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Antti H S Laaksonen 2017-05-29 22:11:00 +03:00
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chapter20.tex
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@ -542,16 +542,16 @@ and there are some edges between the sets:
\end{center}
The size of the cut is the sum of the edges
that go from the set $A$ to the set $B$.
that go from $A$ to $B$.
This is an upper bound for the flow
in the graph, because the flow has to proceed
from the set $A$ to the set $B$.
Thus, a maximum flow is smaller than or equal to
any cut in the graph.
from $A$ to $B$.
Thus, the size of a maximum flow is smaller than or equal to
the size of any cut in the graph.
On the other hand, the Ford–Fulkerson algorithm
produces a flow that is \emph{exactly} as large
as a cut in the graph.
produces a flow whose size is \emph{exactly} as large
as the size of a cut in the graph.
Thus, the flow has to be a maximum flow
and the cut has to be a minimum cut.
@ -773,14 +773,14 @@ from the source to the sink is 1.
\index{maximum matching}
The \key{maximum matching} problem asks to find
a maximum-size set of node pairs of a graph
a maximum-size set of node pairs in an undirected graph
such that each pair is connected with an edge and
each node belongs to at most one pair.
There are polynomial algorithms for finding
maximum matchings in general graphs \cite{edm65},
but such algorithms are complex and do
not appear in programming contests.
but such algorithms are complex and
rarely seen in programming contests.
However, in bipartite graphs,
the maximum matching problem is much easier
to solve, because we can reduce it to the
@ -813,7 +813,7 @@ the groups are $\{1,2,3,4\}$ and $\{5,6,7,8\}$.
\path[draw,thick,-] (4) -- (7);
\end{tikzpicture}
\end{center}
The size of a maximum matching of the graph is 3:
The size of a maximum matching of this graph is 3:
\begin{center}
\begin{tikzpicture}[scale=0.60]
\node[draw, circle] (1) at (2,4.5) {1};
@ -843,8 +843,8 @@ to the maximum flow problem by adding two new nodes
to the graph: a source and a sink.
We also add edges from the source
to each left node and from each right node to the sink.
After this, the maximum flow of the graph
equals the maximum matching of the original graph.
After this, the size of a maximum flow in the graph
equals the size of a maximum matching in the original graph.
For example, the reduction for the above
graph is as follows:
@ -895,11 +895,8 @@ The maximum flow of this graph is as follows:
\node[draw, circle] (a) at (-2,2.25) {\phantom{0}};
\node[draw, circle] (b) at (12,2.25) {\phantom{0}};
%\path[draw,thick,->] (1) -- (5);
%\path[draw,thick,->] (2) -- (7);
\path[draw,thick,->] (3) -- (5);
\path[draw,thick,->] (3) -- (6);
%\path[draw,thick,->] (3) -- (8);
\path[draw,thick,->] (4) -- (7);
\path[draw,thick,->] (a) -- (1);
@ -1007,7 +1004,7 @@ we cannot form such a matching.
Since $X$ contains more nodes than $f(X)$,
there are no pairs for all nodes in $X$.
For example, in the above graph, both nodes 2 and 4
should be connected with node 7 which is not allowed.
should be connected with node 7 which is not possible.
\subsubsection{Kőnig's theorem}
@ -1052,9 +1049,9 @@ with a maximum matching of size 3:
\path[draw=red,thick,-,line width=2pt] (3) -- (6);
\end{tikzpicture}
\end{center}
Kőnig's theorem tells us that the size
Now Kőnig's theorem tells us that the size
of a minimum node cover is also 3.
It can be constructed as follows:
Such a cover can be constructed as follows:
\begin{center}
\begin{tikzpicture}[scale=0.60]
@ -1116,7 +1113,7 @@ set is as follows:
\index{path cover}
A \key{path cover} is a set of paths of a graph
A \key{path cover} is a set of paths in a graph
such that each node of the graph belongs to at least one path.
It turns out that in directed, acyclic graphs,
we can reduce the problem of finding a minimum
@ -1177,12 +1174,12 @@ by constructing a \emph{matching graph} where each node
of the original graph is represented by
two nodes: a left node and a right node.
There is an edge from a left node to a right node
if there is a such an edge in the original graph.
if there is such an edge in the original graph.
In addition, the matching graph contains a source and a sink,
and there are edges from the source to all
left nodes and from all right nodes to the sink.
A maximum matching of the resulting graph corresponds
A maximum matching in the resulting graph corresponds
to a minimum node-disjoint path cover in
the original graph.
For example, the following matching graph