ford fulkerson algorithm

src/SUMMARY.md

@@ -100,8 +100,8 @@
         - [Hamiltonian paths](hamiltonian_path.md)
         - [De Bruijn sequences](de_bruijn_sequences.md)
         - [Knight’s tours](knight_s_tour.md)
-    <!-- - [Flows and cuts](README.md) -->
-<!--         - [Ford–Fulkerson algorithm](README.md) -->
+    - [Flows and cuts](flows_and_cuts.md)
+        - [Ford–Fulkerson algorithm](ford_fulkerson_algorithm.md)
 <!--         - [Disjoint paths](README.md) -->
 <!--         - [Maximum matchings](README.md) -->
 <!--         - [Path covers](README.md) -->

src/flows_and_cuts.md

@@ -0,0 +1,150 @@
+# Flows and cuts
+
+In this chapter, we focus on the following
+two problems:
+
+- **Finding a maximum flow**:
+What is the maximum amount of flow we can
+send from a node to another node?
+- **Finding a minimum cut**:
+What is a minimum-weight set of edges
+that separates two nodes of the graph?
+
+The input for both these problems is a directed,
+weighted graph that contains two special nodes:
+the _source_ is a node with no incoming edges,
+and the _sink_ is a node with no outgoing edges.
+
+As an example, we will use the following graph
+where node 1 is the source and node 6
+is the sink:
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.9]
+\node[draw, circle] (1) at (1,2) {1};
+\node[draw, circle] (2) at (3,3) {2};
+\node[draw, circle] (3) at (5,3) {3};
+\node[draw, circle] (4) at (7,2) {6};
+\node[draw, circle] (5) at (3,1) {4};
+\node[draw, circle] (6) at (5,1) {5};
+\path[draw,thick,->] (1) -- node[font=\small,label=5] {} (2);
+\path[draw,thick,->] (2) -- node[font=\small,label=6] {} (3);
+\path[draw,thick,->] (3) -- node[font=\small,label=5] {} (4);
+\path[draw,thick,->] (1) -- node[font=\small,label=below:4] {} (5);
+\path[draw,thick,->] (5) -- node[font=\small,label=below:1] {} (6);
+\path[draw,thick,->] (6) -- node[font=\small,label=below:2] {} (4);
+\path[draw,thick,<-] (2) -- node[font=\small,label=left:3] {} (5);
+\path[draw,thick,->] (3) -- node[font=\small,label=left:8] {} (6);
+\end{tikzpicture}
+</script>
+
+## Maximum Flow
+
+In the **maximum flow** problem,
+our task is to send as much flow as possible
+from the source to the sink.
+The weight of each edge is a capacity that
+restricts the flow
+that can go through the edge.
+In each intermediate node,
+the incoming and outgoing
+flow has to be equal.
+
+For example, the maximum size of a flow
+in the example graph is 7.
+The following picture shows how we can
+route the flow:
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.9]
+\node[draw, circle] (1) at (1,2) {1};
+\node[draw, circle] (2) at (3,3) {2};
+\node[draw, circle] (3) at (5,3) {3};
+\node[draw, circle] (4) at (7,2) {6};
+\node[draw, circle] (5) at (3,1) {4};
+\node[draw, circle] (6) at (5,1) {5};
+\path[draw,thick,->] (1) -- node[font=\small,label=3/5] {} (2);
+\path[draw,thick,->] (2) -- node[font=\small,label=6/6] {} (3);
+\path[draw,thick,->] (3) -- node[font=\small,label=5/5] {} (4);
+\path[draw,thick,->] (1) -- node[font=\small,label=below:4/4] {} (5);
+\path[draw,thick,->] (5) -- node[font=\small,label=below:1/1] {} (6);
+\path[draw,thick,->] (6) -- node[font=\small,label=below:2/2] {} (4);
+\path[draw,thick,<-] (2) -- node[font=\small,label=left:3/3] {} (5);
+\path[draw,thick,->] (3) -- node[font=\small,label=left:1/8] {} (6);
+\end{tikzpicture}
+</script>
+
+The notation $v/k$ means
+that a flow of $v$ units is routed through
+an edge whose capacity is $k$ units.
+The size of the flow is $7$,
+because the source sends $3+4$ units of flow
+and the sink receives $5+2$ units of flow.
+It is easy see that this flow is maximum,
+because the total capacity of the edges
+leading to the sink is $7$.
+
+## Minimum Cut
+
+In the **minimum cut** problem,
+our task is to remove a set
+of edges from the graph
+such that there will be no path from the source
+to the sink after the removal
+and the total weight of the removed edges
+is minimum.
+
+The minimum size of a cut in the example graph is 7.
+It suffices to remove the edges $2 \rightarrow 3$
+and $4 \rightarrow 5$:
+
+<script type="text/tikz">
+\begin{center}
+\begin{tikzpicture}[scale=0.9]
+\node[draw, circle] (1) at (1,2) {1};
+\node[draw, circle] (2) at (3,3) {2};
+\node[draw, circle] (3) at (5,3) {3};
+\node[draw, circle] (4) at (7,2) {6};
+\node[draw, circle] (5) at (3,1) {4};
+\node[draw, circle] (6) at (5,1) {5};
+\path[draw,thick,->] (1) -- node[font=\small,label=5] {} (2);
+\path[draw,thick,->] (2) -- node[font=\small,label=6] {} (3);
+\path[draw,thick,->] (3) -- node[font=\small,label=5] {} (4);
+\path[draw,thick,->] (1) -- node[font=\small,label=below:4] {} (5);
+\path[draw,thick,->] (5) -- node[font=\small,label=below:1] {} (6);
+\path[draw,thick,->] (6) -- node[font=\small,label=below:2] {} (4);
+\path[draw,thick,<-] (2) -- node[font=\small,label=left:3] {} (5);
+\path[draw,thick,->] (3) -- node[font=\small,label=left:8] {} (6);
+
+\path[draw=red,thick,-,line width=2pt] (4-.3,3-.3) -- (4+.3,3+.3);
+\path[draw=red,thick,-,line width=2pt] (4-.3,3+.3) -- (4+.3,3-.3);
+\path[draw=red,thick,-,line width=2pt] (4-.3,1-.3) -- (4+.3,1+.3);
+\path[draw=red,thick,-,line width=2pt] (4-.3,1+.3) -- (4+.3,1-.3);
+\end{tikzpicture}
+\end{center}
+</script>
+
+After removing the edges,
+there will be no path from the source to the sink.
+The size of the cut is $7$,
+because the weights of the removed edges
+are $6$ and $1$.
+The cut is minimum, because there is no valid
+way to remove edges from the graph such that
+their total weight would be less than $7$.
+
+It is not a coincidence that
+the maximum size of a flow
+and the minimum size of a cut
+are the same in the above example.
+It turns out that a maximum flow
+and a minimum cut are
+_always_ equally large,
+so the concepts are two sides of the same coin.
+
+Next we will discuss the Ford–Fulkerson
+algorithm that can be used to find
+the maximum flow and minimum cut of a graph.
+The algorithm also helps us to understand
+_why_ they are equally large.
+