Inoptimization theory,maximum flow problems involve finding a feasible flow through aflow 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 thecirculation problem. The maximum value of an s-t flow (i.e., flow fromsource s tosink t) is equal to the minimum capacity of ans-t cut (i.e., cut severing s from t) in the network, as stated in themax-flow min-cut theorem.

The maximum flow problem was first formulated in 1954 byT. E. Harris and F. S. Ross as a simplified model of Soviet railway traffic flow.^[1][2][3]

In 1955,Lester R. Ford, Jr. andDelbert R. Fulkerson created the first known algorithm, theFord–Fulkerson algorithm.^[4][5] In their 1955 paper,^[4] Ford and Fulkerson wrote that the problem of Harris and Ross is formulated as follows (see^[1] p. 5):

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.

In their bookFlows in Networks,^[5] in 1962, Ford and Fulkerson wrote:

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].

where [11] refers to the 1955 secret reportFundamentals of a Method for Evaluating Rail net Capacities by Harris and Ross^[3] (see^[1] p. 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; thepush-relabel algorithm ofGoldberg andTarjan; and the binary blocking flow algorithm of Goldberg and Rao. The algorithms of Sherman^[6] and Kelner, Lee, Orecchia and Sidford,^[7][8] respectively, find an approximately optimal maximum flow but only work in undirected graphs.

In 2013James B. Orlin published a paper describing an${\displaystyle O(|V||E|)}$ algorithm.^[9]

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${\displaystyle O(|E{|}^{1+o(1)})}$ for theminimum-cost flow problem of which for the maximum flow problem is a particular case.^[10][11] For thesingle-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.^[12][13] Both algorithms were deemed best papers at the 2022Symposium on Foundations of Computer Science.^[14][15]

First we establish some notation:

• Let${\displaystyle N=(V,E)}$ be a flow network with${\displaystyle s,t\in V}$ being the source and the sink of${\displaystyle N}$ respectively.
• If${\displaystyle g}$ is a function on the edges of${\displaystyle N}$ then its value on${\displaystyle (u,v)\in E}$ is denoted by${\displaystyle g_{uv}}$ or${\displaystyle g(u,v).}$

Definition. Thecapacity of an edge is the maximum amount of flow that can pass through an edge. Formally it is a map${\displaystyle c:E\to {\mathbb{R}}^{+}.}$

Definition. Aflow is a map${\displaystyle f:E\to \mathbb{R}}$ that satisfies the following:

• Capacity constraint. The flow of an edge cannot exceed its capacity, in other words:${\displaystyle f_{uv}\le c_{uv}}$ for all${\displaystyle (u,v)\in E.}$
• 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:

${\displaystyle \mathrm{\forall }v\in V\setminus \{s,t\}:\sum _{u:(u,v)\in E,f_{uv}>0}{f}_{uv}=\sum _{u:(v,u)\in E,f_{vu}>0}{f}_{vu}.}$

Remark. Flows are skew symmetric:${\displaystyle f_{uv}=-f_{vu}}$ for all${\displaystyle (u,v)\in E.}$

Definition. Thevalue of flow is the amount of flow passing from the source to the sink. Formally for a flow${\displaystyle f:E\to {\mathbb{R}}^{+}}$ it is given by:

${\displaystyle |f|=\sum _{v:(s,v)\in E}{f}_{sv}=\sum _{u:(u,t)\in E}{f}_{ut}.}$

Definition. Themaximum flow problem is to route as much flow as possible from the source to the sink, in other words find the flow${\displaystyle f_{\text{max}}}$ 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${\displaystyle x}$ units of flow on edge${\displaystyle u}$ in one maximum flow, and${\displaystyle y>x}$ units of flow on${\displaystyle u}$ in another maximum flow, then for each${\displaystyle \mathrm{\Delta }\in [0,y-x]}$ we can send${\displaystyle x+\mathrm{\Delta }}$ units on${\displaystyle u}$ 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${\displaystyle \mathrm{\Delta }}$ values for each pair${\displaystyle x,y}$.

The following tables show the historical development of algorithms for solving the maximum flow problem. Many of the listed publications include similar tables comparing their results to earlier works.

Astrongly-polynomial time algorithm has polynomial time bounds that depend only on the number of inputs, but do not depend on the magnitude of these numbers. Here, the inputs are the vertices (numbered below as${\displaystyle V}$) and edges (numbered as${\displaystyle E}$). The complexity of each algorithm is stated usingbig O notation.

Method	Year	Complexity	Description
Edmonds–Karp algorithm[16]	1969	${\displaystyle O(V{E}^2)}$	A specialization of Ford–Fulkerson, finding augmenting paths withbreadth-first search.
Dinic's algorithm[17]	1970	${\displaystyle O(V^{2}E)}$ (arbitrary weights) ${\displaystyle O(min\{V^{2/3},E^{1/2}\}E)}$ (unit weights)	Repeated phases that build a "layered" subgraph ofresidual graph edges belonging to shortest paths, usingbreadth-first search, and then find a blocking flow (a maximum flow in this layered graph) in time${\displaystyle O(VE)}$ per phase. The shortest path length increases in each phase so there are at most${\displaystyle V-1}$ phases.
Karzanov's algorithm[18]	1974	${\displaystyle O(V^3)}$	A predecessor to thepush-relabel algorithm using preflows (flow functions allowing excess at vertices) to find a blocking flow in each phase of Dinic's algorithm in time${\displaystyle O(V^2)}$ per phase. The first cubic-time flow algorithm.
Cherkassky's algorithm[19]	1977	${\displaystyle O(V^2\sqrt{E})}$	Combines the blocking flow methods of Dinic (within blocks of consecutive BFS layers) and Karzanov (to combine blocks). The first subcubic strongly polynomial time bound for sparse graphs. Remained best for some values of${\displaystyle E}$ until KRT 1988.
Malhotra, Kumar, and Maheshwari[20]	1978	${\displaystyle O(V^3)}$	Not an improvement in complexity over Karzanov, but a simplification. Finds blocking flows by repeatedly finding a "reference node" of the layered graph and a flow that saturates all its incoming or outgoing edges, in time proportional to the number of nodes plus the number of saturated edges.
Galil's algorithm[21]	1978	${\displaystyle O(V^{5/3}{E}^{2/3})}$	Modifies Cherkasky's algorithm by replacing the method for finding flows within blocks of consecutive layers.
Galil, Naamad, andShiloach[22][23]	1978	${\displaystyle O(VE(\log V{)}^2)}$	Usestree contraction on a breadth-first search forest of the layered graph to speed up blocking flows. The first of many${\displaystyle O(VE\mathrm{polylog}V)}$ algorithms, still the best polynomial exponents for a strongly polynomial algorithm.
Blocking flow withlink/cut trees.[24]	1981	${\displaystyle O(VE\log V)}$	Introduces thelink/cut tree data structure and uses it to find augmenting paths in layered networks in logarithmicamortized time per path.
Push–relabel algorithm with link/cut trees[25]	1986	${\displaystyle O\left(VE\log \frac{V^2}{E}\right)}$	The push-relabel algorithm maintains a preflow, and a height function estimating residual distance to the sink. It modifies the preflow by pushing excess to lower-height vertices and increases the height function at vertices without residual edges to lower heights, until all excess returns to the source. Link/cut trees allow pushes along paths rather than one edge at a time.
Cheriyan and Hagerup[26]	1989	randomized,${\displaystyle O(VE+V^2(\log V{)}^2)}$ with high probability	Push-relabel on a subgraph to which one edge is added at a time, prioritizing pushes of high excess amounts, with randomly permutedadjacency lists
Alon[27]	1989	${\displaystyle O(VE+V^{8/3}\log V)}$	Derandomization of Cheriyan and Hagerup
Cheriyan, Hagerup, and Mehlhorn[28]	1990	${\displaystyle {\displaystyle O\left(\frac{V^3}{\log V}\right)}}$	Uses Alon's derandomization of Cheriyan and Hagerup with ideas related to theMethod of Four Russians to speed up the search for height-reducing edges on which to push excess.
King, Rao, and Tarjan[29]	1992	${\displaystyle O(VE+V^{2+\epsilon })}$ for any${\displaystyle \epsilon >0}$	Another derandomization of Cheriyan and Hagerup. Preliminary version of King, Rao, and Tarjan 1994 with weaker bounds.
Phillips and Westbrook[30]	1993	${\displaystyle O(VE{\log }_{E/V}V+V(\log V{)}^{2+\epsilon })}$ for any${\displaystyle \epsilon >0}$	Improved from King, Rao, and Tarjan 1992 using similar ideas.
King, Rao, and Tarjan[31]	1994	${\displaystyle O\left(VE{\log }_{\frac{E}{V\log V}}V\right)}$	Improved from Phillips and Westbrook using similar ideas.
Orlin[9]	2013	${\displaystyle O(VE)}$	Applies a pseudopolynomial algorithm of Goldberg and Rao to a compressed network, maintained using data structures for dynamic transitive closure. Takes time${\displaystyle O(VE+E^{31/16}(\log V{)}^2)}$, which simplifies to${\displaystyle O(VE)}$ for${\displaystyle E=O(V^{16/15-\epsilon })}$, while previous bounds simplify to${\displaystyle O(VE)}$ for${\displaystyle E=\mathrm{\Omega }(V^{1+ϵ})}$.
Orlin and Gong[32]	2021	${\displaystyle {\displaystyle O\left(\frac{VE\log V}{\log \log V+\log \frac{E}{V}}\right)}}$	Based on a pseudopolynomial algorithm of Ahuja, Orlin, and Tarjan. Faster than King, Rao, and Tarjan and does not use link/cut trees, but not faster than Orlin + KRT.

In parallel to the development of strongly-polynomial flow algorithms, there has been a long line ofpseudo-polynomial and weakly polynomial time bounds, whose running time depends on the magnitude of the input capacities. Here, the value${\displaystyle U}$ refers to the largest edge capacity after rescaling all capacities tointeger values. (If the network containsirrational capacities, this rescaling may not be possible and these algorithms may not produce exact solutions or may fail to converge even to an approximate solution.) The difference between pseudo-polynomial and weakly polynomial is that a pseudo-polynomial bound may be polynomial in${\displaystyle U}$, but for a weakly polynomial bound it can be polynomial only in${\displaystyle \log U}$.

Method	Year	Complexity	Description
Ford–Fulkerson algorithm[33]	1956	${\displaystyle O(EU)}$	As long as there is an open path through the residual graph, send the minimum of the residual capacities on that path.
Binary blocking flow algorithm[34]	1998	${\displaystyle O\left(E\cdot min\{V^{2/3},E^{1/2}\}\cdot \log \frac{V^2}{E}\cdot \log U\right)}$
Kathuria-Liu-Sidford algorithm[35]	2020	${\displaystyle E^{4/3+o(1)}{U}^{1/3}}$	Interior point methods and edge boosting using${\displaystyle {\ell }_p}$-norm flows. Builds on earlier algorithm of Madry, which achieved runtime${\displaystyle \tilde{O}(E^{10/7}{U}^{1/7})}$.[36]
BLNPSSSW / BLLSSSW algorithm[37] [38]	2020	${\displaystyle \tilde{O}((E+V^{3/2})\log U)}$	Interior point methods and dynamic maintenance of electric flows with expander decompositions.
Gao-Liu-Peng algorithm[39]	2021	${\displaystyle \tilde{O}(E^{\frac{3}{2}-\frac{1}{328}}\log U)}$	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[10]	2022	${\displaystyle O(E^{1+o(1)}\log U)}$ The exact complexity is not stated clearly in the paper, but appears to be${\displaystyle E\exp O({\log }^{7/8}E\log \log E)\log U}$	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${\displaystyle E^{1+o(1)}}$ approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized${\displaystyle E^{o(1)}}$ time using a dynamic data structure.
Bernstein, Blikstad, Saranurak, Tu[40]	2024	${\displaystyle O(n^{2+o(1)}\log U)}$ randomized algorithm when the edge capacities come from the set${\displaystyle \{1,\dots ,U\}}$	The algorithm is a variant of the push-relabel algorithm by introducing theweighted 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${\displaystyle n^{o(1)}}$ factor) comes from the difficulty in applying directed expander hierarchies to the computation ofsparse cuts, which do not allow for natural dynamic updating.

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 themax-flow min-cut theorem, but that the flow onevery edge is integral. This is crucial for manycombinatorial 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.

Given a network${\displaystyle N=(V,E)}$ with a set of sources${\displaystyle S=\{s_1,\dots ,s_n\}}$ and a set of sinks${\displaystyle T=\{t_1,\dots ,t_m\}}$ instead of only one source and one sink, we are to find the maximum flow across${\displaystyle N}$. We can transform the multi-source multi-sink problem into a maximum flow problem by adding aconsolidated source connecting to each vertex in${\displaystyle S}$ and aconsolidated sinkconnected by each vertex in${\displaystyle T}$ (also known assupersource andsupersink) with infinite capacity on each edge (See Fig. 4.1.1.).

Given abipartite graph${\displaystyle G=(X\cup Y,E)}$, we are to find amaximum cardinality matching in${\displaystyle G}$, 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${\displaystyle N=(X\cup Y\cup \{s,t\},E^{\prime })}$, where

1. ${\displaystyle E^{\prime }}$ contains the edges in${\displaystyle G}$ directed from${\displaystyle X}$ to${\displaystyle Y}$.
2. ${\displaystyle (s,x)\in E^{\prime }}$ for each${\displaystyle x\in X}$ and${\displaystyle (y,t)\in E^{\prime }}$ for each${\displaystyle y\in Y}$.
3. ${\displaystyle c(e)=1}$ for each${\displaystyle e\in E^{\prime }}$ (See Fig. 4.3.1).

Then the value of the maximum flow in${\displaystyle N}$ is equal to the size of the maximum matching in${\displaystyle G}$, and a maximum cardinality matching can be found by taking those edges that have flow${\displaystyle 1}$ in an integral max-flow.

Given adirected acyclic graph${\displaystyle G=(V,E)}$, we are to find the minimum number ofvertex-disjoint paths to cover each vertex in${\displaystyle V}$. We can construct a bipartite graph${\displaystyle G^{\prime }=(V_{\text{out}}\cup V_{\text{in}},E^{\prime })}$ from${\displaystyle G}$, where

1. ${\displaystyle V_{\text{out}}=\{v_{\text{out}}\mid v\in V\wedge v\text{ has outgoing edge(s)}\}}$
2. ${\displaystyle V_{\text{in}}=\{v_{\text{in}}\mid v\in V\wedge v\text{ has incoming edge(s)}\}}$
3. ${\displaystyle E^{\prime }=\{(u_{\text{out}},v_{\text{in}})\in V_{out}\times V_{in}\mid (u,v)\in E\}}$.

Then it can be shown that${\displaystyle G^{\prime }}$ has a matching${\displaystyle M}$ of size${\displaystyle m}$ if and only if${\displaystyle G}$ has a vertex-disjoint path cover${\displaystyle C}$ containing${\displaystyle m}$ edges and${\displaystyle n-m}$ paths, where${\displaystyle n}$ is the number of vertices in${\displaystyle G}$. Therefore, the problem can be solved by finding the maximum cardinality matching in${\displaystyle G^{\prime }}$ instead.

Assume we have found a matching${\displaystyle M}$ of${\displaystyle G^{\prime }}$, and constructed the cover${\displaystyle C}$ from it. Intuitively, if two vertices${\displaystyle u_{\mathrm{o}\mathrm{u}\mathrm{t}},v_{\mathrm{i}\mathrm{n}}}$ are matched in${\displaystyle M}$, then the edge${\displaystyle (u,v)}$ is contained in${\displaystyle C}$. Clearly the number of edges in${\displaystyle C}$ is${\displaystyle m}$. To see that${\displaystyle C}$ is vertex-disjoint, consider the following:

1. Each vertex${\displaystyle v_{\text{out}}}$ in${\displaystyle G^{\prime }}$ can either benon-matched in${\displaystyle M}$, in which case there are no edges leaving${\displaystyle v}$ in${\displaystyle C}$; or it can bematched, in which case there is exactly one edge leaving${\displaystyle v}$ in${\displaystyle C}$. In either case, no more than one edge leaves any vertex${\displaystyle v}$ in${\displaystyle C}$.
2. Similarly for each vertex${\displaystyle v_{\text{in}}}$ in${\displaystyle G^{\prime }}$ – if it is matched, there is a single incoming edge into${\displaystyle v}$ in${\displaystyle C}$; otherwise${\displaystyle v}$ has no incoming edges in${\displaystyle C}$.

Thus no vertex has two incoming or two outgoing edges in${\displaystyle C}$, which means all paths in${\displaystyle C}$ are vertex-disjoint.

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

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

${\displaystyle \sum _{i\in V}{f}_{iv}\le c(v)\mathrm{\forall }v\in V\mathrm{\setminus }\{s,t\}.}$

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

Given a directed graph${\displaystyle G=(V,E)}$ and two vertices${\displaystyle s}$ and${\displaystyle t}$, we are to find the maximum number of paths from${\displaystyle s}$ to${\displaystyle t}$. 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${\displaystyle N=(V,E)}$ from${\displaystyle G}$, with${\displaystyle s}$ and${\displaystyle t}$ being the source and the sink of${\displaystyle N}$ respectively, and assigning each edge a capacity of${\displaystyle 1}$. In this network, the maximum flow is${\displaystyle k}$ iff there are${\displaystyle k}$ edge-disjoint paths.

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

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${\displaystyle k}$, or at most${\displaystyle k}$. Most variants of this problem areNP-complete, except for small values of${\displaystyle k}$.^[41]

Aclosure of a directed graph is a set of verticesC, such that no edges leaveC. Theclosure problem is the task of finding the maximum-weight or minimum-weight closure in a vertex-weighted directed graph. It may be solved inpolynomial time using a reduction to the maximum flow problem.

In thebaseball elimination problem there aren teams competing in a league. At a specific stage of the league season,w_i is the number of wins andr_i is the number of games left to play for teami andr_ij is the number of games left against teamj. 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^[42] proposed a method which reduces this problem to maximum network flow. In this method a network is created to determine whether teamk is eliminated.

LetG = (V,E) be a network withs,t ∈V being the source and the sink respectively. One adds a game node_ij – which represents the number of plays between these two teams. We also add a team node for each team and connect each game node{i,j} withi <j toV, and connects each of them froms by an edge with capacityr_ij – which represents the number of plays between these two teams. We also add a team node for each team and connect each game node{i,j} with two team nodesi andj to ensure one of them wins. One does not need to restrict the flow value on these edges. Finally, edges are made from team nodei to the sink nodet and the capacity ofw_k +r_k –w_i is set to prevent teami from winning more thanw_k +r_k.LetS be the set of all teams participating in the league and let

.

In this method it is claimed teamk is not eliminatedif and only if a flow value of sizer(S − {k}) exists in networkG. In the mentioned article it is proved that this flow value is the maximum flow value froms tot.

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 flightsF 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 mostk crews.

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

LetG = (V,E) be a network withs,t ∈V as the source and the sink nodes. For the source and destination of every flighti, one adds two nodes toV, nodes_i as the source and noded_i as the destination node of flighti. One also adds the following edges toE:

1. An edge with capacity [0, 1] betweens and eachs_i.
2. An edge with capacity [0, 1] between eachd_i andt.
3. An edge with capacity [1, 1] between each pair ofs_i andd_i.
4. An edge with capacity [0, 1] between eachd_i ands_j, if sources_j is reachable with a reasonable amount of time and cost from the destination of flighti.
5. An edge with capacity [0,∞] betweens andt.

In the mentioned method, it is claimed and proved that finding a flow value ofk inG betweens andt is equal to finding a feasible schedule for flight setF with at mostk crews.^[43]

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 graphG' = (A ∪B,E) is created where each flight has a copy in setA and setB. If the same plane can perform flightj after flighti,i∈A is connected toj∈B. A matching inG' induces a schedule forF and obviously maximum bipartite matching in this graph produces an airline schedule with minimum number of crews.^[43] As it is mentioned in the Application part of this article, the maximum cardinality bipartite matching is an application of maximum flow 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 capacityc 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.

1. Add a source nodes and add edges from it to every factory nodef_i with capacityp_i wherep_i is the production rate of factoryf_i.
2. Add a sink nodet and add edges from all villagesv_i tot with capacityd_i whered_i is the demand rate of villagev_i.

LetG = (V,E) be this new network. There exists a circulation that satisfies the demand if and only if :

Maximum flow value(G)${\displaystyle =\sum _{i\in v}{d}_i}$.

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.^[44]

In their book, Kleinberg and Tardos present an algorithm forsegmenting an image.^[46] 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_i ≥ 0 is the likelihood that pixeli belongs to the foreground,b_i ≥ 0 in the likelihood that pixeli belongs to the background, andp_ij is the penalty if two adjacent pixelsi andj 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

${\displaystyle q(A,B)=\sum _{i\in A}{a}_i+\sum _{i\in B}{b}_i-\sum _{\begin{array}{c}i,j\text{ adjacent}\\ |A\cap \{i,j\}|=1\end{array}}{p}_{ij}}$,

Indeed, for pixels inA (considered as the foreground), we gaina_i; for all pixels inB (considered as the background), we gainb_i. On the border, between two adjacent pixelsi andj, we loosep_ij. It is equivalent to minimize the quantity

${\displaystyle q^{\prime }(A,B)=\sum _{i\in A}{b}_i+\sum _{i\in B}{a}_i+\sum _{\begin{array}{c}i,j\text{ adjacent}\\ |A\cap \{i,j\}|=1\end{array}}{p}_{ij}}$

because

${\displaystyle q(A,B)=\sum _{i\in A\cup B}{a}_i+\sum _{i\in A\cup B}{b}_i-q^{\prime }(A,B).}$

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 pixeli by an edge of weighta_i. We connect the pixeli to the sink by an edge of weightb_i. We connect pixeli to pixelj with weightp_ij. Now, it remains to compute a minimum cut in that network (or equivalently a maximum flow). The last figure shows a minimum cut.

1. In theminimum-cost flow problem, each edge (u,v) also has acost-coefficienta_uv in addition to its capacity. If the flow through the edge isf_uv, then the total cost isa_uvf_uv. It is required to find a flow of a given sized, 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 bydisjunctive constraints: anegative disjunctive constraint says that a certain pair of edges cannot simultaneously have a nonzero flow; apositive disjunctive constraints says that, in a certain pair of edges, at least one must have a nonzero flow. With negative constraints, the problem becomesstrongly NP-hard even for simple networks. With positive constraints, the problem is polynomial if fractional flows are allowed, but may bestrongly NP-hard when the flows must be integral.^[47]

• Joseph Cheriyan andKurt Mehlhorn (1999). "An analysis of the highest-level selection rule in the preflow-push max-flow algorithm".Information Processing Letters.69 (5):239–242.CiteSeerX 10.1.1.42.8563.doi:10.1016/S0020-0190(99)00019-8.
• Daniel D. Sleator andRobert E. Tarjan (1983)."A data structure for dynamic trees"(PDF).Journal of Computer and System Sciences.26 (3):362–391.doi:10.1016/0022-0000(83)90006-5.ISSN 0022-0000.
• Eugene Lawler (2001). "4. Network Flows".Combinatorial Optimization: Networks and Matroids. Dover. pp. 109–177.ISBN 978-0-486-41453-9.

• Network flow problem
• Computational problems in graph theory

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