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\begin{document}
\lecture{17} {Matching in Bipartite Graphs}{Hardeep Singh Guru}

\section{Matching}

\begin{Def}
A \em{Matching} in a graph is a set of edges, no two of which share
an end-point.
\end{Def}

Let us look at an example. In the following graph the marked edges
form a matching as no two share a common vertex.

\begin{figure}[h]
 \centering
 \includegraphics[width=1in,height=1.7in]{lec17h_img1.jpg}
 \caption{Matching}
 \label{Matching}
\end{figure}

\section{Algorithm}

We now look at an algorithm which takes as input a bipartite graph
and outputs a matching of maximum size. The basic idea of the
algorithm lies in iteratively increasing the size of the matching by
one at every stage until a matching of maximum size is obtained.

Let $M$ be the current set of edges in the matching and let $M_0$ be
any maximum matching.

Now, consider the graph induced by $M$ $\oplus$ $M_0$. We can
clearly see that this graph would consist of connected components
with the degree of each vertex in these being either zero, one or
two. The degree would be zero for the end points of the edges which
are matched in both $M$ and $M_0$ or in neither. The degree will be
one in the case when there is a matching edge incident on that
vertex in only one of the matching and two when two edges incident
on the vertex are matched, one in $M$ and the other in $M_0$.

\begin{figure}[h!]
 \centering
 \includegraphics[width=4.5in,height=.8in]{lec17h_img2.jpg}
 \caption{Connected Components}
 \label{Connected Components}
\end{figure}

It can also be seen that the connected components of $M$ $\oplus$
$M_0$ are either paths or cycles.

What can we say about the cycles? We can see that the cycles would
necessarily need to be of even length.

Hence, we see that the connected components are either paths or
cycles of even length.

Now, let us consider a matching $M$ which has size exactly one less
the size of the maximum matching $M_0$.

\newtheorem{claim}{Claim}
\begin{claim}
$M$ $\oplus$ $M_0$ will contain exactly one path of odd length.
\end{claim}

Suppose that there are two odd length paths. They can be either one
of the following types.

\begin{figure}[h!]
 \centering
 \includegraphics[width=2.2in,height=1in]{lec17h_img3.jpg}
 \caption{Odd Length Paths}
 \label{Odd Length Paths}
\end{figure}

Consider the path of first type where the first edge in the path
belongs to $M$. This cannot be the case since we can otherwise
switch the matched edges in $M$ and $M_0$ which belong to this path
and hence increase the size of $M_0$ by one.

Also, the paths of type two starting with $M_0$ cannot be more than
one in number because then we can switch the matched edges belonging
to $M$ and $M_0$ in all these paths and hence get a matching of size
greater then the size of the matching in $M_0$ which is a
contradiction since $M_0$ has been assumed to be the maximum
matching.

Similarly one can prove the following claim.

\newtheorem{claim2}{Claim}
\begin{claim}
If $M$ is not a maximum there is a path in the graph which starts
and ends at vertices that are unmatched (i.e., do not have any edge
of $M$ incident on them) and alternate edges in the path are from
$M$. That is the first, third, $\dots$, last edges are not in $M$
while the second, fourth, $\dots$ are in $M$.
\end{claim}

For a proof note that such a path exists in the graph induced by $M$
$\oplus$ $M_0$. All other connected components in $M$ $\oplus$ $M_0$
have at least as many edges from $M$ as from $M_0$. Such an path is
called an augmenting path. Suppose we find an augmenting path, then
we can increase the size of $M$ by ``switching" the matched and the
unmatched edges. That is remove the matched edges in the path from
$M$ and introduce in the unmatched edges on the path in $M_0$.

\begin{figure}[h!tb]
 \centering
 \includegraphics[width=4.5in,height=2in]{lec17h_img4.jpg}
 \caption{Augmenting Path}
 \label{Augmenting Path}
\end{figure}

The broad outline of our algorithm is now clear. In each iteration,
find an augmenting path and increase the size of the matching. We
next see how to find such an augmenting path.

We start with a vertex which has no matched edges and label it as
``even". The other end of the edges from this vertex are labeled as
``odd" as shown in the figure. At the ``odd" level we go through
only the matched edges and at ``even" level we go through all the
edges. Suppose we find an odd vertex that is unmatched, we have
found the path that we are looking for which consists of alternating
matched and unmatched edges and starts and ends with an unmatched
vertex. Hence we can switch the matched and unmatched edges along
this path and increase the size of the matching by one. We
iteratively repeat this procedure until no augmenting path can be
found in the graph. The matching thus obtained will be of maximum
size.

Also, we can ignore back edges since edges from an ``even" vertex
will be only to an ``odd" vertex (the graph is bipartite).

Next time we will prove that the algorithm is indeed correct.

\end{document}