View source for Maximum flow problem

{{Short description|Computational problem in graph theory}}
{{Use dmy dates|date=May 2022}}
[[File:Pets flow.svg|alt=Flow network for the problem: Each human (r''i'') is willing to adopt a cat (w''i''1) and/or a dog (w''i''2). However each pet (p''i'') 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.|thumb|252x252px|Flow network for the problem: Each human (r''i'') is willing to adopt a cat (w''i''1) and/or a dog (w''i''2). However each pet (p''i'') 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.]]
In [[Optimization (mathematics)|optimization theory]], '''maximum flow problems''' involve finding a feasible flow through a [[flow network]] that obtains the maximum possible flow rate.

The maximum flow problem can be seen as a special case of more [[complex network]] flow problems, such as the [[circulation problem]]. The maximum value of an s-t flow (i.e., flow from [[Glossary of graph theory#Direction|source]] s to [[Glossary of graph theory#Direction|sink]] t) is equal to the minimum capacity of an [[Cut (graph theory)|s-t cut]] (i.e., cut severing s from t) in the network, as stated in the [[max-flow min-cut theorem]].

==History==
The maximum flow problem was first formulated in 1954 by [[Ted Harris (mathematician)|T. E. Harris]] and F. S. Ross as a simplified model of Soviet railway traffic flow.<ref name=":0">{{Cite journal | last1 = Schrijver | first1 = A. | title = On the history of the transportation and maximum flow problems | doi = 10.1007/s101070100259 | journal = Mathematical Programming | volume = 91 | issue = 3 | pages = 437–445 | year = 2002 | citeseerx = 10.1.1.23.5134 | s2cid = 10210675 }}</ref><ref>{{Cite book | doi = 10.1007/0-387-25837-X_5 | first1 =  Saul I. | last1 = Gass| first2 = Arjang A. | last2 = Assad | chapter = Mathematical, algorithmic and professional developments of operations research from 1951 to 1956 | title = An Annotated Timeline of Operations Research | series = International Series in Operations Research & Management Science | volume = 75 | pages = 79–110 | year = 2005 | isbn = 978-1-4020-8116-3 }}</ref><ref name=":2">{{cite journal | first1 = T. E. | last1 = Harris | author-link1 = Ted Harris (mathematician) | first2 = F. S. | last2 = Ross | year = 1955 | title = Fundamentals of a Method for Evaluating Rail Net Capacities | journal = Research Memorandum| url = http://www.dtic.mil/dtic/tr/fulltext/u2/093458.pdf| archive-url = https://web.archive.org/web/20140108021700/http://www.dtic.mil/dtic/tr/fulltext/u2/093458.pdf| url-status = dead| archive-date = 8 January 2014}}</ref>
In 1955, [[Lester R. Ford, Jr.]] and [[D. R. Fulkerson|Delbert R. Fulkerson]] created the first known algorithm, the [[Ford–Fulkerson algorithm]].<ref name=":1">{{Cite journal | last1 = Ford | first1 = L. R. | author-link1 = L. R. Ford, Jr.| last2 = Fulkerson | first2 = D. R. | author-link2 = D. R. Fulkerson| doi = 10.4153/CJM-1956-045-5 | title = Maximal flow through a network | journal = [[Canadian Journal of Mathematics]]| volume = 8 | pages = 399–404 | year = 1956 | doi-access = free }}</ref><ref name=":3">Ford, L.R., Jr.; Fulkerson, D.R., ''Flows in Networks'', Princeton University Press (1962).</ref> In their 1955 paper,<ref name=":1" /> Ford and Fulkerson wrote that the problem of Harris and Ross is formulated as follows (see<ref name=":0" /> p.&nbsp;5):<blockquote>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 other.</blockquote>In their book ''Flows in Networks'',<ref name=":3" /> in 1962, Ford and Fulkerson wrote:<blockquote>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 [11].</blockquote>where [11] refers to the 1955 secret report ''Fundamentals of a Method for Evaluating Rail net Capacities'' by Harris and Ross<ref name=":2" /> (see<ref name=":0" /> p.&nbsp;5).

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 [[Push–relabel maximum flow algorithm|push-relabel algorithm]] of [[Andrew V. Goldberg|Goldberg]] and [[Robert Tarjan|Tarjan]]; and the binary blocking flow algorithm of Goldberg and Rao. The algorithms of Sherman<ref>{{Cite book | last = Sherman | first = Jonah | chapter = Nearly Maximum Flows in Nearly Linear Time | doi = 10.1109/FOCS.2013.36 | title = Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science | pages = 263–269 | year = 2013 | arxiv = 1304.2077 | isbn = 978-0-7695-5135-7 | s2cid = 14681906 }}</ref> and Kelner, Lee, Orecchia and Sidford,<ref>{{Cite book | last1 = Kelner | first1 = J. A. | last2 = Lee | first2 = Y. T. | last3 = Orecchia | first3 = L. | last4 = Sidford | first4 = A. | chapter = An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations | doi = 10.1137/1.9781611973402.16 | title = Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms | pages = 217 | year = 2014 | isbn = 978-1-61197-338-9 | chapter-url = http://math.mit.edu/~kelner/Publications/Docs/klos_maxflow_main.pdf | arxiv = 1304.2338 | s2cid = 10733914 | url-status = dead | archive-url = https://web.archive.org/web/20160303170302/http://math.mit.edu/~kelner/Publications/Docs/klos_maxflow_main.pdf | archive-date =2016-03-03 }}</ref><ref>{{cite web | url = http://web.mit.edu/newsoffice/2013/new-algorithm-can-dramatically-streamline-solutions-to-the-max-flow-problem-0107.html | title = New algorithm can dramatically streamline solutions to the 'max flow' problem | first = Helen | last = Knight | date = 7 January 2014 | access-date = 8 January 2014 | publisher = MIT News}}</ref> respectively, find an approximately optimal maximum flow but only work in undirected graphs.

In 2013 [[James B. Orlin]] published a paper describing an <math>O(|V| |E|)</math> algorithm.<ref name="orlin" />

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 <math>O(|E|^{1+o(1)})</math> for the [[Minimum-cost flow problem|minimum-cost flow]] problem of which for the maximum flow problem is a particular case.<ref name="almost linear" /><ref>{{Cite web |last=Klarreich |first=Erica |date=2022-06-08 |title=Researchers Achieve 'Absurdly Fast' Algorithm for Network Flow |url=https://www.quantamagazine.org/researchers-achieve-absurdly-fast-algorithm-for-network-flow-20220608/ |access-date=2022-06-08 |website=Quanta Magazine |language=en}}</ref> For the [[Single source shortest path problem|single-source shortest path (SSSP) problem]] with negative weights another particular case of minimum-cost flow problem an algorithm in almost-linear time has also been reported.<ref>{{Cite arXiv |last1=Bernstein |first1=Aaron |last2=Nanongkai |first2=Danupon |last3=Wulff-Nilsen |first3=Christian |date=2022-10-30 |title=Negative-Weight Single-Source Shortest Paths in Near-linear Time |class=cs.DS |eprint=2203.03456 }}</ref><ref>{{Cite web |last=Brubaker |first=Ben |date=2023-01-18 |title=Finally, a Fast Algorithm for Shortest Paths on Negative Graphs |url=https://www.quantamagazine.org/finally-a-fast-algorithm-for-shortest-paths-on-negative-graphs-20230118/ |access-date=2023-01-25 |website=Quanta Magazine |language=en}}</ref> Both algorithms were deemed best papers at the 2022 [[Symposium on Foundations of Computer Science]].<ref>{{Cite web |title=FOCS 2022 |url=https://focs2022.eecs.berkeley.edu/awards.html |access-date=2023-01-25 |website=focs2022.eecs.berkeley.edu}}</ref><ref>{{Cite web |last=Santosh |first=Nagarakatte |title=FOCS 2022 Best Paper Award for Prof. Aaron Bernstein's Paper |url=https://www.cs.rutgers.edu/news-events/news/news-item/focs-2022-best-paper-award-for-prof-aaron-bernstein-s-paper |access-date=2023-01-25 |website=www.cs.rutgers.edu |language=en-gb}}</ref>

==Definition==
[[File:Simpe_flow_network.svg|thumb|upright=0.8|A flow network, with source ''s'' and sink ''t''. The numbers next to the edges are the capacities.]]

First we establish some notation:
* Let <math>N = (V, E)</math> be a flow network with <math>s, t \in V</math> being the source and the sink of <math>N</math> respectively. 
* If <math>g</math> is a function on the edges of <math>N</math> then its value on <math>(u,v) \in E</math> is denoted by <math>g_{uv}</math> or <math>g(u,v).</math>

'''Definition.''' The '''capacity''' of an edge is the maximum amount of flow that can pass through an edge. Formally it is a map <math>c: E \to \R^+.</math>

'''Definition.''' A '''flow''' is a map <math>f : E \to \R</math> that satisfies the following:
*''Capacity constraint''. The flow of an edge cannot exceed its capacity, in other words: <math>f_{uv} \leq c_{uv}</math> for all <math>(u, v) \in E.</math>
* ''Conservation of flows.'' 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:
::<math>\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}.</math>

''Remark''. Flows are skew symmetric: <math>f_{uv} = -f_{vu}</math> for all <math>(u, v) \in E.</math>

'''Definition.''' The '''value of flow''' is the amount of flow passing from the source to the sink. Formally for a flow <math>f : E \to \R^+</math> it is given by:
:<math>|f| = \sum_{v:\ (s,v) \in E} f_{sv} = \sum_{u:\ (u,t) \in E} f_{ut}.</math>

'''Definition.''' The '''maximum flow problem''' is to route as much flow as possible from the source to the sink, in other words find the flow <math>f_\textrm{max}</math> with maximum value.

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 words, if we send <math>x</math> units of flow on edge <math>u</math> in one maximum flow, and <math>y > x</math> units of flow on <math>u</math> in another maximum flow, then for each <math>\Delta \in [0, y-x]</math> we can send <math>x+\Delta</math> units on <math>u</math> 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 <math>\Delta</math> values for each pair <math>x, y</math>.

==Algorithms==
The following table lists algorithms for solving the maximum flow problem.
Here, <math>V</math> and <math>E</math> denote the number of vertices and edges of the network.
The value <math>U</math> refers to the largest edge capacity after rescaling all capacities to integer values
(if the network contains [[Irrational number|irrational]] capacities, <math>U</math> may be infinite).

{| class="wikitable"
! Method
! Year
! Complexity
! Description
|-
| [[Linear programming]]
|
|
| Constraints given by the definition of a [[flow network|legal flow]]. See the [[Max-flow min-cut theorem#Linear program formulation|linear program]] here.
|-
| [[Ford–Fulkerson algorithm]]
| 1955
| <math>O(EU)</math>
| As long as there is an open path through the residual graph, send the minimum of the residual capacities on that path.
|-
| [[Edmonds–Karp algorithm]]
| 1970
| <math>O(VE^2)</math>
| A specialization of Ford–Fulkerson, finding augmenting paths with [[breadth-first search]].
|-
| [[Dinic's algorithm]]
| 1970
| <math>O(V^2E)</math>
| In each phase the algorithms builds a layered graph with [[breadth-first search]] on the [[residual graph]]. The maximum flow in a layered graph can be calculated in <math>O(VE)</math> time, and the maximum number of phases is <math>V-1</math>. In networks with unit capacities, Dinic's algorithm terminates in <math>O(\min\{V^{2/3}, E^{1/2}\}E)</math> time.
|-
| MKM (Malhotra, Kumar, Maheshwari) algorithm<ref>{{Cite journal | last1 = Malhotra | first1 =V.M.| last2 = Kumar | first2 = M. Pramodh| last3 = Maheshwari | first3 = S.N.| doi = 10.1016/0020-0190(78)90016-9| title = An <math>O(|V|^3)</math> algorithm for finding maximum flows in networks| journal = Information Processing Letters| volume = 7| issue = 6| pages = 277–278 | year = 1978| url =https://eprints.utas.edu.au/160/1/iplFlow.pdf}}</ref>
| 1978
| <math>O(V^3)</math>
| A modification of Dinic's algorithm with a different approach to constructing blocking flows. Refer to the [[#{{harvid|Malhotra|Kumar|Maheshwari|1978}}|original paper]].
|-
| [[Dinic's algorithm]] with dynamic trees
| 1983
| <math>O(VE \log V)</math>
| The [[dynamic trees]] data structure speeds up the maximum flow computation in the layered graph to <math>O(VE \log V)</math>.
|-
| General [[push–relabel algorithm]]<ref name="goldberg1988"/>
| 1986
| <math>O(V^2E)</math>
| 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 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.
|-
| [[Push–relabel algorithm]] with ''FIFO'' vertex selection rule<ref name="goldberg1988"/>
| 1988
| <math>O(V^3)</math>
| 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.
|-
|-
| [[Push–relabel algorithm]] with ''maximum distance'' vertex selection rule<ref name="cheriyan1988">{{cite book
 | last1 = Cheriyan
 | first1 = J.
 | last2 = Maheshwari
 | first2 = S. N.
 | title = Foundations of Software Technology and Theoretical Computer Science
 | chapter = Analysis of preflow push algorithms for maximum network flow
 | series = Lecture Notes in Computer Science
 | volume = 338
 | year = 1988
 | pages = 30–48
 | issn = 0302-9743
 | doi = 10.1007/3-540-50517-2_69
| isbn = 978-3-540-50517-4
 }}</ref>
| 1988
| <math>O(V^2 \sqrt{E})</math>
| Push-relabel algorithm variant which always selects the most distant vertex from <math>s</math> or <math>t</math> (i.e. the highest label vertex) but otherwise proceeds as the FIFO algorithm.
|-
| [[Push-relabel algorithm]] with dynamic trees<ref name="goldberg1988">{{Cite journal
 | last1 = Goldberg | first1 = A. V. | author-link1 = Andrew V. Goldberg
 | last2 = Tarjan | first2 = R. E. | author-link2 = Robert E. Tarjan
 | doi = 10.1145/48014.61051
 | title = A new approach to the maximum-flow problem
 | journal = [[Journal of the ACM]]
 | volume = 35
 | issue = 4
 | pages = 921
 | year = 1988
 | s2cid = 52152408
| doi-access = free
 }}</ref>
| 1988
| <math>O\left( VE \log \frac{V^2}{E} \right)</math>
| 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 ''path'' instead of a single edge.
|-
| KRT (King, Rao, Tarjan)'s algorithm<ref>{{Cite journal | last1 = King | first1 = V.| last2 = Rao | first2 = S.| last3 = Tarjan | first3 = R.| doi = 10.1006/jagm.1994.1044| title = A Faster Deterministic Maximum Flow Algorithm| journal = Journal of Algorithms| volume = 17| issue = 3| pages = 447–474| year = 1994| s2cid = 15493}}</ref>
| 1994
| <math>O\left( VE \log_{\frac{E}{V \log V}} V \right)</math>
|
|-
| Binary blocking flow algorithm<ref>{{Cite journal | last1 = Goldberg | first1 = A. V. | author-link1 = Andrew V. Goldberg| last2 = Rao | first2 = S. | doi = 10.1145/290179.290181 | title = Beyond the flow decomposition barrier | journal = [[Journal of the ACM]]| volume = 45 | issue = 5 | pages = 783 | year = 1998 | s2cid = 96030 | doi-access = free }}</ref>
| 1998
| <math>O\left( E \cdot \min\{V^{2/3}, E^{1/2}\} \cdot \log \frac{V^2}{E} \cdot \log U \right)</math>
|
|-
| James B Orlin's + KRT (King, Rao, Tarjan)'s algorithm<ref name="orlin">{{Cite book | last1 = Orlin | first1 = James B.| title = Proceedings of the forty-fifth annual ACM symposium on Theory of Computing| doi = 10.1145/2488608.2488705| chapter = Max flows in O(nm) time, or better| pages = 765–774| year = 2013| isbn = 9781450320290| citeseerx = 10.1.1.259.5759| s2cid = 207205207}}</ref>
| 2013
| <math>O(VE)</math>
| Orlin's algorithm solves max-flow in <math>O(VE)</math> time for <math>E \leq O(V^{\frac{16}{15} - \epsilon})</math> while KRT solves it in <math>O(VE)</math> for <math>E > V^{1+\epsilon}</math>.
|-
| Kathuria-Liu-Sidford algorithm<ref>{{cite book |last1=Kathuria |first1=T. |last2=Liu |first2=Y.P. |last3=Sidford |first3=A. |title=Unit Capacity Maxflow in Almost <math> O(m^{4/3}) </math> Time |date= 16–19 November 2020 |publisher=IEEE |location=Durham, NC, USA |pages=119–130}}</ref>
| 2020
| <math> E^{4/3+o(1)}U^{1/3} </math>
| Interior point methods and edge boosting using <math>\ell_p</math>-norm flows. Builds on earlier algorithm of Madry, which achieved runtime <math> \tilde O(E^{10/7}U^{1/7}) </math>.<ref>{{cite book |last1=Madry |first1=Aleksander |title=Computing Maximum Flow with Augmenting Electrical Flows |date=9-11 October 2016 |publisher=IEEE |location=New Brunswick, New Jersey |pages=593–602}}</ref>
|-
| BLNPSSSW / BLLSSSW algorithm<ref>{{cite book |last1=Brand |first1=J. vd |last2=Lee |first2=Y.T. |last3=Nanongkai |first3=D. |last4=Peng |first4=R. |last5=Saranurak |first5=T. |last6=Sidford |first6=A. |last7=Song |first7=Z. |last8=Wang |first8=D. |title=Bipartite Matching in Nearly-linear Time on Moderately Dense Graphs |date= 16–19 November 2020 |publisher=IEEE |location=Durham, NC, USA |pages=919–930}}</ref>
<ref>{{cite arXiv |last1=Brand |first1=J. vd |last2=Lee |first2=Y.T. |last3=Liu |first3=Y.P. |last4=Saranurak |first4=T. |last5=Sidford |first5=A |last6=Song |first6=Z. |last7=Wang |first7=D. |title=Minimum Cost Flows, MDPs, and ℓ1-Regression in Nearly Linear Time for Dense Instances |year=2021 |class=cs.DS |eprint=2101.05719}}</ref>
| 2020
| <math>\tilde O((E + V^{3/2}) \log U)</math>
| Interior point methods and dynamic maintenance of electric flows with expander decompositions.
|-
| Gao-Liu-Peng algorithm<ref>{{Cite arXiv | last1 = Gao | first1 = Y.| last2 = Liu | first2 = Y.P.| last3 = Peng | first3 = R.| title = Fully Dynamic Electrical Flows: Sparse Maxflow Faster Than Goldberg-Rao| year = 2021| class = cs.DS| eprint = 2101.07233}}</ref>
| 2021
| <math>\tilde O(E^{\frac 32 - \frac 1{328}} \log U)</math>
| 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 [Mądry JACM ‘16]. This entails designing data structures that, in limited settings, return edges with large electric energy in a graph undergoing resistance updates.
|-
| Chen, Kyng, Liu, Peng, Gutenberg and Sachdeva's algorithm<ref name="almost linear">{{Cite arXiv | last1 = Chen | first1 = L.| last2 = Kyng | first2 = R.| last3 = Liu | first3 = Y.P.| last4 = Gutenberg | first4 = M.P. | last5 = Sachdeva | first5 = S. | title = Maximum Flow and Minimum-Cost Flow in Almost-Linear Time| year = 2022| class = cs.DS| eprint = 2203.00671}}</ref>
| 2022
|<math>O(E^{1+o(1)} \log U)</math>
The exact complexity is not stated clearly in the paper, but appears to be <math>E \exp O(\log^{7/8} E \log \log E) \log U</math>
| 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 <math>E^{1+o(1)}</math> approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized <math>E^{o(1)}</math> time using a dynamic data structure.
|-
| Bernstein, Blikstad, Saranurak, Tu<ref>{{Cite arXiv | last1 = Bernstein | first1 = A. | last2 = Blikstad | first2 = J. | last3 = Saranurak | first3 = T. | last4 = Tu | first4 = T. | title = Maximum Flow by Augmenting Paths in <math>n^{2 + o(1)}</math> Time | year = 2024 | class = cs.DS | eprint = 2406.03648}}</ref>
| 2024
|<math>O(n^{2 + o(1)} \log U)</math> randomized algorithm when the edge capacities come from the set <math>\{1, \dots, U\}</math>
| The algorithm is a variant of the push-relabel algorithm by introducing the ''weighted'' variant. The paper establishes a weight function on directed and acyclic graphs (DAG), and attempts to imitate it on general graphs using directed expander hierarchies, which induce a natural vertex ordering that produces the weight function similar to that of the DAG special case. The randomization aspect (and subsequently, the <math>n^{o(1)}</math> factor) comes from the difficulty in applying directed expander hierarchies to the computation of ''sparse cuts'', which do not allow for natural dynamic updating.
|}

For additional algorithms, see {{harvtxt|Goldberg|Tarjan|1988}}.

==Integral flow theorem==
The integral flow theorem states that
: '''If each edge in a flow network has integral capacity, then there exists an integral maximal flow.'''

The claim is not only that the value of the flow is an integer, which follows directly from the [[max-flow min-cut theorem]], but that the flow on ''every edge'' is integral. This is crucial for many [[discrete mathematics|combinatorial]] 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.

==Application==

===Multi-source multi-sink maximum flow problem===
[[File:Multi-source multi-sink flow problem.svg|thumb|right|Fig. 4.1.1. Transformation of a multi-source multi-sink maximum flow problem into a single-source single-sink maximum flow problem]]
Given a network <math>N = (V, E)</math> with a set of sources <math>S = \{s_1, \ldots, s_n\}</math> and a set of sinks <math>T = \{t_1, \ldots, t_m\}</math> instead of only one source and one sink, we are to find the maximum flow across <math>N</math>. We can transform the multi-source multi-sink problem into a maximum flow problem by adding a ''consolidated source'' connecting to each vertex in <math>S</math> and a ''consolidated sink ''connected by each vertex in <math>T</math> (also known as ''supersource'' and ''supersink'') with infinite capacity on each edge (See Fig. 4.1.1.).

===Maximum cardinality bipartite matching===
[[File:Maximum bipartite matching to max flow.svg|thumb|right|Fig. 4.3.1. Transformation of a maximum bipartite matching problem into a maximum flow problem]]
Given a [[bipartite graph]] <math>G = (X \cup Y, E)</math>, we are to find a [[maximum cardinality matching]] in <math>G</math>, 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 <math>N = (X \cup Y \cup \{s,t\}, E')</math>, where
# <math>E'</math> contains the edges in <math>G</math> directed from <math>X</math> to <math>Y</math>.
# <math>(s,x) \in E'</math> for each <math>x \in X</math> and <math>(y,t) \in E'</math> for each <math>y \in Y</math>.
# <math>c(e) = 1</math> for each <math>e \in E'</math> (See Fig. 4.3.1).

Then the value of the maximum flow in <math>N</math> is equal to the size of the maximum matching in <math>G</math>, and a maximum cardinality matching can be found by taking those edges that have flow <math>1</math> in an integral max-flow.

===Minimum path cover in directed acyclic graph===
Given a [[directed acyclic graph]] <math>G = (V, E)</math>, we are to find the minimum number of [[Path (graph theory)|vertex-disjoint paths]] to cover each vertex in <math>V</math>. We can construct a bipartite graph <math>G' = (V_\textrm{out} \cup V_\textrm{in}, E')</math> from <math>G</math>, where
# <math>V_\textrm{out} = \{ v_\textrm{out} \mid v \in V \land v \text{ has outgoing edge(s)} \}</math>
# <math>V_\textrm{in} = \{ v_\textrm{in} \mid v \in V \land v \text{ has incoming edge(s)} \}</math>
# <math>E' = \{(u_\textrm{out}, v_\textrm{in}) \in V_{out} \times V_{in} \mid (u, v) \in E \}</math>.

Then it can be shown that <math>G'</math> has a matching <math>M</math> of size <math>m</math> if and only if <math>G</math> has a vertex-disjoint path cover <math>C</math> containing <math>m</math> edges and <math>n-m</math> paths, where <math>n</math> is the number of vertices in <math>G</math>. Therefore, the problem can be solved by finding the maximum cardinality matching in <math>G'</math> instead.

Assume we have found a matching <math>M</math> of <math>G'</math>, and constructed the cover <math>C</math> from it. Intuitively, if two vertices <math>u_\mathrm{out}, v_\mathrm{in}</math> are matched in <math>M</math>, then the edge <math>(u, v)</math> is contained in <math>C</math>. Clearly the number of edges in <math>C</math> is <math>m</math>. To see that <math>C</math> is vertex-disjoint, consider the following:
# Each vertex <math>v_\textrm{out}</math> in <math>G'</math> can either be ''non-matched'' in <math>M</math>, in which case there are no edges leaving <math>v</math> in <math>C</math>; or it can be ''matched'', in which case there is exactly one edge leaving <math>v</math> in <math>C</math>. In either case, no more than one edge leaves any vertex <math>v</math> in <math>C</math>.
# Similarly for each vertex <math>v_\textrm{in}</math> in <math>G'</math> – if it is matched, there is a single incoming edge into <math>v</math> in <math>C</math>; otherwise <math>v</math> has no incoming edges in <math>C</math>.
Thus no vertex has two incoming or two outgoing edges in <math>C</math>, which means all paths in <math>C</math> are vertex-disjoint.

To show that the cover <math>C</math> has size <math>n-m</math>, we start with an empty cover and build it incrementally. To add a vertex <math>u</math> 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 <math>(u,v) \in E</math> and some path in the cover starts at <math>v</math>, or <math>(v,u) \in E</math> and some path ends at <math>v</math>. 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 <math>n</math> vertices, the sum of the number of paths and edges in the cover is <math>n</math>. Therefore, if the number of edges in the cover is <math>m</math>, the number of paths is <math>n-m</math>.

===Maximum flow with vertex capacities===
[[File:Node splitting.svg|thumb|right|Fig. 4.4.1. Transformation of a maximum flow problem with vertex capacities constraint into the original maximum flow problem by node splitting]]

Let <math>N = (V, E)</math> be a network. Suppose there is capacity at each node in addition to edge capacity, that is, a mapping <math>c: V\to \R^+,</math> such that the flow <math>f</math> has to satisfy not only the capacity constraint and the conservation of flows, but also the vertex capacity constraint

:<math> \sum_{i\in V} f_{iv} \le c(v) \qquad \forall v \in V \backslash \{s,t\}.</math>

In other words, the amount of flow passing through a vertex cannot exceed its capacity. To find the maximum flow across <math>N</math>, we can transform the problem into the maximum flow problem in the original sense by expanding <math>N</math>. First, each <math>v\in V</math> is replaced by <math>v_{\text{in}}</math> and <math>v_{\text{out}}</math>, where <math>v_{\text{in}}</math> is connected by edges going into <math>v</math> and <math>v_{\text{out}}</math> is connected to edges coming out from <math>v</math>, then assign capacity <math>c(v)</math> to the edge connecting <math>v_{\text{in}}</math> and <math>v_{\text{out}}</math> (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.

===Maximum number of paths from s to t===
Given a directed graph <math>G = (V, E)</math> and two vertices <math>s</math> and <math>t</math>, we are to find the maximum number of paths from <math>s</math> to <math>t</math>. This problem has several variants:

1. The paths must be edge-disjoint. This problem can be transformed to a maximum flow problem by constructing a network <math>N = (V, E)</math> from <math>G</math>, with <math>s</math> and <math>t</math> being the source and the sink of <math>N</math> respectively, and assigning each edge a capacity of <math>1</math>. In this network, the maximum flow is <math>k</math> iff there are <math>k</math> edge-disjoint paths.

2. The paths must be independent, i.e., vertex-disjoint (except for <math>s</math> and <math>t</math>). We can construct a network <math>N = (V, E)</math> from <math>G</math> with vertex capacities, where the capacities of all vertices and all edges are <math>1</math>. Then the value of the maximum flow is equal to the maximum number of independent paths from <math>s</math> to <math>t</math>.

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 <math>k</math>, or at most <math>k</math>. Most variants of this problem are NP-complete, except for small values of <math>k</math>.<ref>{{Cite journal|last1=Itai|first1=A.|last2=Perl|first2=Y.|last3=Shiloach|first3=Y.|year=1982|title=The complexity of finding maximum disjoint paths with length constraints|journal=Networks|language=en|volume=12|issue=3|pages=277–286|doi=10.1002/net.3230120306|issn=1097-0037}}</ref>

=== Closure problem ===
{{Main|Closure problem}}
A '''closure''' of a directed graph is a set of vertices ''C'', such that no edges leave ''C''. The '''closure problem''' is the task of finding the maximum-weight or minimum-weight closure in a vertex-weighted directed graph. It may be solved in [[Time complexity|polynomial time]] using a reduction to the maximum flow problem.

==Real world applications==

===Baseball elimination===
[[File:Baseball Elimination Problem.png|thumb|Construction of network flow for baseball elimination problem]]
In the [[baseball]] elimination problem there are ''n'' teams competing in a league. At a specific stage of the league season, ''w''<sub>''i''</sub> is the number of wins and ''r''<sub>''i''</sub> is the number of games left to play for team ''i'' and ''r''<sub>ij</sub> is the number of games left against team ''j''. 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<ref>{{Cite journal | last1 = Schwartz | first1 = B. L. | title = Possible Winners in Partially Completed Tournaments | doi = 10.1137/1008062 | journal = [[SIAM Review]]| jstor = 2028206| volume = 8 | issue = 3 | pages = 302–308 | year = 1966 | bibcode = 1966SIAMR...8..302S }}</ref> proposed a method which reduces this problem to maximum network flow. In this method a network is created to determine whether team ''k'' is eliminated.

Let ''G'' = (''V'', ''E'') be a network with {{math|''s'',''t'' ∈ ''V''}} being the source and the sink respectively. One adds a game node<sub>''ij''</sub> – which represents the number of plays between these two teams. We also add a team node for each team and connect each game node {{math|{{mset|''i'', ''j''}}}} with ''i'' < ''j'' to ''V'', and connects each of them from ''s'' by an edge with capacity ''r''<sub>''ij''</sub> – which represents the number of plays between these two teams. We also add a team node for each team and connect each game node {{math|{{mset|''i'', ''j''}}}} with two team nodes ''i'' and ''j'' 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 ''i'' to the sink node ''t'' and the capacity of {{math|''w''<sub>''k''</sub> + ''r''<sub>''k''</sub> – ''w''<sub>''i''</sub>}} is set to prevent team ''i'' from winning more than {{math|''w''<sub>''k''</sub> + ''r''<sub>''k''</sub>}}.
Let ''S'' be the set of all teams participating in the league and let 
:<math>r(S - \{k\}) = \sum_{i,j \in \{S-\{k\}\} \atop i < j} r_{ij}</math>. 
In this method it is claimed team ''k'' is not eliminated if and only if a flow value of size ''r''(''S'' − {''k''}) exists in network ''G''. In the mentioned article it is proved that this flow value is the maximum flow value from ''s'' to ''t''.

===Airline scheduling===
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 ''F'' 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 ''k'' crews.

To solve this problem one uses a variation of the [[circulation problem]] called bounded circulation which is the generalization of [[flow network|network flow]] problems, with the added constraint of a lower bound on edge flows.

Let ''G'' = (''V'', ''E'') be a network with {{math|''s'',''t'' ∈ ''V''}} as the source and the sink nodes. For the source and destination of every flight ''i'', one adds two nodes to ''V'', node ''s''<sub>''i''</sub> as the source and node ''d''<sub>''i''</sub> as the destination node of flight ''i''. One also adds the following edges to ''E'':
# An edge with capacity [0, 1] between ''s'' and each ''s''<sub>''i''</sub>.
# An edge with capacity [0, 1] between each ''d''<sub>''i''</sub> and ''t''.
# An edge with capacity [1, 1] between each pair of ''s''<sub>''i''</sub> and ''d''<sub>''i''</sub>.
# An edge with capacity [0, 1] between each ''d''<sub>''i''</sub> and ''s''<sub>''j''</sub>, if source ''s''<sub>''j''</sub> is reachable with a reasonable amount of time and cost from the destination of flight ''i''.
# An edge with capacity [0, [[∞]]] between ''s'' and ''t''.

In the mentioned method, it is claimed and proved that finding a flow value of ''k'' in ''G'' between ''s'' and ''t'' is equal to finding a feasible schedule for flight set ''F'' with at most ''k'' crews.<ref name="ITA">{{cite book | author = [[Thomas H. Cormen]], [[Charles E. Leiserson]], [[Ronald L. Rivest]], and [[Clifford Stein]] | title=Introduction to Algorithms, Second Edition | chapter = 26. Maximum Flow | year = 2001 | publisher = MIT Press and McGraw-Hill | isbn = 978-0-262-03293-3 | pages=643–668| title-link=Introduction to Algorithms }}</ref>

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 {{math|1={{var|G'}} = (''A'' ∪ ''B'', ''E'')}} is created where each flight has a copy in set ''A'' and set ''B''. If the same plane can perform flight ''j'' after flight ''i'', {{math|''i''∈''A''}} is connected to {{math|''j''∈''B''}}. A matching in {{mvar|G'}} induces a schedule for ''F'' and obviously maximum bipartite matching in this graph produces an airline schedule with minimum number of crews.<ref name="ITA"/> As it is mentioned in the Application part of this article, the maximum cardinality bipartite matching is an application of maximum flow problem.

===Circulation–demand problem===
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 {{mvar|c}} 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.
# Add a source node {{mvar|s}} and add edges from it to every factory node {{mvar|f<sub>i</sub>}} with capacity {{mvar|p<sub>i</sub>}} where {{mvar|p<sub>i</sub>}} is the production rate of factory {{mvar|f<sub>i</sub>}}.
# Add a sink node {{mvar|t}} and add edges from all villages {{mvar|v<sub>i</sub>}} to {{mvar|t}} with capacity {{mvar|d<sub>i</sub>}} where {{mvar|d<sub>i</sub>}} is the demand rate of village {{mvar|v<sub>i</sub>}}.
Let ''G'' = (''V'', ''E'') be this new network. There exists a circulation that satisfies the demand if and only if :
: {{math|Maximum flow value(''G'')}} <math> = \sum_{i \in v} d_i </math>.
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.

The problem can be extended by adding a lower bound on the flow on some edges.<ref>{{Cite web|url=https://www.cs.cmu.edu/~ckingsf/bioinfo-lectures/flowext.pdf|title=Max-flow extensions: circulations with demands|last=Carl Kingsford}}</ref>

=== Image segmentation ===
[[File:Maxflow imagesegmentation image.png|thumb|Source image of size 8x8.|alt=|left|110x110px]]
[[File:Maxflow imagesegmentation network.png|thumb|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<sub>i</sub> is high when the pixel is green, b<sub>i</sub> when the pixel is not green. The penalty p<sub>ij</sub> are all equal.<ref>{{Cite web|url=https://gitlab.com/francois.schwarzentruber/imagesegmentationwithmaxflow|title=Project imagesegmentationwithmaxflow, that contains the source code to produce these illustrations.|website=GitLab|language=en|url-status=dead|archive-url=https://web.archive.org/web/20191222173132/https://gitlab.com/francois.schwarzentruber/imagesegmentationwithmaxflow|archive-date=2019-12-22|access-date=2019-12-22}}</ref>]]
In their book, Kleinberg and Tardos present an algorithm for [[image segmentation|segmenting]] an image.<ref>{{Cite web|url=https://www.pearson.com/us/higher-education/program/Kleinberg-Algorithm-Design/PGM319216.html |title=Algorithm Design|website=pearson.com|language=en|access-date=2019-12-21}}</ref> 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: ''a<sub>i</sub>'' ≥ 0 is the likelihood that pixel ''i'' belongs to the foreground, ''b<sub>i</sub>'' ≥ 0 in the likelihood that pixel ''i'' belongs to the background, and ''p<sub>ij</sub>'' is the penalty if two adjacent pixels ''i'' and ''j'' are placed one in the foreground and the other in the background. The goal is to find a partition (''A'', ''B'') of the set of pixels that maximize the following quantity

:<math>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}</math>,

Indeed, for pixels in ''A'' (considered as the foreground), we gain ''a<sub>i</sub>''; for all pixels in ''B'' (considered as the background), we gain ''b<sub>i</sub>''. On the border, between two adjacent pixels ''i'' and ''j'', we loose ''p<sub>ij</sub>''. It is equivalent to minimize the quantity

:<math>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}</math>

because

:<math>q(A, B) = \sum_{i \in A\cup B} a_i + \sum_{i \in A\cup B} b_i - q'(A, B).</math>

[[File:Maxflow imagesegmentation result.png|thumb|Minimum cut displayed on the network (triangles VS circles).]]

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 ''i'' by an edge of weight ''a<sub>i</sub>''.  We connect the pixel ''i'' to the sink by an edge of weight ''b<sub>i</sub>''. We connect pixel ''i'' to pixel ''j'' with weight ''p<sub>ij</sub>''. Now, it remains to compute a minimum cut in that network (or equivalently a maximum flow). The last figure shows a minimum cut.

== Extensions ==
1. In the '''[[minimum-cost flow problem]]''', each edge (''u'',v) also has a '''cost-coefficient''' ''a<sub>uv</sub>'' in addition to its capacity. If the flow through the edge is ''f<sub>uv</sub>'', then the total cost is ''a<sub>uv</sub>f<sub>uv</sub>.''  It is required to find a flow of a given size ''d'', 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.

2. The maximum-flow problem can be augmented by '''disjunctive constraints''': a ''negative disjunctive constraint'' says that a certain pair of edges cannot simultaneously have a nonzero flow; a ''positive disjunctive constraints'' says that, in a certain pair of edges, at least one must have a nonzero flow. With negative constraints, the problem becomes [[strongly NP-hard]] even for simple networks. With positive constraints, the problem is polynomial if fractional flows are allowed, but may be [[strongly NP-hard]] when the flows must be integral.<ref>{{Cite journal|last1=Schauer|first1=Joachim|last2=Pferschy|first2=Ulrich|date=1 July 2013|title=The maximum flow problem with disjunctive constraints|journal=Journal of Combinatorial Optimization|language=en|volume=26|issue=1|pages=109–119|doi=10.1007/s10878-011-9438-7|issn=1382-6905|citeseerx=10.1.1.414.4496|s2cid=6598669}}</ref>

== References ==
{{reflist}}

==Further reading==
* {{cite journal
 | author = [[Joseph Cheriyan]] and [[Kurt Mehlhorn]]
 | title = An analysis of the highest-level selection rule in the preflow-push max-flow algorithm
 | journal = Information Processing Letters
 | volume = 69
 | year = 1999
 | pages = 239–242
 | doi = 10.1016/S0020-0190(99)00019-8
 | issue = 5
| citeseerx = 10.1.1.42.8563
 }}
* {{cite journal
 | author = [[Daniel D. Sleator]] and [[Robert E. Tarjan]]
 | title = A data structure for dynamic trees
 | journal = Journal of Computer and System Sciences
 | volume = 26
 | year = 1983
 | issn = 0022-0000
 | pages = 362–391
 | doi = 10.1016/0022-0000(83)90006-5
 | url = https://www.cs.cmu.edu/~sleator/papers/dynamic-trees.pdf
 | issue = 3
}}
* {{cite book | author =  Eugene Lawler | author-link = Eugene Lawler | title = Combinatorial Optimization: Networks and Matroids | chapter = 4. Network Flows | year = 2001 | publisher = Dover | isbn = 978-0-486-41453-9 | pages = 109–177}}

{{DEFAULTSORT:Maximum Flow Problem}}
[[Category:Network flow problem]]
[[Category:Computational problems in graph theory]]