Ford–Fulkerson algorithm

The Ford–Fulkerson method or Ford–Fulkerson algorithm (FFA) is a greedy algorithm that computes the maximum flow in a flow network. It is sometimes called a "method" instead of an "algorithm" as the approach to finding augmenting paths in a residual graph is not fully specified or it is specified in several implementations with different running times. It was published in 1956 by L. R. Ford Jr. and D. R. Fulkerson. The name "Ford–Fulkerson" is often also used for the Edmonds–Karp algorithm, which is a fully defined implementation of the Ford–Fulkerson method.

The idea behind the algorithm is as follows: as long as there is a path from the source (start node) to the sink (end node), with available capacity on all edges in the path, we send flow along one of the paths. Then we find another path, and so on. A path with available capacity is called an augmenting path.

Algorithm

Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G(V,E)} be a graph, and for each edge from u to v, let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c(u,v)} be the capacity and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(u,v)} be the flow. We want to find the maximum flow from the source s to the sink t. After every step in the algorithm the following is maintained:

Capacity constraints	$\forall (u,v)\in E:\ f(u,v)\leq c(u,v)$	The flow along an edge cannot exceed its capacity.
Skew symmetry	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \forall (u,v)\in E:\ f(u,v)=-f(v,u)}	The net flow from u to v must be the opposite of the net flow from v to u (see example).
Flow conservation	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \forall u\in V:u\neq s{\text{ and }}u\neq t\Rightarrow \sum _{w\in V}f(u,w)=0}	The net flow to a node is zero, except for the source, which "produces" flow, and the sink, which "consumes" flow.
Value(f)	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sum _{(s,u)\in E}f(s,u)=\sum _{(v,t)\in E}f(v,t)}	The flow leaving from s must be equal to the flow arriving at t.

This means that the flow through the network is a legal flow after each round in the algorithm. We define the residual network Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G_{f}(V,E_{f})} to be the network with capacity Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c_{f}(u,v)=c(u,v)-f(u,v)} and no flow. Notice that it can happen that a flow from v to u is allowed in the residual network, though disallowed in the original network: if Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(u,v)>0} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c(v,u)=0} then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c_{f}(v,u)=c(v,u)-f(v,u)=f(u,v)>0} .

Inputs Given a Network Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G=(V,E)} with flow capacity c, a source node s, and a sink node t

Output Compute a flow f from s to t of maximum value

1. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(u,v)\leftarrow 0} for all edges $(u,v)$
2. While there is a path p from s to t in Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G_{f}} , such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c_{f}(u,v)>0} for all edges Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (u,v)\in p} :
   1. Find Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c_{f}(p)=\min\{c_{f}(u,v):(u,v)\in p\}}
   2. For each edge Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (u,v)\in p}
      1. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(u,v)\leftarrow f(u,v)+c_{f}(p)} (Send flow along the path)
      2. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(v,u)\leftarrow f(v,u)-c_{f}(p)} (The flow might be "returned" later)

The path in step 2 can be found with, for example, breadth-first search (BFS) or depth-first search in Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G_{f}(V,E_{f})} . The former is known as the Edmonds–Karp algorithm.

When no more paths in step 2 can be found, s will not be able to reach t in the residual network. If S is the set of nodes reachable by s in the residual network, then the total capacity in the original network of edges from S to the remainder of V is on the one hand equal to the total flow we found from s to t, and on the other hand serves as an upper bound for all such flows. This proves that the flow we found is maximal. See also Max-flow Min-cut theorem.

If the graph Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G(V,E)} has multiple sources and sinks, we act as follows: Suppose that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T=\{t\mid t{\text{ is a sink}}\}} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S=\{s\mid s{\text{ is a source}}\}} . Add a new source $s^{*}$ with an edge Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (s^{*},s)} from $s^{*}$ to every node Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle s\in S} , with capacity Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c(s^{*},s)=d_{s}=\sum _{(s,u)\in E}c(s,u)} . And add a new sink Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle t^{*}} with an edge Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (t,t^{*})} from every node Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle t\in T} to Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle t^{*}} , with capacity Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c(t,t^{*})=d_{t}=\sum _{(v,t)\in E}c(v,t)} . Then apply the Ford–Fulkerson algorithm.

Also, if a node u has capacity constraint Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d_{u}} , we replace this node with two nodes Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle u_{\mathrm {in} },u_{\mathrm {out} }} , and an edge $(u_{\mathrm {in} },u_{\mathrm {out} })$, with capacity Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c(u_{\mathrm {in} },u_{\mathrm {out} })=d_{u}} . We can then apply the Ford–Fulkerson algorithm.

Complexity

By adding the flow augmenting path to the flow already established in the graph, the maximum flow will be reached when no more flow augmenting paths can be found in the graph. However, there is no certainty that this situation will ever be reached, so the best that can be guaranteed is that the answer will be correct if the algorithm terminates. In the case that the algorithm does not terminate, the flow might not converge towards the maximum flow. However, this situation only occurs with irrational flow values. When the capacities are integers, the runtime of Ford–Fulkerson is bounded by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle O(Ef)} (see big O notation), where $E$ is the number of edges in the graph and $f$ is the maximum flow in the graph. This is because each augmenting path can be found in Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle O(E)} time and increases the flow by an integer amount of at least $1$, with the upper bound $f$.

A variation of the Ford–Fulkerson algorithm with guaranteed termination and a runtime independent of the maximum flow value is the Edmonds–Karp algorithm, which runs in Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle O(VE^{2})} time.

Integer flow example

The following example shows the first steps of Ford–Fulkerson in a flow network with 4 nodes, source $A$ and sink $D$. This example shows the worst-case behaviour of the algorithm. In each step, only a flow of $1$ is sent across the network. If breadth-first-search were used instead, only two steps would be needed.

Step	Flow network
Initial flow network.	Error creating thumbnail:
Flow is sent along the augmenting path Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A,B,C,D} . Here, the bottleneck is the $B$–$C$ edge, so only one unit of flow is possible.	Error creating thumbnail:
Here, one unit of flow is sent along the augmenting path Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A,C,B,D} . In this case, flow is "pushed back" from $C$ to $B$. The flow into $C$ that originally came from $B$ now comes from $A$, and $B$ is now free to send flow to $D$ directly. As a result, the $B$–$C$ edge is left with zero flow, but the overall flow increases by one.	Error creating thumbnail:
1998 intermediate steps are omitted here.
The final flow network, with a total flow of 2000 units.	Error creating thumbnail:

Non-terminating example

Error creating thumbnail:

Consider the flow network shown on the right, with source Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle s} , sink Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle t} , capacities of edges Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle e_{1}=1} , Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle e_{2}=r=({\sqrt {5}}-1)/2} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle e_{3}=1} , and the capacity of all other edges some integer $M\geq 2$. The constant Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r} was chosen so, that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r^{2}=1-r} . We use augmenting paths according to the following table, where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p_{1}=\{s,v_{4},v_{3},v_{2},v_{1},t\}} , Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p_{2}=\{s,v_{2},v_{3},v_{4},t\}} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p_{3}=\{s,v_{1},v_{2},v_{3},t\}} .

Step	Augmenting path	Sent flow	Residual capacities
			$e_{1}$	$e_{2}$	$e_{3}$
0			Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r^{0}=1}	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r}	$1$
1	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{s,v_{2},v_{3},t\}}	$1$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r^{0}}	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r^{1}}	$0$
2	$p_{1}$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r^{1}}	$r^{0}-r^{1}=r^{2}$	$0$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r^{1}}
3	$p_{2}$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r^{1}}	$r^{2}$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r^{1}}	$0$
4	$p_{1}$	$r^{2}$	$0$	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r^{1}-r^{2}=r^{3}}	$r^{2}$
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r^2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r^2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r^3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}

Note that after step 1 as well as after step 5, the residual capacities of edges Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_3} are in the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r^n} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r^{n+1}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} , respectively, for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \in \mathbb{N}} . This means that we can use augmenting paths Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_2} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_3} infinitely many times and residual capacities of these edges will always be in the same form. Total flow in the network after step 5 is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 + 2(r^1 + r^2)} . If we continue to use augmenting paths as above, the total flow converges to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textstyle 1 + 2\sum_{i=1}^\infty r^i = 3 + 2r} . However, note that there is a flow of value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2M + 1} , by sending Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} units of flow along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sv_1t} , 1 unit of flow along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sv_2v_3t} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} units of flow along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sv_4t} . Therefore, the algorithm never terminates and the flow does not converge to the maximum flow.

Another non-terminating example based on the Euclidean algorithm is given by Backman & Huynh (2018), where they also show that the worst case running-time of the Ford-Fulkerson algorithm on a network Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(V,E)} in ordinal numbers is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega^{\Theta(|E|)}} .

Python implementation of the Edmonds–Karp algorithm

import collections


class Graph:
    """
    This class represents a directed graph using
    adjacency matrix representation.
    """

    def __init__(self, graph):
        self.graph = graph  # residual graph
        self.row = len(graph)

    def bfs(self, s, t, parent):
        """
        Returns true if there is a path from
        source 's' to sink 't' in residual graph.
        Also fills parent[] to store the path.
        """

        # Mark all the vertices as not visited
        visited = [False] * self.row

        # Create a queue for BFS
        queue = collections.deque()

        # Mark the source node as visited and enqueue it
        queue.append(s)
        visited[s] = True

        # Standard BFS loop
        while queue:
            u = queue.popleft()

            # Get all adjacent vertices of the dequeued vertex u
            # If an adjacent has not been visited, then mark it
            # visited and enqueue it
            for ind, val in enumerate(self.graph[u]):
                if (visited[ind] == False) and (val > 0):
                    queue.append(ind)
                    visited[ind] = True
                    parent[ind] = u

        # If we reached sink in BFS starting from source, then return
        # true, else false
        return visited[t]

    # Returns the maximum flow from s to t in the given graph
    def edmonds_karp(self, source, sink):
        # This array is filled by BFS and to store path
        parent = [-1] * self.row

        max_flow = 0  # There is no flow initially

        # Augment the flow while there is path from source to sink
        while self.bfs(source, sink, parent):
            # Find minimum residual capacity of the edges along the
            # path filled by BFS. Or we can say find the maximum flow
            # through the path found.
            path_flow = float("Inf")
            s = sink
            while s != source:
                path_flow = min(path_flow, self.graph[parent[s]][s])
                s = parent[s]

            # Add path flow to overall flow
            max_flow += path_flow

            # update residual capacities of the edges and reverse edges
            # along the path
            v = sink
            while v != source:
                u = parent[v]
                self.graph[u][v] -= path_flow
                self.graph[v][u] += path_flow
                v = parent[v]

        return max_flow

See also

• Berge's theorem
• Approximate max-flow min-cut theorem
• Turn restriction routing
• Dinic's algorithm

References

• Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). "Section 26.2: The Ford–Fulkerson method". Introduction to Algorithms (Second ed.). MIT Press and McGraw–Hill. pp. 651–664. ISBN 0-262-03293-7.
• George T. Heineman; Gary Pollice; Stanley Selkow (2008). "Chapter 8:Network Flow Algorithms". Algorithms in a Nutshell. Oreilly Media. pp. 226–250. ISBN 978-0-596-51624-6.
• Jon Kleinberg; Éva Tardos (2006). "Chapter 7:Extensions to the Maximum-Flow Problem". Algorithm Design. Pearson Education. pp. 378–384. ISBN 0-321-29535-8.
• Samuel Gutekunst (2019). ENGRI 1101. Cornell University.
• Backman, Spencer; Huynh, Tony (2018). "Transfinite Ford–Fulkerson on a finite network". Computability. 7 (4): 341–347. arXiv:1504.04363. doi:10.3233/COM-180082. S2CID 15497138.

External links

• A tutorial explaining the Ford–Fulkerson method to solve the max-flow problem
• Another Java animation
• Java Web Start application

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