The algorithm performs two basic operations, the \emph{push} and
\emph{relabel} operations, while always maintaining a valid preflow $f$
-and distance labeling $d$.
+and a valid distance labeling $d$.
\subsubsection{The push procedure}
$(u, v) \in E_f$ such that $d(u) = d(v) + 1$, i.e.\ to a vertex with a
smaller distance label. We call such edges \emph{admissible}. See
Algorithm \ref{alg:push} for a pseudocode implementation of the push
-operation. Assuming that $f$ is a valid preflow, it is easy to verify
-that $f$ remains a valid preflow after running the push procedure on
-some admissible edge $(u, v)$. The capacity constraint is fulfilled
-since we at most increase the flow along $(u, v)$ with the residual
-capacity $c_f(u,v)$. The preflow constraint is fulfilled since the
-excess $e$ increases for $v$, remains non-negative for $u$ and remains
-the same for all other vertices.
+operation.
+
+Assuming that $f$ is a valid preflow, it is easy to verify that $f$
+remains a valid preflow after running the push procedure on some
+admissible edge $(u, v)$. The capacity constraint is fulfilled since we
+at most increase the flow along $(u, v)$ with the residual capacity
+$c_f(u,v)$. The preflow constraint is fulfilled since the excess $e$
+increases for $v$, remains non-negative for $u$ and remains the same for
+all other vertices.
The distance labels are not changed during the push procedure, however,
the residual network is changed. The edge $(u, v)$ might disappear, and
d(u) &\leq d(v) + 1 \\
d(v) &\leq d(u) + 1,
\end{align}
-and $d$ remains a valid labeling. \fixme{not very clear}
+and $d$ remains a valid labeling. \fixme{not very clear?}
\fixme{Change height $h$ to distance $d$.}
\label{alg:relabel}
\end{algorithm}
+\fixme{Relabel: What if the set is empty?}
+
If $d$ was a valid labeling before running the relabel procedure, then
we still have $d(u) \leq d(v) + 1$ for all neighbors $v$ of $u$ in the
-residual graph, and $d$ remains a valid labeling.
+residual graph, and $d$ remains a valid labeling. The capacity
+constraint and preflow constraint remain valid assuming they were
+fulfilled before the procedure was started.
\subsubsection{Putting it all together}
-The idea is now to initialize a valid preflow and distance labeling, and
-then run the push and relabel procedures when possible until a maximum
-preflow is obtained.
-
+In the first phase of the algorithm we initialize a valid preflow and
+distance labeling by saturating all edges out of the source $s$, and
+then setting its distance label $d(s) = N$. We then run the push and
+relabel when possible until a maximum preflow is obtained. This phase is
+finished when there are no more active nodes.
+
+\fixme{What is saturated?}
+
+A node $u$ can only be successfully relabeled to obtain a new label if
+the edges outgoing edges of $u$ in the residual network have changed.
+This is why the push and relabel procedures often are combined into a
+\emph{discharge} procedure as shown in Algorithm \ref{alg:discharge}.
+When it is run on an active node $u$, as much as possible of the excess
+flow is pushed to other nodes before a relabeling is done.
\begin{algorithm}
\begin{algorithmic}
\Function{Discharge}{$u$}
\label{alg:discharge}
\end{algorithm}
+In the second phase of the algorithm this preflow is turned into a
+maximum flow by sending excess flow from inside the network back to the
+source. We can skip this part of the algorithm, as we are only
+interested in finding a minimum cut, and not the maximum flow.
+
+The following theorem allows us to find a minimum cut after the firt
+phase of the algorithm is finished.
+\begin{theorem}[Cut identification]
+ Given a network $G = (V, E, c)$, assume that the first phase of
+ the push-relabel algorithm has terminated and no more active nodes
+ remain. Then there exists an $x \in \mathbb{N} \cap \left(0,
+ N \right)$ such that there is no vertex with label $x$, and the
+ vertex sets $S = \{ u : d(u) > x\}$ and $T = V - S$ define a minimum
+ cut $C = (S, T)$ in $G$.
+\end{theorem}
+\begin{proof}
+ \fixme{Sketch of proof, could be nicer, shorter (split up?) and more
+ rigorous maybe.}
+
+ There are $N$ nodes, the source has label $N$ and the sink has label
+ $0$, so the $N - 2$ remaining vertices can not occupy all the $N-1$
+ other labels, and there must exist an $x$ as described.
+
+ As no vertex has label $x$, the set $T$ contains all (and only)
+ vertices with labels less than $x$.
+
+ Assume there was an edge non-saturated edge $(u, v) \in E_f$ such
+ that $u \in S$ and $v \in T$. From the construction of $S$ and $T$,
+ we have $d(v) \leq d(u) + 2$, which contradicts the valid labeling
+ constraint stating that $d(u) \leq d(v) + 1$ for every residual
+ edge. Therefore, no such non-saturated edge from $S$ to $T$ can
+ exist. From the construction of $E_f$ we now know that all edges
+ from $S$ to $T$ are saturated, and all edges from $T$ to $S$ have
+ no flow. This means that the capacity of the cut is equal to the
+ flow through the cut, i.e.\ $c(S, T) = f(S, T)$.
+
+ Since the first phase of the algorithm has terminated, there can be
+ no active vertices, and therefore no excess in $T$, except for the
+ sink. If all flow excess in vertices in $S$ is returned to the
+ source, we can apply the max-flow min-cut theorem to conclude that
+ $C = (S, T)$ is a minimum $s$-$t$-cut, since the cut capacity is
+ equal to the flow from $s$ to $t$.
+\end{proof}
+
+\subsubsection{Complexity}
+Bleep bloop.
+
\subsubsection{Heuristics}
+Different heuristics exists that can speed up the algorithm
+considerably. Being heuristics, they are not guaranteed to work, and
+might perform differently on different kinds of graphs. The most used
+heuristics are the gap- and global relabeling heuristics, both
\begin{algorithm}
\begin{algorithmic}
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