Ford–Fulkerson algorithm

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 $G (V, E)$ be a graph, and for each edge from u to v, let $c (u, v)$ be the capacity and $f (u, v)$ 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:

This means that the flow through the network is a legal flow after each round in the algorithm. We define the residual network $G_{f} (V, E_{f})$ to be the network with capacity $c_{f} (u, v) = c (u, v) - f (u, v)$ 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 $f (u, v) > 0$ and $c (v, u) = 0$ then $c_{f} (v, u) = c (v, u) - f (v, u) = f (u, v) > 0$ .

<b>Algorithm</b> Ford<span typeof="mw:Entity">–</span>Fulkerson

Inputs Given a Network

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

$f (u, v) \leftarrow 0$ for all edges $(u, v)$
While there is a path p from s to t in Gf, such that cf(u,v)>0 for all edges (u,v)∈p:
1. Find $c_{f} (p) = min {c_{f} (u, v) : (u, v) \in p}$
2. For each edge (u,v)∈p
  1. $f (u, v) \leftarrow f (u, v) + c_{f} (p)$ (Send flow along the path)
  2. $f (v, u) \leftarrow f (v, u) - c_{f} (p)$ (The flow might be "returned" later)

"←" denotes assignment. For instance, "largest ← item" means that the value of largest changes to the value of item.
"return" terminates the algorithm and outputs the following value.

The path in step 2 can be found with, for example, breadth-first search (BFS) or depth-first search in $G_{f} (V, E_{f})$ . 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 $G (V, E)$ has multiple sources and sinks, we act as follows: Suppose that $T = {t ∣ t is a sink}$ and $S = {s ∣ s is a source}$ . Add a new source $s^{*}$ with an edge $(s^{*}, s)$ from $s^{*}$ to every node $s \in S$ , with capacity $c (s^{*}, s) = d_{s} = \sum_{(s, u) \in E} c (s, u)$ . And add a new sink $t^{*}$ with an edge $(t, t^{*})$ from every node $t \in T$ to $t^{*}$ , with capacity $c (t, t^{*}) = d_{t} = \sum_{(v, t) \in E} c (v, t)$ . Then apply the Ford–Fulkerson algorithm.

Also, if a node u has capacity constraint $d_{u}$ , we replace this node with two nodes $u_{i n}, u_{o u t}$ , and an edge $(u_{i n}, u_{o u t})$ , with capacity $c (u_{i n}, u_{o u t}) = d_{u}$ . We can then apply the Ford–Fulkerson algorithm.

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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 $O (E f)$ (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 $O (E)$ 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 $O (V E^{2})$ 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.

Non-terminating example

Consider the flow network shown on the right, with source $s$ , sink $t$ , capacities of edges $e_{1} = 1$ , $e_{2} = r = (\sqrt{5} - 1) / 2$ and $e_{3} = 1$ , and the capacity of all other edges some integer $M \geq 2$ . The constant $r$ was chosen so, that $r^{2} = 1 - r$ . We use augmenting paths according to the following table, where $p_{1} = {s, v_{4}, v_{3}, v_{2}, v_{1}, t}$ , $p_{2} = {s, v_{2}, v_{3}, v_{4}, t}$ and $p_{3} = {s, v_{1}, v_{2}, v_{3}, t}$ .

Note that after step 1 as well as after step 5, the residual capacities of edges $e_{1}$ , $e_{2}$ and $e_{3}$ are in the form $r^{n}$ , $r^{n + 1}$ and $0$ , respectively, for some $n \in N$ . This means that we can use augmenting paths $p_{1}$ , $p_{2}$ , $p_{1}$ and $p_{3}$ 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 $1 + 2 (r^{1} + r^{2})$ . If we continue to use augmenting paths as above, the total flow converges to $1 + 2 \sum_{i = 1}^{\infty} r^{i} = 3 + 2 r$ . However, note that there is a flow of value $2 M + 1$ , by sending $M$ units of flow along $s v_{1} t$ , 1 unit of flow along $s v_{2} v_{3} t$ , and $M$ units of flow along $s v_{4} t$ . 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 $G (V, E)$ in ordinal numbers is $ω^{Θ (| E |)}$ .

Python implementation of the Edmonds–Karp algorithm

<span id="mwAUE"></span><span class="linenos" data-line="1" id="mwAUI"></span><span class="kn" id="mwAUM">import</span><span class="w" id="mwAUQ"> </span><span class="nn" id="mwAUU">collections</span> <span class="linenos" data-line="2" id="mwAUY"></span> <span class="linenos" data-line="3" id="mwAUc"></span> <span class="linenos" data-line="4" id="mwAUg"></span><span class="k" id="mwAUk">class</span><span class="w" id="mwAUo"> </span><span class="nc" id="mwAUs">Graph</span><span class="p" id="mwAUw">:</span> <span class="linenos" data-line="5" id="mwAU0"></span><span class="w" id="mwAU4"> </span><span class="sd" id="mwAU8">"""</span> <span class="linenos" data-line="6" id="mwAVA"></span><span class="sd" id="mwAVE"> This class represents a directed graph using</span> <span class="linenos" data-line="7" id="mwAVI"></span><span class="sd" id="mwAVM"> adjacency matrix representation.</span> <span class="linenos" data-line="8" id="mwAVQ"></span><span class="sd" id="mwAVU"> """</span> <span class="linenos" data-line="9" id="mwAVY"></span> <span class="linenos" data-line="10" id="mwAVc"></span> <span class="k" id="mwAVg">def</span><span class="w" id="mwAVk"> </span><span class="fm" id="mwAVo">__init__</span><span class="p" id="mwAVs">(</span><span class="bp" id="mwAVw">self</span><span class="p" id="mwAV0">,</span> <span class="n" id="mwAV4">graph</span><span class="p" id="mwAV8">):</span> <span class="linenos" data-line="11" id="mwAWA"></span> <span class="bp" id="mwAWE">self</span><span class="o" id="mwAWI">.</span><span class="n" id="mwAWM">graph</span> <span class="o" id="mwAWQ">=</span> <span class="n" id="mwAWU">graph</span> <span class="c1" id="mwAWY"># residual graph</span> <span class="linenos" data-line="12" id="mwAWc"></span> <span class="bp" id="mwAWg">self</span><span class="o" id="mwAWk">.</span><span class="n" id="mwAWo">row</span> <span class="o" id="mwAWs">=</span> <span class="nb" id="mwAWw">len</span><span class="p" id="mwAW0">(</span><span class="n" id="mwAW4">graph</span><span class="p" id="mwAW8">)</span> <span class="linenos" data-line="13" id="mwAXA"></span> <span class="linenos" data-line="14" id="mwAXE"></span> <span class="k" id="mwAXI">def</span><span class="w" id="mwAXM"> </span><span class="nf" id="mwAXQ">bfs</span><span class="p" id="mwAXU">(</span><span class="bp" id="mwAXY">self</span><span class="p" id="mwAXc">,</span> <span class="n" id="mwAXg">s</span><span class="p" id="mwAXk">,</span> <span class="n" id="mwAXo">t</span><span class="p" id="mwAXs">,</span> <span class="n" id="mwAXw">parent</span><span class="p" id="mwAX0">):</span> <span class="linenos" data-line="15" id="mwAX4"></span><span class="w" id="mwAX8"> </span><span class="sd" id="mwAYA">"""</span> <span class="linenos" data-line="16" id="mwAYE"></span><span class="sd" id="mwAYI"> Returns true if there is a path from</span> <span class="linenos" data-line="17" id="mwAYM"></span><span class="sd" id="mwAYQ"> source 's' to sink 't' in residual graph.</span> <span class="linenos" data-line="18" id="mwAYU"></span><span class="sd" id="mwAYY"> Also fills parent[] to store the path.</span> <span class="linenos" data-line="19" id="mwAYc"></span><span class="sd" id="mwAYg"> """</span> <span class="linenos" data-line="20" id="mwAYk"></span> <span class="linenos" data-line="21" id="mwAYo"></span> <span class="c1" id="mwAYs"># Mark all the vertices as not visited</span> <span class="linenos" data-line="22" id="mwAYw"></span> <span class="n" id="mwAY0">visited</span> <span class="o" id="mwAY4">=</span> <span class="p" id="mwAY8">[</span><span class="kc" id="mwAZA">False</span><span class="p" id="mwAZE">]</span> <span class="o" id="mwAZI">*</span> <span class="bp" id="mwAZM">self</span><span class="o" id="mwAZQ">.</span><span class="n" id="mwAZU">row</span> <span class="linenos" data-line="23" id="mwAZY"></span> <span class="linenos" data-line="24" id="mwAZc"></span> <span class="c1" id="mwAZg"># Create a queue for BFS</span> <span class="linenos" data-line="25" id="mwAZk"></span> <span class="n" id="mwAZo">queue</span> <span class="o" id="mwAZs">=</span> <span class="n" id="mwAZw">collections</span><span class="o" id="mwAZ0">.</span><span class="n" id="mwAZ4">deque</span><span class="p" id="mwAZ8">()</span> <span class="linenos" data-line="26" id="mwAaA"></span> <span class="linenos" data-line="27" id="mwAaE"></span> <span class="c1" id="mwAaI"># Mark the source node as visited and enqueue it</span> <span class="linenos" data-line="28" id="mwAaM"></span> <span class="n" id="mwAaQ">queue</span><span class="o" id="mwAaU">.</span><span class="n" id="mwAaY">append</span><span class="p" id="mwAac">(</span><span class="n" id="mwAag">s</span><span class="p" id="mwAak">)</span> <span class="linenos" data-line="29" id="mwAao"></span> <span class="n" id="mwAas">visited</span><span class="p" id="mwAaw">[</span><span class="n" id="mwAa0">s</span><span class="p" id="mwAa4">]</span> <span class="o" id="mwAa8">=</span> <span class="kc" id="mwAbA">True</span> <span class="linenos" data-line="30" id="mwAbE"></span> <span class="linenos" data-line="31" id="mwAbI"></span> <span class="c1" id="mwAbM"># Standard BFS loop</span> <span class="linenos" data-line="32" id="mwAbQ"></span> <span class="k" id="mwAbU">while</span> <span class="n" id="mwAbY">queue</span><span class="p" id="mwAbc">:</span> <span class="linenos" data-line="33" id="mwAbg"></span> <span class="n" id="mwAbk">u</span> <span class="o" id="mwAbo">=</span> <span class="n" id="mwAbs">queue</span><span class="o" id="mwAbw">.</span><span class="n" id="mwAb0">popleft</span><span class="p" id="mwAb4">()</span> <span class="linenos" data-line="34" id="mwAb8"></span> <span class="linenos" data-line="35" id="mwAcA"></span> <span class="c1" id="mwAcE"># Get all adjacent vertices of the dequeued vertex u</span> <span class="linenos" data-line="36" id="mwAcI"></span> <span class="c1" id="mwAcM"># If an adjacent has not been visited, then mark it</span> <span class="linenos" data-line="37" id="mwAcQ"></span> <span class="c1" id="mwAcU"># visited and enqueue it</span> <span class="linenos" data-line="38" id="mwAcY"></span> <span class="k" id="mwAcc">for</span> <span class="n" id="mwAcg">ind</span><span class="p" id="mwAck">,</span> <span class="n" id="mwAco">val</span> <span class="ow" id="mwAcs">in</span> <span class="nb" id="mwAcw">enumerate</span><span class="p" id="mwAc0">(</span><span class="bp" id="mwAc4">self</span><span class="o" id="mwAc8">.</span><span class="n" id="mwAdA">graph</span><span class="p" id="mwAdE">[</span><span class="n" id="mwAdI">u</span><span class="p" id="mwAdM">]):</span> <span class="linenos" data-line="39" id="mwAdQ"></span> <span class="k" id="mwAdU">if</span> <span class="p" id="mwAdY">(</span><span class="n" id="mwAdc">visited</span><span class="p" id="mwAdg">[</span><span class="n" id="mwAdk">ind</span><span class="p" id="mwAdo">]</span> <span class="o" id="mwAds">==</span> <span class="kc" id="mwAdw">False</span><span class="p" id="mwAd0">)</span> <span class="ow" id="mwAd4">and</span> <span class="p" id="mwAd8">(</span><span class="n" id="mwAeA">val</span> <span class="o" id="mwAeE">></span> <span class="mi" id="mwAeI">0</span><span class="p" id="mwAeM">):</span> <span class="linenos" data-line="40" id="mwAeQ"></span> <span class="n" id="mwAeU">queue</span><span class="o" id="mwAeY">.</span><span class="n" id="mwAec">append</span><span class="p" id="mwAeg">(</span><span class="n" id="mwAek">ind</span><span class="p" id="mwAeo">)</span> <span class="linenos" data-line="41" id="mwAes"></span> <span class="n" id="mwAew">visited</span><span class="p" id="mwAe0">[</span><span class="n" id="mwAe4">ind</span><span class="p" id="mwAe8">]</span> <span class="o" id="mwAfA">=</span> <span class="kc" id="mwAfE">True</span> <span class="linenos" data-line="42" id="mwAfI"></span> <span class="n" id="mwAfM">parent</span><span class="p" id="mwAfQ">[</span><span class="n" id="mwAfU">ind</span><span class="p" id="mwAfY">]</span> <span class="o" id="mwAfc">=</span> <span class="n" id="mwAfg">u</span> <span class="linenos" data-line="43" id="mwAfk"></span> <span class="linenos" data-line="44" id="mwAfo"></span> <span class="c1" id="mwAfs"># If we reached sink in BFS starting from source, then return</span> <span class="linenos" data-line="45" id="mwAfw"></span> <span class="c1" id="mwAf0"># true, else false</span> <span class="linenos" data-line="46" id="mwAf4"></span> <span class="k" id="mwAf8">return</span> <span class="n" id="mwAgA">visited</span><span class="p" id="mwAgE">[</span><span class="n" id="mwAgI">t</span><span class="p" id="mwAgM">]</span> <span class="linenos" data-line="47" id="mwAgQ"></span> <span class="linenos" data-line="48" id="mwAgU"></span> <span class="c1" id="mwAgY"># Returns the maximum flow from s to t in the given graph</span> <span class="linenos" data-line="49" id="mwAgc"></span> <span class="k" id="mwAgg">def</span><span class="w" id="mwAgk"> </span><span class="nf" id="mwAgo">edmonds_karp</span><span class="p" id="mwAgs">(</span><span class="bp" id="mwAgw">self</span><span class="p" id="mwAg0">,</span> <span class="n" id="mwAg4">source</span><span class="p" id="mwAg8">,</span> <span class="n" id="mwAhA">sink</span><span class="p" id="mwAhE">):</span> <span class="linenos" data-line="50" id="mwAhI"></span> <span class="c1" id="mwAhM"># This array is filled by BFS and to store path</span> <span class="linenos" data-line="51" id="mwAhQ"></span> <span class="n" id="mwAhU">parent</span> <span class="o" id="mwAhY">=</span> <span class="p" id="mwAhc">[</span><span class="o" id="mwAhg">-</span><span class="mi" id="mwAhk">1</span><span class="p" id="mwAho">]</span> <span class="o" id="mwAhs">*</span> <span class="bp" id="mwAhw">self</span><span class="o" id="mwAh0">.</span><span class="n" id="mwAh4">row</span> <span class="linenos" data-line="52" id="mwAh8"></span> <span class="linenos" data-line="53" id="mwAiA"></span> <span class="n" id="mwAiE">max_flow</span> <span class="o" id="mwAiI">=</span> <span class="mi" id="mwAiM">0</span> <span class="c1" id="mwAiQ"># There is no flow initially</span> <span class="linenos" data-line="54" id="mwAiU"></span> <span class="linenos" data-line="55" id="mwAiY"></span> <span class="c1" id="mwAic"># Augment the flow while there is path from source to sink</span> <span class="linenos" data-line="56" id="mwAig"></span> <span class="k" id="mwAik">while</span> <span class="bp" id="mwAio">self</span><span class="o" id="mwAis">.</span><span class="n" id="mwAiw">bfs</span><span class="p" id="mwAi0">(</span><span class="n" id="mwAi4">source</span><span class="p" id="mwAi8">,</span> <span class="n" id="mwAjA">sink</span><span class="p" id="mwAjE">,</span> <span class="n" id="mwAjI">parent</span><span class="p" id="mwAjM">):</span> <span class="linenos" data-line="57" id="mwAjQ"></span> <span class="c1" id="mwAjU"># Find minimum residual capacity of the edges along the</span> <span class="linenos" data-line="58" id="mwAjY"></span> <span class="c1" id="mwAjc"># path filled by BFS. Or we can say find the maximum flow</span> <span class="linenos" data-line="59" id="mwAjg"></span> <span class="c1" id="mwAjk"># through the path found.</span> <span class="linenos" data-line="60" id="mwAjo"></span> <span class="n" id="mwAjs">path_flow</span> <span class="o" id="mwAjw">=</span> <span class="nb" id="mwAj0">float</span><span class="p" id="mwAj4">(</span><span class="s2" id="mwAj8">"Inf"</span><span class="p" id="mwAkA">)</span> <span class="linenos" data-line="61" id="mwAkE"></span> <span class="n" id="mwAkI">s</span> <span class="o" id="mwAkM">=</span> <span class="n" id="mwAkQ">sink</span> <span class="linenos" data-line="62" id="mwAkU"></span> <span class="k" id="mwAkY">while</span> <span class="n" id="mwAkc">s</span> <span class="o" id="mwAkg">!=</span> <span class="n" id="mwAkk">source</span><span class="p" id="mwAko">:</span> <span class="linenos" data-line="63" id="mwAks"></span> <span class="n" id="mwAkw">path_flow</span> <span class="o" id="mwAk0">=</span> <span class="nb" id="mwAk4">min</span><span class="p" id="mwAk8">(</span><span class="n" id="mwAlA">path_flow</span><span class="p" id="mwAlE">,</span> <span class="bp" id="mwAlI">self</span><span class="o" id="mwAlM">.</span><span class="n" id="mwAlQ">graph</span><span class="p" id="mwAlU">[</span><span class="n" id="mwAlY">parent</span><span class="p" id="mwAlc">[</span><span class="n" id="mwAlg">s</span><span class="p" id="mwAlk">]][</span><span class="n" id="mwAlo">s</span><span class="p" id="mwAls">])</span> <span class="linenos" data-line="64" id="mwAlw"></span> <span class="n" id="mwAl0">s</span> <span class="o" id="mwAl4">=</span> <span class="n" id="mwAl8">parent</span><span class="p" id="mwAmA">[</span><span class="n" id="mwAmE">s</span><span class="p" id="mwAmI">]</span> <span class="linenos" data-line="65" id="mwAmM"></span> <span class="linenos" data-line="66" id="mwAmQ"></span> <span class="c1" id="mwAmU"># Add path flow to overall flow</span> <span class="linenos" data-line="67" id="mwAmY"></span> <span class="n" id="mwAmc">max_flow</span> <span class="o" id="mwAmg">+=</span> <span class="n" id="mwAmk">path_flow</span> <span class="linenos" data-line="68" id="mwAmo"></span> <span class="linenos" data-line="69" id="mwAms"></span> <span class="c1" id="mwAmw"># update residual capacities of the edges and reverse edges</span> <span class="linenos" data-line="70" id="mwAm0"></span> <span class="c1" id="mwAm4"># along the path</span> <span class="linenos" data-line="71" id="mwAm8"></span> <span class="n" id="mwAnA">v</span> <span class="o" id="mwAnE">=</span> <span class="n" id="mwAnI">sink</span> <span class="linenos" data-line="72" id="mwAnM"></span> <span class="k" id="mwAnQ">while</span> <span class="n" id="mwAnU">v</span> <span class="o" id="mwAnY">!=</span> <span class="n" id="mwAnc">source</span><span class="p" id="mwAng">:</span> <span class="linenos" data-line="73" id="mwAnk"></span> <span class="n" id="mwAno">u</span> <span class="o" id="mwAns">=</span> <span class="n" id="mwAnw">parent</span><span class="p" id="mwAn0">[</span><span class="n" id="mwAn4">v</span><span class="p" id="mwAn8">]</span> <span class="linenos" data-line="74" id="mwAoA"></span> <span class="bp" id="mwAoE">self</span><span class="o" id="mwAoI">.</span><span class="n" id="mwAoM">graph</span><span class="p" id="mwAoQ">[</span><span class="n" id="mwAoU">u</span><span class="p" id="mwAoY">][</span><span class="n" id="mwAoc">v</span><span class="p" id="mwAog">]</span> <span class="o" id="mwAok">-=</span> <span class="n" id="mwAoo">path_flow</span> <span class="linenos" data-line="75" id="mwAos"></span> <span class="bp" id="mwAow">self</span><span class="o" id="mwAo0">.</span><span class="n" id="mwAo4">graph</span><span class="p" id="mwAo8">[</span><span class="n" id="mwApA">v</span><span class="p" id="mwApE">][</span><span class="n" id="mwApI">u</span><span class="p" id="mwApM">]</span> <span class="o" id="mwApQ">+=</span> <span class="n" id="mwApU">path_flow</span> <span class="linenos" data-line="76" id="mwApY"></span> <span class="n" id="mwApc">v</span> <span class="o" id="mwApg">=</span> <span class="n" id="mwApk">parent</span><span class="p" id="mwApo">[</span><span class="n" id="mwAps">v</span><span class="p" id="mwApw">]</span> <span class="linenos" data-line="77" id="mwAp0"></span> <span class="linenos" data-line="78" id="mwAp4"></span> <span class="k" id="mwAp8">return</span> <span class="n" id="mwAqA">max_flow</span>

Algorithm

Complexity

Integer flow example

Non-terminating example

Python implementation of the Edmonds–Karp algorithm

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