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\begin{document}

	\section{Residual Graphs}
	As our network gets bigger, finding a maximum flow by hand becomes much more difficult. It will be convenient to have an algorithm that finds a maximal flow in any network.

	\vspace{1ex}

	The first thing we'll need to construct such an algorithm is a \textit{residual graph}.

	\vspace{2ex}
	\hrule

	\begin{center}
		\begin{minipage}[t]{0.48\textwidth}
		We'll start with the following network and flow:
		\begin{center}
			\begin{tikzpicture}[node distance = 20mm]
				% Nodes
				\begin{scope}[layer = nodes]
					\node[main] (S) {$S$};
					\node[main] (A) [above right of = S] {$A$};
					\node[main] (B) [below right of = S] {$B$};
					\node[main] (T) [above right of = B] {$T$};
				\end{scope}

				% Edges
				\draw[-&gt;]
					(S) edge node[label] {$1$} (A)
					(A) edge node[label] {$3$} (T)
					(B) edge node[label] {$2$} (A)
					(S) edge node[label] {$2$} (B)
					(B) edge node[label] {$1$} (T)
				;

				% Flow
				\draw[path]
					(S) -- node[above left, flow] {$(1)$} (A)
					-- node[above right, flow] {$(1)$} (T)
				;

			\end{tikzpicture}
		\end{center}
		\end{minipage}
		\hfill
		\begin{minipage}[t]{0.48\textwidth}
		First, we'll copy all nodes and \say{unused} edges:
		\begin{center}
			\begin{tikzpicture}[node distance = 20mm]
				% Nodes
				\begin{scope}[layer = nodes]
					\node[main] (S) {$S$};
					\node[main] (A) [above right of = S] {$A$};
					\node[main] (B) [below right of = S] {$B$};
					\node[main] (T) [above right of = B] {$T$};
				\end{scope}

				% Edges
				\draw[-&gt;]
					(B) edge node[label] {$2$} (A)
					(S) edge node[label] {$2$} (B)
					(B) edge node[label] {$1$} (T)
				;
			\end{tikzpicture}
		\end{center}
		\end{minipage}
	\end{center}

	\hrule

	\begin{center}
		\begin{minipage}[t]{0.48\textwidth}
		Then, we'll add the unused capacity of \say{used} edges: (Note that $3 - 1 = 2$)
		\begin{center}
			\begin{tikzpicture}[node distance = 20mm]
				% Nodes
				\begin{scope}[layer = nodes]
					\node[main] (S) {$S$};
					\node[main] (A) [above right of = S] {$A$};
					\node[main] (B) [below right of = S] {$B$};
					\node[main] (T) [above right of = B] {$T$};
				\end{scope}

				% Edges
				\draw[-&gt;]
					(A) edge node[label] {$2$} (T)
					(B) edge node[label] {$2$} (A)
					(S) edge node[label] {$2$} (B)
					(B) edge node[label] {$1$} (T)
				;
			\end{tikzpicture}
		\end{center}
		\end{minipage}
		\hfill
		\begin{minipage}[t]{0.48\textwidth}
		Finally, we'll add \say{used} capacity as edges in the opposite direction:
		\begin{center}
			\begin{tikzpicture}[node distance = 20mm]
				% Nodes
				\begin{scope}[layer = nodes]
					\node[main] (S) {$S$};
					\node[main] (A) [above right of = S] {$A$};
					\node[main] (B) [below right of = S] {$B$};
					\node[main] (T) [above right of = B] {$T$};
				\end{scope}

				% Edges
				\draw[-&gt;]
					(A) edge node[label] {$1$} (S)
					(T) edge [bend right] node[label] {$1$} (A)
					(A) edge [bend right] node[label] {$2$} (T)
					(B) edge node[label] {$2$} (A)
					(S) edge node[label] {$2$} (B)
					(B) edge node[label] {$1$} (T)
				;
			\end{tikzpicture}
		\end{center}
		This graph is the residual of the original flow.
	\end{minipage}
	\end{center}

	\hrule
	\vspace{3ex}

	You can think of the residual graph as a \say{list of possible changes} to the original flow. \\
	There are two ways we can change a flow:
	\begin{itemize}
		\item We can add flow along a path
		\item We can remove flow along another path
	\end{itemize}

	\vspace{1ex}

	A residual graph captures both of these actions, showing us where we can add flow (forward edges) and where we can remove it (reverse edges). Note that \say{removing} flow along an edge is equivalent to adding flow in the opposite direction.


	\vfill
	\pagebreak

	\problem{}&lt;FindResidual&gt;
	Construct the residual of this flow.

	\begin{center}
		\begin{tikzpicture}[node distance = 25mm]
			% Nodes
			\begin{scope}[layer = nodes]
				\node[main] (S) {$S$};
				\node[main] (A) [above right of = S] {$A$};
				\node[main] (B) [below right of = S] {$B$};
				\node[main] (T) [above right of = B] {$T$};
			\end{scope}

			% Edges
			\draw[-&gt;]
				(S) edge node[label] {$2$} (A)
				(A) edge node[label] {$1$} (T)
				(A) edge node[label] {$3$} (B)
				(S) edge node[label] {$1$} (B)
				(B) edge node[label] {$2$} (T)
			;

			% Flow
			\draw[path]
				(S)
				-- node[above left, flow] {$(2)$} (A)
				-- node[left, flow] {$(2)$} (B)
				-- node[below right, flow] {$(2)$} (T)
			;
		\end{tikzpicture}
	\end{center}

	\begin{solution}
		\begin{center}
			\begin{tikzpicture}[node distance = 25mm]
				% Nodes
				\begin{scope}[layer = nodes]
					\node[main] (S) {$S$};
					\node[main] (A) [above right of = S] {$A$};
					\node[main] (B) [below right of = S] {$B$};
					\node[main] (T) [above right of = B] {$T$};
				\end{scope}

				% Edges
				\draw[-&gt;]
					(A) edge node[label] {$2$} (S)
					(A) edge node[label] {$1$} (T)
					(A) edge[out=295,in=65] node[label] {$1$} (B)
					(B) edge[out=115,in=245] node[label] {$2$} (A)
					(S) edge node[label] {$1$} (B)
					(T) edge node[label] {$2$} (B)
				;
			\end{tikzpicture}
		\end{center}
	\end{solution}

	\vfill
	\problem{}
	Is the flow in \ref{FindResidual} maximal? \\
	If it isn't, find a maximal flow. \\
	\hint{Look at the residual graph. Can we add flow along another path?}

	\vfill
	\pagebreak

	\problem{}
	Show that...
	\begin{enumerate}
		\item A maximal flow exists in every network with integral\footnotemark{} edge weights.
		\item Every edge in this flow carries an integral amount of flow
	\end{enumerate}

	\footnotetext{Integral = \say{integer} as an adjective.}

	\vfill


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