179:
5163:
5160:
5159:
5136:
5134:
5131:
5130:
5112:
5111:
5107:
5105:
5102:
5101:
5081:
5078:
5077:
5061:
5058:
5057:
5041:
5038:
5037:
5021:
5018:
5017:
4989:
4986:
4985:
4969:
4966:
4965:
4964:are matched in
4944:
4943:
4939:
4923:
4922:
4918:
4916:
4913:
4912:
4896:
4893:
4892:
4873:
4871:
4868:
4867:
4851:
4848:
4847:
4825:
4823:
4820:
4819:
4803:
4800:
4799:
4783:
4780:
4779:
4757:
4754:
4753:
4737:
4734:
4733:
4717:
4714:
4713:
4697:
4694:
4693:
4692:if and only if
4677:
4674:
4673:
4657:
4654:
4653:
4652:has a matching
4634:
4632:
4629:
4628:
4577:
4573:
4558:
4554:
4541:
4540:
4536:
4526:
4525:
4521:
4504:
4502:
4499:
4498:
4478:
4453:
4452:
4448:
4435:
4434:
4430:
4428:
4425:
4424:
4404:
4379:
4378:
4374:
4361:
4360:
4356:
4354:
4351:
4350:
4331:
4328:
4327:
4305:
4295:
4294:
4290:
4280:
4279:
4275:
4261:
4259:
4256:
4255:
4239:
4236:
4235:
4197:
4194:
4193:
4187:
4166:
4163:
4162:
4146:
4143:
4142:
4126:
4123:
4122:
4099:
4091:
4088:
4087:
4056:
4053:
4052:
4028:
4025:
4024:
4005:
3985:
3982:
3981:
3959:
3956:
3955:
3936:
3916:
3913:
3912:
3894:
3891:
3890:
3874:
3871:
3870:
3854:
3851:
3850:
3831:
3829:
3826:
3825:
3800:
3759:
3756:
3755:
3739:
3736:
3735:
3691:
3688:
3687:
3685:bipartite graph
3673:
3659:(also known as
3644:
3641:
3640:
3620:
3617:
3616:
3596:
3593:
3592:
3572:
3568:
3553:
3549:
3538:
3535:
3534:
3514:
3510:
3495:
3491:
3480:
3477:
3476:
3442:
3439:
3438:
3427:
3422:
3392:
3351:
3347:
3345:
3342:
3341:
3309:
3305:
3303:
3300:
3299:
3242:
3238:
3234:
3217:
3214:
3213:
3168:
3164:
3156:
3153:
3152:
3104:
3091:
3090:
3086:
3072:
3071:
3069:
3066:
3065:
3017:
3013:
3009:
2986:
2985:
2983:
2980:
2979:
2943:
2939:
2935:
2925:
2921:
2917:
2903:
2902:
2900:
2897:
2896:
2879:
2875:
2873:
2870:
2869:
2846:
2842:
2838:
2813:
2809:
2805:
2803:
2800:
2799:
2766:
2762:
2754:
2751:
2750:
2722:
2719:
2718:
2686:
2685:
2681:
2667:
2664:
2663:
2635:
2632:
2631:
2601:
2598:
2597:
2547:
2543:
2541:
2519:
2515:
2511:
2498:
2494:
2490:
2477:
2473:
2468:
2465:
2464:
2414:
2409:
2405:
2398:
2394:
2389:
2386:
2385:
2342:
2338:
2336:
2323:
2319:
2314:
2311:
2310:
2281:
2278:
2277:
2261:
2258:
2257:
2234:
2228:
2224:
2216:
2213:
2212:
2173:
2169:
2161:
2158:
2157:
2114:
2110:
2102:
2099:
2098:
2048:
2045:
2044:
2001:
1998:
1997:
1958:
1954:
1946:
1943:
1942:
1902:
1898:
1894:
1881:
1877:
1873:
1859:
1856:
1855:
1833:
1830:
1829:
1801:
1798:
1797:
1764:
1760:
1752:
1749:
1748:
1710:
1706:
1695:
1692:
1691:
1649:
1646:
1645:
1580:
1577:
1576:
1556:
1553:
1552:
1536:
1533:
1532:
1516:
1513:
1512:
1509:
1482:
1479:
1478:
1462:
1459:
1458:
1442:
1439:
1438:
1416:
1413:
1412:
1372:
1369:
1368:
1352:
1349:
1348:
1326:
1323:
1322:
1306:
1303:
1302:
1286:
1283:
1282:
1260:
1259:
1255:
1253:
1250:
1249:
1216:
1212:
1179:
1163:
1159:
1126:
1114:
1106:
1104:
1101:
1100:
1080:
1075:
1074:
1060:
1057:
1056:
1010:
1007:
1006:
986:
982:
967:
963:
961:
958:
957:
926:
922:
905:
901:
870:
854:
850:
833:
829:
798:
758:
755:
754:
706:
703:
702:
682:
678:
666:
662:
660:
657:
656:
633:
619:
616:
615:
586:
581:
580:
566:
563:
562:
518:
515:
514:
494:
490:
488:
485:
484:
450:
447:
446:
430:
427:
426:
410:
407:
406:
387:
384:
383:
355:
352:
351:
317:
314:
313:
290:
238:
233:
232:
224:
216:
213:
212:
190:
182:
177:
169:
161:
158:
157:
86:
17:
12:
11:
5:
9730:
9720:
9719:
9714:
9700:
9699:
9693:
9677:
9659:(3): 362â391.
9634:
9621:10.1.1.42.8563
9614:(5): 239â242.
9594:
9591:
9588:
9587:
9550:(1): 109â119.
9534:
9509:
9479:
9461:
9440:
9426:Clifford Stein
9402:
9375:(3): 302â308.
9357:
9338:(3): 277â286.
9322:
9301:
9280:
9265:
9250:
9234:
9229:
9225:
9221:
9217:
9213:
9210:
9191:
9145:
9126:(3): 447â474.
9110:
9095:
9069:
9014:
9003:(6): 277â278.
8979:
8974:
8969:
8964:
8960:
8956:
8953:
8934:
8908:
8883:
8857:
8836:
8810:
8786:
8771:
8732:
8709:
8687:
8644:
8629:
8593:
8577:
8532:
8497:
8490:
8464:
8442:10.1.1.23.5134
8435:(3): 437â445.
8413:
8412:
8410:
8407:
8373:
8369:
8362:
8355:
8336:
8333:
8327:
8312:
8301:
8284:
8283:
8272:
8269:
8266:
8263:
8260:
8257:
8253:
8250:
8246:
8241:
8237:
8231:
8228:
8225:
8222:
8219:
8215:
8211:
8206:
8202:
8196:
8193:
8190:
8187:
8184:
8180:
8176:
8173:
8170:
8167:
8164:
8161:
8158:
8144:
8143:
8130:
8127:
8123:
8114:
8111:
8107:
8103:
8100:
8097:
8094:
8091:
8088:
8085:
8081:
8077:
8076:
8071: adjacent
8068:
8065:
8062:
8059:
8058:
8055:
8051:
8046:
8042:
8036:
8033:
8030:
8026:
8022:
8017:
8013:
8007:
8004:
8001:
7997:
7993:
7990:
7987:
7984:
7981:
7978:
7974:
7971:
7954:
7939:
7928:
7919:
7918:
7904:
7901:
7897:
7888:
7885:
7881:
7877:
7874:
7871:
7868:
7865:
7862:
7859:
7855:
7851:
7850:
7845: adjacent
7842:
7839:
7836:
7833:
7832:
7829:
7825:
7820:
7816:
7810:
7807:
7804:
7800:
7796:
7791:
7787:
7781:
7778:
7775:
7771:
7767:
7764:
7761:
7758:
7755:
7752:
7749:
7717:
7706:
7695:
7684:are all equal.
7681:
7677:
7673:
7657:
7654:
7647:
7646:
7632:
7628:
7622:
7619:
7616:
7612:
7608:
7575:
7574:
7569:
7562:
7555:
7552:with capacity
7544:
7535:
7530:
7523:
7516:
7513:with capacity
7509:
7491:
7488:
7424:
7393:
7392:
7381:
7371:
7362:
7353:
7347:
7341:
7332:
7326:
7316:
7310:
7304:
7279:
7270:
7206:
7203:
7171:
7170:
7156:
7153:
7149:
7141:
7138:
7135:
7130:
7127:
7124:
7121:
7118:
7115:
7112:
7109:
7106:
7103:
7100:
7094:
7090:
7087:
7084:
7081:
7078:
7075:
7072:
7069:
7066:
7044:
7035:
7020:
7011:
7002:
6964:
6927:
6886:
6873:
6864:
6841:
6838:
6836:
6833:
6809:Main article:
6806:
6803:
6790:
6770:
6750:
6727:
6707:
6687:
6667:
6647:
6644:
6641:
6638:
6635:
6632:
6629:
6609:
6589:
6566:
6557:iff there are
6546:
6526:
6506:
6486:
6466:
6446:
6426:
6423:
6420:
6417:
6414:
6411:
6408:
6385:
6365:
6345:
6325:
6305:
6302:
6299:
6296:
6293:
6290:
6287:
6275:
6272:
6253:
6226:
6205:
6202:
6199:
6196:
6176:
6150:
6129:
6103:
6076:
6049:
6028:
6025:
6022:
6013:. First, each
6002:
5982:
5971:
5970:
5959:
5956:
5953:
5950:
5947:
5944:
5941:
5938:
5935:
5932:
5929:
5925:
5922:
5919:
5916:
5913:
5908:
5905:
5901:
5895:
5892:
5889:
5885:
5861:
5841:
5836:
5831:
5826:
5823:
5820:
5817:
5797:
5794:
5791:
5788:
5785:
5782:
5779:
5759:
5756:
5743:
5740:
5737:
5717:
5697:
5677:
5657:
5637:
5634:
5631:
5628:
5625:
5622:
5619:
5599:
5579:
5576:
5573:
5570:
5567:
5564:
5561:
5541:
5521:
5518:
5515:
5495:
5472:
5452:
5441:
5440:
5428:
5408:
5388:
5368:
5347:
5344:
5315:
5303:
5291:
5271:
5251:
5231:
5207:
5187:
5167:
5154:can either be
5142:
5139:
5110:
5085:
5076:. To see that
5065:
5045:
5025:
5005:
5002:
4999:
4996:
4993:
4973:
4950:
4947:
4942:
4938:
4932:
4929:
4926:
4921:
4900:
4879:
4876:
4855:
4831:
4828:
4807:
4787:
4767:
4764:
4761:
4741:
4721:
4701:
4681:
4661:
4640:
4637:
4625:
4624:
4612:
4609:
4606:
4603:
4600:
4597:
4594:
4591:
4588:
4583:
4580:
4576:
4572:
4567:
4564:
4561:
4557:
4553:
4550:
4539:
4535:
4524:
4520:
4517:
4514:
4510:
4507:
4496:
4485:
4477:
4474:
4471:
4468:
4465:
4462:
4451:
4447:
4444:
4433:
4422:
4411:
4403:
4400:
4397:
4394:
4391:
4388:
4377:
4373:
4370:
4359:
4335:
4315:
4311:
4308:
4304:
4293:
4289:
4278:
4274:
4271:
4267:
4264:
4243:
4219:
4216:
4213:
4210:
4207:
4204:
4201:
4186:
4183:
4170:
4150:
4130:
4119:
4118:
4105:
4102:
4098:
4095:
4075:
4072:
4069:
4066:
4063:
4060:
4050:
4038:
4035:
4032:
4011:
4008:
4004:
4001:
3998:
3995:
3992:
3989:
3969:
3966:
3963:
3942:
3939:
3935:
3932:
3929:
3926:
3923:
3920:
3910:
3898:
3878:
3869:directed from
3858:
3837:
3834:
3810:
3806:
3803:
3799:
3796:
3793:
3790:
3787:
3784:
3781:
3778:
3775:
3772:
3769:
3766:
3763:
3743:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3672:
3669:
3648:
3624:
3600:
3580:
3575:
3571:
3567:
3564:
3561:
3556:
3552:
3548:
3545:
3542:
3522:
3517:
3513:
3509:
3506:
3503:
3498:
3494:
3490:
3487:
3484:
3464:
3461:
3458:
3455:
3452:
3449:
3446:
3426:
3423:
3421:
3418:
3402:
3401:
3391:
3388:
3378:
3377:
3363:
3360:
3357:
3354:
3350:
3327:
3324:
3321:
3318:
3315:
3312:
3308:
3296:
3284:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3260:
3257:
3254:
3249:
3245:
3241:
3237:
3233:
3230:
3227:
3224:
3221:
3200:
3197:
3194:
3191:
3186:
3183:
3180:
3177:
3174:
3171:
3167:
3163:
3160:
3150:
3147:
3143:
3142:
3138:
3127:
3124:
3121:
3118:
3111:
3108:
3103:
3098:
3095:
3089:
3085:
3079:
3076:
3063:
3060:
3056:
3055:
3052:
3041:
3038:
3035:
3032:
3029:
3024:
3020:
3016:
3012:
3008:
3005:
3002:
2999:
2993:
2990:
2977:
2974:
2968:
2967:
2955:
2950:
2946:
2942:
2938:
2932:
2928:
2924:
2920:
2916:
2910:
2907:
2882:
2878:
2866:
2853:
2849:
2845:
2841:
2835:
2832:
2829:
2826:
2823:
2820:
2816:
2812:
2808:
2797:
2794:
2790:
2789:
2775:
2772:
2769:
2765:
2761:
2758:
2738:
2735:
2732:
2729:
2726:
2706:
2701:
2698:
2693:
2690:
2684:
2680:
2677:
2674:
2671:
2651:
2648:
2645:
2642:
2639:
2628:
2617:
2614:
2611:
2608:
2605:
2595:
2592:
2588:
2587:
2585:
2573:
2569:
2566:
2563:
2560:
2555:
2550:
2546:
2540:
2537:
2534:
2531:
2526:
2522:
2518:
2514:
2510:
2505:
2501:
2497:
2493:
2489:
2486:
2483:
2480:
2476:
2472:
2462:
2459:
2455:
2454:
2452:
2440:
2436:
2433:
2426:
2423:
2420:
2417:
2413:
2408:
2404:
2401:
2397:
2393:
2383:
2380:
2376:
2375:
2368:
2356:
2350:
2345:
2341:
2335:
2332:
2329:
2326:
2322:
2318:
2308:
2305:
2298:
2297:
2285:
2265:
2254:
2243:
2238:
2231:
2227:
2223:
2220:
2210:
2207:
2196:
2195:
2192:
2181:
2176:
2172:
2168:
2165:
2155:
2152:
2141:
2140:
2136:
2125:
2122:
2117:
2113:
2109:
2106:
2096:
2093:
2086:
2085:
2073:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2037:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
1995:
1992:
1985:
1984:
1981:original paper
1977:
1966:
1961:
1957:
1953:
1950:
1940:
1937:
1933:
1932:
1920:
1917:
1914:
1909:
1905:
1901:
1897:
1893:
1888:
1884:
1880:
1876:
1872:
1869:
1866:
1863:
1843:
1840:
1837:
1817:
1814:
1811:
1808:
1805:
1794:residual graph
1786:
1775:
1772:
1767:
1763:
1759:
1756:
1746:
1743:
1737:
1736:
1729:
1718:
1713:
1709:
1705:
1702:
1699:
1689:
1686:
1680:
1679:
1676:
1665:
1662:
1659:
1656:
1653:
1643:
1640:
1634:
1633:
1630:linear program
1622:
1620:
1618:
1612:
1611:
1608:
1605:
1602:
1584:
1560:
1540:
1520:
1508:
1505:
1492:
1489:
1486:
1466:
1446:
1426:
1423:
1420:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1356:
1336:
1333:
1330:
1310:
1290:
1258:
1239:
1238:
1227:
1222:
1219:
1215:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1185:
1182:
1178:
1174:
1169:
1166:
1162:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1132:
1129:
1125:
1121:
1117:
1113:
1109:
1083:
1078:
1073:
1070:
1067:
1064:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
992:
989:
985:
981:
978:
973:
970:
966:
951:
950:
949:
948:
937:
932:
929:
925:
919:
916:
911:
908:
904:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
869:
865:
860:
857:
853:
847:
844:
839:
836:
832:
828:
825:
822:
819:
816:
813:
810:
807:
804:
801:
797:
792:
789:
786:
783:
780:
777:
774:
771:
768:
765:
762:
749:
748:
742:
731:
728:
725:
722:
719:
716:
713:
710:
688:
685:
681:
677:
672:
669:
665:
636:
632:
629:
626:
623:
594:
589:
584:
579:
576:
573:
570:
552:
551:
540:
537:
534:
531:
528:
525:
522:
500:
497:
493:
483:is denoted by
472:
469:
466:
463:
460:
457:
454:
434:
414:
403:
391:
371:
368:
365:
362:
359:
339:
336:
333:
330:
327:
324:
321:
289:
286:
261:
256:
253:
250:
247:
244:
241:
236:
231:
227:
223:
220:
197:
193:
189:
185:
180:
176:
172:
168:
165:
154:James B. Orlin
116:In their book
85:
82:
15:
9:
6:
4:
3:
2:
9729:
9718:
9715:
9713:
9710:
9709:
9707:
9696:
9690:
9686:
9682:
9681:Eugene Lawler
9678:
9674:
9670:
9666:
9662:
9658:
9654:
9647:
9643:
9639:
9635:
9631:
9627:
9622:
9617:
9613:
9609:
9605:
9604:Kurt Mehlhorn
9601:
9597:
9596:
9583:
9579:
9575:
9571:
9567:
9563:
9558:
9553:
9549:
9545:
9538:
9523:
9519:
9513:
9497:
9493:
9489:
9483:
9472:
9465:
9457:
9451:
9443:
9437:
9433:
9432:
9427:
9423:
9419:
9415:
9409:
9407:
9398:
9394:
9390:
9386:
9382:
9378:
9374:
9370:
9369:
9361:
9353:
9349:
9345:
9341:
9337:
9333:
9326:
9317:
9312:
9305:
9296:
9291:
9284:
9276:
9269:
9261:
9254:
9246:
9227:
9223:
9219:
9215:
9208:
9195:
9187:
9183:
9178:
9173:
9169:
9165:
9164:
9159:
9155:
9149:
9141:
9137:
9133:
9129:
9125:
9121:
9114:
9106:
9102:
9098:
9092:
9088:
9084:
9080:
9073:
9065:
9061:
9056:
9051:
9047:
9043:
9042:
9037:
9033:
9032:Tarjan, R. E.
9029:
9023:
9021:
9019:
9010:
9006:
9002:
8998:
8991:
8972:
8962:
8951:
8938:
8923:
8919:
8912:
8897:
8893:
8887:
8872:
8868:
8861:
8852:
8847:
8840:
8825:
8821:
8814:
8805:
8800:
8793:
8791:
8782:
8778:
8774:
8772:9781450320290
8768:
8764:
8760:
8755:
8750:
8746:
8739:
8737:
8720:
8713:
8702:
8698:
8694:
8690:
8684:
8680:
8676:
8671:
8666:
8662:
8655:
8648:
8640:
8636:
8632:
8626:
8622:
8618:
8613:
8608:
8604:
8597:
8590:
8584:
8582:
8572:
8567:
8563:
8559:
8558:
8553:
8549:
8545:
8539:
8537:
8525:
8521:
8514:
8510:
8509:Harris, T. E.
8504:
8502:
8493:
8487:
8483:
8479:
8475:
8468:
8460:
8456:
8452:
8448:
8443:
8438:
8434:
8430:
8423:
8421:
8419:
8414:
8406:
8404:
8400:
8396:
8392:
8388:
8383:
8381:
8377:
8365:
8358:
8352:
8348:
8345:, each edge (
8344:
8343:
8332:
8330:
8323:
8319:
8315:
8308:
8304:
8297:
8288:
8270:
8264:
8261:
8258:
8251:
8248:
8244:
8239:
8235:
8229:
8226:
8223:
8220:
8217:
8213:
8209:
8204:
8200:
8194:
8191:
8188:
8185:
8182:
8178:
8174:
8168:
8165:
8162:
8156:
8149:
8148:
8147:
8128:
8125:
8121:
8112:
8109:
8098:
8095:
8092:
8086:
8083:
8066:
8063:
8060:
8053:
8049:
8044:
8040:
8034:
8031:
8028:
8024:
8020:
8015:
8011:
8005:
8002:
7999:
7995:
7991:
7985:
7982:
7979:
7972:
7969:
7961:
7960:
7959:
7957:
7950:
7946:
7942:
7935:
7931:
7924:
7902:
7899:
7895:
7886:
7883:
7872:
7869:
7866:
7860:
7857:
7840:
7837:
7834:
7827:
7823:
7818:
7814:
7808:
7805:
7802:
7798:
7794:
7789:
7785:
7779:
7776:
7773:
7769:
7765:
7759:
7756:
7753:
7747:
7740:
7739:
7738:
7736:
7732:
7728:
7724:
7720:
7713:
7709:
7702:
7698:
7691:
7670:
7662:
7653:
7650:
7630:
7626:
7620:
7617:
7614:
7610:
7606:
7597:
7592:
7591:
7590:
7588:
7584:
7580:
7536:
7501:
7500:
7499:
7487:
7484:
7475:
7471:
7465:
7461:
7456:
7453:after flight
7452:
7448:
7444:
7438:
7434:
7430:
7420:
7418:
7415:with at most
7414:
7410:
7406:
7402:
7398:
7390:
7386:
7382:
7379:
7374:
7370:
7365:
7361:
7356:
7352:
7348:
7344:
7340:
7335:
7331:
7327:
7324:
7319:
7315:
7311:
7307:
7303:
7299:
7295:
7294:
7293:
7291:
7287:
7282:
7278:
7273:
7269:
7265:
7261:
7256:
7252:
7248:
7243:
7239:
7235:
7230:
7228:
7224:
7219:
7217:
7213:
7202:
7200:
7196:
7192:
7188:
7184:
7180:
7176:
7154:
7151:
7147:
7139:
7136:
7133:
7122:
7116:
7113:
7107:
7104:
7101:
7098:
7092:
7088:
7079:
7073:
7070:
7064:
7057:
7056:
7055:
7053:
7047:
7043:
7038:
7034:
7029:
7023:
7019:
7014:
7010:
7005:
7001:
6996:
6992:
6988:
6984:
6978:
6974:
6967:
6963:
6959:
6955:
6951:
6947:
6941:
6937:
6930:
6924:
6920:
6916:
6911:
6907:
6903:
6898:
6896:
6892:
6885:
6881:
6876:
6872:
6867:
6863:
6859:
6855:
6846:
6832:
6830:
6826:
6822:
6818:
6812:
6802:
6788:
6768:
6761:, or at most
6748:
6739:
6725:
6705:
6685:
6665:
6642:
6639:
6636:
6630:
6627:
6607:
6587:
6578:
6564:
6544:
6524:
6504:
6484:
6464:
6444:
6421:
6418:
6415:
6409:
6406:
6397:
6383:
6363:
6343:
6323:
6300:
6297:
6294:
6288:
6285:
6271:
6251:
6224:
6200:
6194:
6174:
6148:
6127:
6101:
6074:
6047:
6026:
6023:
6020:
6000:
5980:
5957:
5951:
5948:
5945:
5936:
5933:
5930:
5920:
5914:
5911:
5906:
5903:
5899:
5893:
5890:
5887:
5883:
5875:
5874:
5873:
5859:
5839:
5834:
5821:
5818:
5815:
5792:
5789:
5786:
5780:
5777:
5764:
5755:
5741:
5738:
5735:
5715:
5695:
5675:
5655:
5635:
5632:
5626:
5623:
5620:
5597:
5577:
5574:
5568:
5565:
5562:
5539:
5519:
5516:
5513:
5493:
5484:
5470:
5450:
5426:
5406:
5386:
5366:
5345:
5342:
5313:
5304:
5289:
5269:
5249:
5229:
5221:
5205:
5185:
5165:
5157:
5140:
5137:
5108:
5099:
5098:
5097:
5083:
5063:
5043:
5023:
5000:
4997:
4994:
4971:
4940:
4936:
4919:
4898:
4877:
4874:
4853:
4844:
4829:
4826:
4805:
4785:
4778:paths, where
4765:
4762:
4759:
4739:
4719:
4699:
4679:
4659:
4638:
4635:
4607:
4604:
4598:
4595:
4592:
4586:
4581:
4578:
4574:
4570:
4565:
4562:
4559:
4555:
4551:
4537:
4533:
4522:
4512:
4508:
4505:
4497:
4475:
4472:
4469:
4466:
4463:
4460:
4449:
4442:
4431:
4423:
4401:
4398:
4395:
4392:
4389:
4386:
4375:
4368:
4357:
4349:
4348:
4347:
4333:
4309:
4306:
4302:
4291:
4287:
4276:
4269:
4265:
4262:
4241:
4233:
4214:
4211:
4208:
4202:
4199:
4192:
4182:
4168:
4148:
4128:
4103:
4100:
4096:
4093:
4073:
4070:
4064:
4058:
4051:
4036:
4033:
4030:
4009:
4006:
4002:
3996:
3993:
3990:
3967:
3964:
3961:
3940:
3937:
3933:
3927:
3924:
3921:
3911:
3896:
3876:
3856:
3835:
3832:
3824:
3823:
3822:
3804:
3801:
3797:
3791:
3788:
3785:
3779:
3776:
3773:
3770:
3764:
3761:
3741:
3733:
3714:
3711:
3708:
3705:
3702:
3696:
3693:
3686:
3677:
3668:
3666:
3662:
3646:
3638:
3622:
3614:
3598:
3573:
3569:
3565:
3562:
3559:
3554:
3550:
3543:
3540:
3515:
3511:
3507:
3504:
3501:
3496:
3492:
3485:
3482:
3459:
3456:
3453:
3447:
3444:
3431:
3417:
3415:
3414:combinatorial
3411:
3407:
3400:
3397:
3396:
3395:
3387:
3385:
3358:
3352:
3348:
3322:
3316:
3313:
3310:
3306:
3297:
3295:
3282:
3279:
3276:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3247:
3243:
3239:
3235:
3228:
3225:
3222:
3219:
3195:
3192:
3189:
3181:
3175:
3172:
3169:
3165:
3158:
3151:
3148:
3145:
3144:
3139:
3122:
3119:
3116:
3109:
3106:
3101:
3096:
3093:
3087:
3074:
3064:
3061:
3058:
3057:
3053:
3036:
3033:
3030:
3022:
3018:
3014:
3010:
3006:
3003:
2988:
2978:
2975:
2973:
2970:
2969:
2948:
2944:
2940:
2936:
2930:
2926:
2922:
2918:
2905:
2880:
2876:
2867:
2851:
2847:
2843:
2839:
2830:
2824:
2821:
2818:
2814:
2810:
2806:
2798:
2795:
2792:
2791:
2773:
2770:
2767:
2763:
2759:
2756:
2733:
2730:
2724:
2699:
2696:
2691:
2688:
2682:
2675:
2672:
2669:
2646:
2643:
2637:
2629:
2612:
2609:
2603:
2596:
2593:
2590:
2589:
2586:
2571:
2567:
2564:
2561:
2558:
2553:
2548:
2544:
2538:
2535:
2532:
2524:
2520:
2516:
2512:
2508:
2503:
2499:
2495:
2491:
2481:
2478:
2474:
2470:
2463:
2460:
2457:
2456:
2453:
2438:
2434:
2431:
2424:
2421:
2418:
2415:
2411:
2406:
2402:
2399:
2395:
2391:
2384:
2381:
2378:
2377:
2373:
2369:
2354:
2348:
2343:
2339:
2333:
2330:
2327:
2324:
2320:
2316:
2309:
2306:
2303:
2300:
2299:
2283:
2263:
2255:
2236:
2229:
2225:
2218:
2211:
2208:
2205:
2201:
2198:
2197:
2193:
2174:
2170:
2163:
2156:
2153:
2150:
2146:
2143:
2142:
2137:
2120:
2115:
2111:
2104:
2097:
2094:
2092:
2088:
2087:
2068:
2065:
2062:
2059:
2056:
2050:
2042:
2041:dynamic trees
2038:
2021:
2018:
2015:
2012:
2009:
2003:
1996:
1993:
1990:
1987:
1986:
1982:
1978:
1959:
1955:
1948:
1941:
1938:
1935:
1934:
1915:
1907:
1903:
1899:
1895:
1891:
1886:
1882:
1878:
1874:
1861:
1841:
1838:
1835:
1812:
1809:
1803:
1795:
1791:
1787:
1770:
1765:
1761:
1754:
1747:
1744:
1742:
1739:
1738:
1734:
1730:
1711:
1707:
1703:
1697:
1690:
1687:
1685:
1682:
1681:
1677:
1660:
1657:
1651:
1644:
1641:
1639:
1636:
1635:
1631:
1627:
1623:
1621:
1619:
1617:
1614:
1613:
1609:
1606:
1603:
1600:
1599:
1596:
1582:
1574:
1558:
1538:
1518:
1504:
1490:
1487:
1484:
1444:
1421:
1418:
1395:
1392:
1389:
1386:
1383:
1377:
1354:
1334:
1331:
1328:
1308:
1288:
1278:
1256:
1247:
1243:
1225:
1220:
1217:
1213:
1207:
1204:
1198:
1195:
1192:
1183:
1180:
1176:
1172:
1167:
1164:
1160:
1154:
1151:
1145:
1142:
1139:
1130:
1127:
1123:
1119:
1111:
1099:
1098:
1097:
1081:
1068:
1065:
1062:
1054:
1053:value of flow
1050:
1046:
1033:
1030:
1027:
1021:
1018:
1015:
990:
987:
983:
979:
976:
971:
968:
964:
955:
935:
930:
927:
923:
917:
914:
909:
906:
902:
898:
895:
892:
886:
883:
880:
874:
871:
867:
863:
858:
855:
851:
845:
842:
837:
834:
830:
826:
823:
820:
814:
811:
808:
802:
799:
795:
790:
784:
781:
778:
769:
766:
763:
753:
752:
751:
750:
746:
743:
729:
726:
723:
717:
714:
711:
686:
683:
679:
675:
670:
667:
663:
654:
651:
650:
649:
627:
624:
621:
613:
609:
605:
592:
587:
574:
571:
568:
560:
556:
538:
532:
529:
526:
520:
498:
495:
491:
470:
467:
461:
458:
455:
432:
412:
404:
402:respectively.
389:
369:
366:
363:
360:
357:
334:
331:
328:
322:
319:
311:
310:
309:
303:
299:
294:
285:
283:
279:
275:
251:
245:
242:
239:
229:
218:
209:
187:
174:
163:
155:
150:
148:
144:
140:
134:
132:
126:
121:
119:
113:
108:
106:
102:
98:
93:
91:
81:
79:
75:
71:
67:
63:
58:
56:
52:
48:
40:
36:
32:
28:
23:
19:
9684:
9656:
9652:
9611:
9607:
9547:
9543:
9537:
9525:. Retrieved
9521:
9512:
9500:. Retrieved
9496:the original
9491:
9482:
9464:
9430:
9372:
9366:
9360:
9335:
9331:
9325:
9304:
9283:
9274:
9268:
9259:
9253:
9200:
9194:
9167:
9161:
9148:
9123:
9119:
9113:
9078:
9072:
9045:
9039:
9000:
8996:
8937:
8925:. Retrieved
8921:
8911:
8899:. Retrieved
8895:
8886:
8874:. Retrieved
8870:
8860:
8839:
8827:. Retrieved
8823:
8813:
8744:
8723:. Retrieved
8712:
8701:the original
8660:
8647:
8602:
8596:
8588:
8561:
8555:
8524:the original
8519:
8473:
8467:
8432:
8428:
8394:
8390:
8386:
8384:
8379:
8367:
8360:
8353:
8350:
8346:
8340:
8338:
8325:
8324:with weight
8321:
8317:
8310:
8306:
8299:
8295:
8293:
8145:
7952:
7948:
7944:
7937:
7933:
7926:
7922:
7920:
7734:
7730:
7726:
7722:
7715:
7711:
7704:
7700:
7693:
7687:
7651:
7648:
7595:
7586:
7582:
7578:
7576:
7493:
7482:
7473:
7469:
7463:
7459:
7454:
7450:
7446:
7442:
7436:
7432:
7428:
7421:
7416:
7412:
7408:
7404:
7400:
7396:
7394:
7388:
7384:
7377:
7372:
7368:
7367:, if source
7363:
7359:
7354:
7350:
7342:
7338:
7333:
7329:
7322:
7317:
7313:
7305:
7301:
7297:
7289:
7285:
7280:
7276:
7271:
7267:
7263:
7259:
7254:
7250:
7246:
7241:
7237:
7233:
7231:
7227:network flow
7220:
7215:
7211:
7208:
7198:
7194:
7190:
7186:
7182:
7178:
7174:
7172:
7051:
7045:
7041:
7036:
7032:
7027:
7021:
7017:
7012:
7008:
7003:
6999:
6994:
6990:
6986:
6982:
6976:
6972:
6965:
6961:
6957:
6953:
6949:
6945:
6939:
6935:
6928:
6922:
6918:
6914:
6909:
6905:
6901:
6899:
6894:
6890:
6883:
6879:
6874:
6870:
6865:
6861:
6857:
6851:
6828:
6824:
6820:
6816:
6814:
6740:
6579:
6398:
6277:
5972:
5769:
5485:
5442:
5399:; otherwise
5219:
5155:
5100:Each vertex
4845:
4626:
4188:
4120:
3682:
3664:
3660:
3636:
3612:
3436:
3409:
3403:
3398:
3393:
3381:
3211:
2972:
2371:
2203:
2148:
1610:Description
1575:capacities,
1510:
1411:we can send
1279:
1245:
1241:
1240:
1052:
1048:
1047:
953:
952:
744:
652:
611:
607:
606:
558:
554:
553:
307:
301:
297:
210:
151:
135:
130:
128:
123:
117:
115:
110:
94:
90:T. E. Harris
87:
59:
55:flow network
50:
44:
38:
34:
30:
26:
18:
9527:21 December
9522:pearson.com
9502:22 December
9368:SIAM Review
8892:"FOCS 2022"
8564:: 399â404.
8544:Ford, L. R.
7951:, we loose
5156:non-matched
4732:containing
3661:supersource
3420:Application
1607:Complexity
1242:Definition.
1049:Definition.
608:Definition.
555:Definition.
208:algorithm.
9706:Categories
9316:2101.07233
9295:2101.05719
9170:(5): 783.
9048:(4): 921.
8927:25 January
8901:25 January
8876:25 January
8851:2203.03456
8804:2203.00671
8721:. MIT News
8409:References
8339:1. In the
8335:Extensions
7690:segmenting
4752:edges and
3410:every edge
1628:. See the
1626:legal flow
1573:irrational
1507:Algorithms
288:Definition
9673:0022-0000
9616:CiteSeerX
9574:1382-6905
9552:CiteSeerX
9450:cite book
9352:1097-0037
9105:0302-9743
8781:207205207
8749:CiteSeerX
8725:8 January
8670:1304.2338
8612:1304.2077
8437:CiteSeerX
8320:to pixel
8245:−
8227:∪
8221:∈
8214:∑
8192:∪
8186:∈
8179:∑
8087:∩
8054:∑
8032:∈
8025:∑
8003:∈
7996:∑
7861:∩
7828:∑
7824:−
7806:∈
7799:∑
7777:∈
7770:∑
7618:∈
7611:∑
7300:and each
7117:−
7108:∈
7093:∑
7074:−
6024:∈
5940:∖
5934:∈
5928:∀
5912:≤
5891:∈
5884:∑
5825:→
5739:−
5633:∈
5575:∈
5517:−
5506:has size
4843:instead.
4763:−
4605:∈
4587:∣
4571:×
4552:∈
4473:∧
4467:∈
4461:∣
4399:∧
4393:∈
4387:∣
4288:∪
4097:∈
4086:for each
4034:∈
4023:for each
4003:∈
3965:∈
3954:for each
3934:∈
3780:∪
3774:∪
3706:∪
3665:supersink
3563:…
3505:…
3280:
3268:
3262:
3253:
3226:
3193:
3120:
3102:−
3078:~
3034:
2992:~
2909:~
2877:ℓ
2774:ϵ
2700:ϵ
2697:−
2673:≤
2662:time for
2565:
2559:⋅
2539:
2533:⋅
2482:⋅
2432:
2422:
2334:
2066:
2019:
1839:−
1465:Δ
1437:units on
1425:Δ
1393:−
1378:∈
1375:Δ
1205:∈
1177:∑
1152:∈
1124:∑
1072:→
1028:∈
980:−
893:∈
868:∑
821:∈
796:∑
773:∖
767:∈
761:∀
724:∈
676:≤
631:→
614:is a map
578:→
468:∈
367:∈
300:and sink
95:In 1955,
9644:(1983).
9332:Networks
9064:52152408
9034:(1988).
8697:10733914
8639:14681906
8550:(1956).
8459:10210675
8252:′
8146:because
7973:′
7445:and set
7403:between
6854:baseball
6093:, where
5346:′
5141:′
4878:′
4830:′
4672:of size
4639:′
4509:′
4346:, where
4310:′
4266:′
4189:Given a
4104:′
4010:′
3941:′
3836:′
3821:, where
3805:′
3683:Given a
2089:General
1005:for all
701:for all
559:capacity
272:for the
152:In 2013
143:Goldberg
9582:6598669
9397:2028206
9377:Bibcode
7419:crews.
7266:, node
7218:crews.
6852:In the
6817:closure
6457:, with
5220:matched
1792:on the
1601:Method
84:History
74:s-t cut
9691:
9671:
9618:
9580:
9572:
9554:
9492:GitLab
9438:
9424:, and
9395:
9350:
9184:
9138:
9103:
9093:
9062:
8829:8 June
8779:
8769:
8751:
8695:
8685:
8637:
8627:
8488:
8457:
8439:
7559:where
7520:where
7050:. Let
6827:. The
3635:and a
1931:time.
1632:here.
1187:
1134:
954:Remark
147:Tarjan
112:other.
66:source
9649:(PDF)
9578:S2CID
9474:(PDF)
9393:JSTOR
9311:arXiv
9290:arXiv
9186:96030
9182:S2CID
9140:15493
9136:S2CID
9060:S2CID
8993:(PDF)
8846:arXiv
8799:arXiv
8777:S2CID
8704:(PDF)
8693:S2CID
8665:arXiv
8657:(PDF)
8635:S2CID
8607:arXiv
8527:(PDF)
8516:(PDF)
8455:S2CID
6948:<
6944:with
6658:from
6437:from
5610:, or
4326:from
3149:2022
3062:2021
2976:2020
2796:2020
2594:2013
2461:1998
2382:1994
2307:1988
2209:1988
2202:with
2154:1988
2147:with
2095:1986
1994:1983
1939:1978
1745:1970
1688:1970
1642:1955
1604:Year
68:s to
9689:ISBN
9669:ISSN
9640:and
9602:and
9570:ISSN
9529:2019
9504:2019
9456:link
9436:ISBN
9348:ISSN
9245:Time
9101:ISSN
9091:ISBN
8944:"An
8929:2023
8903:2023
8878:2023
8831:2022
8767:ISBN
8727:2014
8683:ISBN
8625:ISBN
8486:ISBN
8389:: a
7947:and
7725:and
7577:Let
7407:and
7387:and
7358:and
7337:and
7321:and
7232:Let
7137:<
6985:and
6900:Let
6882:and
6600:and
6477:and
6336:and
6243:and
6140:and
6066:and
5770:Let
3980:and
3663:and
2760:>
2749:for
2372:path
2149:FIFO
2039:The
1531:and
1332:>
1244:The
1051:The
915:>
843:>
612:flow
557:The
312:Let
145:and
99:and
70:sink
9661:doi
9626:doi
9562:doi
9385:doi
9340:doi
9172:doi
9128:doi
9083:doi
9050:doi
9005:doi
8759:doi
8675:doi
8617:doi
8566:doi
8478:doi
8447:doi
7581:= (
7548:to
7427:= (
7399:in
7236:= (
7197:to
7185:â {
6952:to
6904:= (
6718:to
6376:to
6256:out
6153:out
6079:out
5379:in
5334:in
5282:in
5242:in
5198:in
5158:in
5129:in
5114:out
5056:is
4866:of
4528:out
4381:out
4363:out
4282:out
3889:to
3734:in
3277:log
3265:log
3259:log
3236:log
3223:exp
3190:log
3117:log
3110:328
3031:log
2562:log
2536:log
2485:min
2419:log
2407:log
2331:log
2276:or
2063:log
2016:log
1868:min
1262:max
513:or
405:If
141:of
45:In
9708::
9667:.
9657:26
9655:.
9651:.
9624:.
9612:69
9610:.
9576:.
9568:.
9560:.
9548:26
9546:.
9520:.
9490:.
9452:}}
9448:{{
9420:,
9416:,
9405:^
9391:.
9383:.
9371:.
9346:.
9336:12
9334:.
9180:.
9168:45
9166:.
9160:.
9134:.
9124:17
9122:.
9099:.
9089:.
9058:.
9046:35
9044:.
9038:.
9030:;
9017:^
8999:.
8995:.
8920:.
8894:.
8869:.
8822:.
8789:^
8775:.
8765:.
8757:.
8735:^
8691:.
8681:.
8673:.
8659:.
8633:.
8623:.
8615:.
8580:^
8560:.
8554:.
8546:;
8535:^
8518:.
8500:^
8484:.
8453:.
8445:.
8433:91
8431:.
8417:^
8374:uv
8370:uv
8363:uv
8356:uv
8328:ij
7955:ij
7733:,
7718:ij
7682:ij
7585:,
7479:G'
7457:,
7435:,
7431:âȘ
7425:G'
7292::
7253:â
7240:,
7201:.
7040:+
7016:â
7007:+
6975:,
6966:ij
6938:,
6929:ij
6921:â
6908:,
6887:ij
6815:A
6801:.
6738:.
6229:in
6106:in
6052:in
5754:.
5319:in
4543:in
4455:in
4437:in
4297:in
3386:.
2966:.
2923:10
2788:.
2692:15
2689:16
2084:.
1983:.
1735:.
1503:.
610:A
284:.
80:.
49:,
9697:.
9675:.
9663::
9632:.
9628::
9584:.
9564::
9531:.
9506:.
9476:.
9458:)
9444:.
9399:.
9387::
9379::
9373:8
9354:.
9342::
9319:.
9313::
9298:.
9292::
9233:)
9228:3
9224:/
9220:4
9216:m
9212:(
9209:O
9188:.
9174::
9142:.
9130::
9107:.
9085::
9066:.
9052::
9011:.
9007::
9001:7
8978:)
8973:3
8968:|
8963:V
8959:|
8955:(
8952:O
8931:.
8905:.
8880:.
8854:.
8848::
8833:.
8807:.
8801::
8783:.
8761::
8729:.
8677::
8667::
8641:.
8619::
8609::
8574:.
8568::
8562:8
8494:.
8480::
8461:.
8449::
8380:d
8376:.
8372:f
8368:a
8361:f
8354:a
8347:u
8326:p
8322:j
8318:i
8313:i
8311:b
8307:i
8302:i
8300:a
8296:i
8271:.
8268:)
8265:B
8262:,
8259:A
8256:(
8249:q
8240:i
8236:b
8230:B
8224:A
8218:i
8210:+
8205:i
8201:a
8195:B
8189:A
8183:i
8175:=
8172:)
8169:B
8166:,
8163:A
8160:(
8157:q
8129:j
8126:i
8122:p
8113:1
8110:=
8106:|
8102:}
8099:j
8096:,
8093:i
8090:{
8084:A
8080:|
8067:j
8064:,
8061:i
8050:+
8045:i
8041:a
8035:B
8029:i
8021:+
8016:i
8012:b
8006:A
8000:i
7992:=
7989:)
7986:B
7983:,
7980:A
7977:(
7970:q
7953:p
7949:j
7945:i
7940:i
7938:b
7934:B
7929:i
7927:a
7923:A
7917:,
7903:j
7900:i
7896:p
7887:1
7884:=
7880:|
7876:}
7873:j
7870:,
7867:i
7864:{
7858:A
7854:|
7841:j
7838:,
7835:i
7819:i
7815:b
7809:B
7803:i
7795:+
7790:i
7786:a
7780:A
7774:i
7766:=
7763:)
7760:B
7757:,
7754:A
7751:(
7748:q
7735:B
7731:A
7727:j
7723:i
7716:p
7712:i
7707:i
7705:b
7701:i
7696:i
7694:a
7678:i
7674:i
7645:.
7631:i
7627:d
7621:v
7615:i
7607:=
7598:)
7596:G
7587:E
7583:V
7579:G
7573:.
7570:i
7568:v
7563:i
7561:d
7556:i
7554:d
7550:t
7545:i
7543:v
7539:t
7534:.
7531:i
7529:f
7524:i
7522:p
7517:i
7515:p
7510:i
7508:f
7504:s
7496:c
7483:F
7474:B
7472:â
7470:j
7464:A
7462:â
7460:i
7455:i
7451:j
7447:B
7443:A
7439:)
7437:E
7433:B
7429:A
7417:k
7413:F
7409:t
7405:s
7401:G
7397:k
7391:.
7389:t
7385:s
7380:.
7378:i
7373:j
7369:s
7364:j
7360:s
7355:i
7351:d
7346:.
7343:i
7339:d
7334:i
7330:s
7325:.
7323:t
7318:i
7314:d
7309:.
7306:i
7302:s
7298:s
7290:E
7286:i
7281:i
7277:d
7272:i
7268:s
7264:V
7260:i
7255:V
7251:t
7249:,
7247:s
7242:E
7238:V
7234:G
7216:k
7212:F
7199:t
7195:s
7191:G
7187:k
7183:S
7181:(
7179:r
7175:k
7169:.
7155:j
7152:i
7148:r
7140:j
7134:i
7129:}
7126:}
7123:k
7120:{
7114:S
7111:{
7105:j
7102:,
7099:i
7089:=
7086:)
7083:}
7080:k
7077:{
7071:S
7068:(
7065:r
7052:S
7046:k
7042:r
7037:k
7033:w
7028:i
7022:i
7018:w
7013:k
7009:r
7004:k
7000:w
6995:t
6991:i
6987:j
6983:i
6979:}
6977:j
6973:i
6971:{
6962:r
6958:s
6954:V
6950:j
6946:i
6942:}
6940:j
6936:i
6934:{
6923:V
6919:t
6917:,
6915:s
6910:E
6906:V
6902:G
6895:k
6891:j
6884:r
6880:i
6875:i
6871:r
6866:i
6862:w
6858:n
6825:C
6821:C
6789:k
6769:k
6749:k
6726:t
6706:s
6686:1
6666:G
6646:)
6643:E
6640:,
6637:V
6634:(
6631:=
6628:N
6608:t
6588:s
6565:k
6545:k
6525:1
6505:N
6485:t
6465:s
6445:G
6425:)
6422:E
6419:,
6416:V
6413:(
6410:=
6407:N
6384:t
6364:s
6344:t
6324:s
6304:)
6301:E
6298:,
6295:V
6292:(
6289:=
6286:G
6252:v
6225:v
6204:)
6201:v
6198:(
6195:c
6175:v
6149:v
6128:v
6102:v
6075:v
6048:v
6027:V
6021:v
6001:N
5981:N
5958:.
5955:}
5952:t
5949:,
5946:s
5943:{
5937:V
5931:v
5924:)
5921:v
5918:(
5915:c
5907:v
5904:i
5900:f
5894:V
5888:i
5860:f
5840:,
5835:+
5830:R
5822:V
5819::
5816:c
5796:)
5793:E
5790:,
5787:V
5784:(
5781:=
5778:N
5742:m
5736:n
5716:m
5696:n
5676:n
5656:v
5636:E
5630:)
5627:u
5624:,
5621:v
5618:(
5598:v
5578:E
5572:)
5569:v
5566:,
5563:u
5560:(
5540:u
5520:m
5514:n
5494:C
5471:C
5451:C
5439:.
5427:C
5407:v
5387:C
5367:v
5343:G
5314:v
5302:.
5290:C
5270:v
5250:C
5230:v
5206:C
5186:v
5166:M
5138:G
5109:v
5084:C
5064:m
5044:C
5024:C
5004:)
5001:v
4998:,
4995:u
4992:(
4972:M
4949:n
4946:i
4941:v
4937:,
4931:t
4928:u
4925:o
4920:u
4899:C
4875:G
4854:M
4827:G
4806:G
4786:n
4766:m
4760:n
4740:m
4720:C
4700:G
4680:m
4660:M
4636:G
4623:.
4611:}
4608:E
4602:)
4599:v
4596:,
4593:u
4590:(
4582:n
4579:i
4575:V
4566:t
4563:u
4560:o
4556:V
4549:)
4538:v
4534:,
4523:u
4519:(
4516:{
4513:=
4506:E
4484:}
4476:v
4470:V
4464:v
4450:v
4446:{
4443:=
4432:V
4410:}
4402:v
4396:V
4390:v
4376:v
4372:{
4369:=
4358:V
4334:G
4314:)
4307:E
4303:,
4292:V
4277:V
4273:(
4270:=
4263:G
4242:V
4218:)
4215:E
4212:,
4209:V
4206:(
4203:=
4200:G
4169:1
4149:G
4129:N
4101:E
4094:e
4074:1
4071:=
4068:)
4065:e
4062:(
4059:c
4049:.
4037:Y
4031:y
4007:E
4000:)
3997:t
3994:,
3991:y
3988:(
3968:X
3962:x
3938:E
3931:)
3928:x
3925:,
3922:s
3919:(
3909:.
3897:Y
3877:X
3857:G
3833:E
3809:)
3802:E
3798:,
3795:}
3792:t
3789:,
3786:s
3783:{
3777:Y
3771:X
3768:(
3765:=
3762:N
3742:G
3718:)
3715:E
3712:,
3709:Y
3703:X
3700:(
3697:=
3694:G
3647:T
3623:S
3599:N
3579:}
3574:m
3570:t
3566:,
3560:,
3555:1
3551:t
3547:{
3544:=
3541:T
3521:}
3516:n
3512:s
3508:,
3502:,
3497:1
3493:s
3489:{
3486:=
3483:S
3463:)
3460:E
3457:,
3454:V
3451:(
3448:=
3445:N
3362:)
3359:1
3356:(
3353:o
3349:E
3326:)
3323:1
3320:(
3317:o
3314:+
3311:1
3307:E
3283:U
3274:)
3271:E
3256:E
3248:8
3244:/
3240:7
3232:(
3229:O
3220:E
3199:)
3196:U
3185:)
3182:1
3179:(
3176:o
3173:+
3170:1
3166:E
3162:(
3159:O
3126:)
3123:U
3107:1
3097:2
3094:3
3088:E
3084:(
3075:O
3040:)
3037:U
3028:)
3023:2
3019:/
3015:3
3011:V
3007:+
3004:E
3001:(
2998:(
2989:O
2954:)
2949:7
2945:/
2941:1
2937:U
2931:7
2927:/
2919:E
2915:(
2906:O
2881:p
2852:3
2848:/
2844:1
2840:U
2834:)
2831:1
2828:(
2825:o
2822:+
2819:3
2815:/
2811:4
2807:E
2771:+
2768:1
2764:V
2757:E
2737:)
2734:E
2731:V
2728:(
2725:O
2705:)
2683:V
2679:(
2676:O
2670:E
2650:)
2647:E
2644:V
2641:(
2638:O
2616:)
2613:E
2610:V
2607:(
2604:O
2572:)
2568:U
2554:E
2549:2
2545:V
2530:}
2525:2
2521:/
2517:1
2513:E
2509:,
2504:3
2500:/
2496:2
2492:V
2488:{
2479:E
2475:(
2471:O
2439:)
2435:V
2425:V
2416:V
2412:E
2403:E
2400:V
2396:(
2392:O
2355:)
2349:E
2344:2
2340:V
2328:E
2325:V
2321:(
2317:O
2284:t
2264:s
2242:)
2237:E
2230:2
2226:V
2222:(
2219:O
2180:)
2175:3
2171:V
2167:(
2164:O
2124:)
2121:E
2116:2
2112:V
2108:(
2105:O
2072:)
2069:V
2060:E
2057:V
2054:(
2051:O
2025:)
2022:V
2013:E
2010:V
2007:(
2004:O
1965:)
1960:3
1956:V
1952:(
1949:O
1919:)
1916:E
1913:}
1908:2
1904:/
1900:1
1896:E
1892:,
1887:3
1883:/
1879:2
1875:V
1871:{
1865:(
1862:O
1842:1
1836:V
1816:)
1813:E
1810:V
1807:(
1804:O
1774:)
1771:E
1766:2
1762:V
1758:(
1755:O
1717:)
1712:2
1708:E
1704:V
1701:(
1698:O
1664:)
1661:U
1658:E
1655:(
1652:O
1583:U
1559:U
1539:E
1519:V
1491:y
1488:,
1485:x
1445:u
1422:+
1419:x
1399:]
1396:x
1390:y
1387:,
1384:0
1381:[
1355:u
1335:x
1329:y
1309:u
1289:x
1257:f
1226:.
1221:t
1218:u
1214:f
1208:E
1202:)
1199:t
1196:,
1193:u
1190:(
1184::
1181:u
1173:=
1168:v
1165:s
1161:f
1155:E
1149:)
1146:v
1143:,
1140:s
1137:(
1131::
1128:v
1120:=
1116:|
1112:f
1108:|
1082:+
1077:R
1069:E
1066::
1063:f
1034:.
1031:E
1025:)
1022:v
1019:,
1016:u
1013:(
991:u
988:v
984:f
977:=
972:v
969:u
965:f
936:.
931:u
928:v
924:f
918:0
910:u
907:v
903:f
899:,
896:E
890:)
887:u
884:,
881:v
878:(
875::
872:u
864:=
859:v
856:u
852:f
846:0
838:v
835:u
831:f
827:,
824:E
818:)
815:v
812:,
809:u
806:(
803::
800:u
791::
788:}
785:t
782:,
779:s
776:{
770:V
764:v
730:.
727:E
721:)
718:v
715:,
712:u
709:(
687:v
684:u
680:c
671:v
668:u
664:f
635:R
628:E
625::
622:f
593:.
588:+
583:R
575:E
572::
569:c
539:.
536:)
533:v
530:,
527:u
524:(
521:g
499:v
496:u
492:g
471:E
465:)
462:v
459:,
456:u
453:(
433:N
413:g
390:N
370:V
364:t
361:,
358:s
338:)
335:E
332:,
329:V
326:(
323:=
320:N
302:t
298:s
260:)
255:)
252:1
249:(
246:o
243:+
240:1
235:|
230:E
226:|
222:(
219:O
196:)
192:|
188:E
184:|
179:|
175:V
171:|
167:(
164:O
125:.
39:i
35:i
31:i
27:i
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
