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PROFESSOR: The stable match
problem that we just looked at

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is one example of a
bipartite matching problem.

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So the setup with a
bipartite matching problem

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is you've got a simple graph.

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And the vertices are
split into two groups,

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as in the stable
matching problem,

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we can call them the girls
and the boys, the G and the b.

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So the definition
of a bipartite graph

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is a graph where there
are some vertices called

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the left vertices and a
disjoint set of vertices

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called the right vertices.

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And every vertex is
either left or right.

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And edges only go between a
left vertex and a right vertex.

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Now, in this case,
the matching problem

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that we want to
consider is that there

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is a specification
that each girl is

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willing to be paired
with certain boys,

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but not all of them.

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So we can specify that
by adding edges where,

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if this is the first
girl on the list,

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and she is willing to be
paired with the second boy

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and the last boy.

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And that's what those
two edges indicate.

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So edges are signaling
compatibility constraints

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on matching up the
girls and the boys.

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And what we're
trying to accomplish

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is getting all of the girls
matched with a unique boy--

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match each girl to a
unique compatible boy.

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So there's an
example of a match,

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where there is one highlighted
magenta edge out of each girl.

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And they go to different boys.

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So formally, we want a
bijection from the girls

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to the boys that follows edges.

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Well, let's look at a case
where I can't find a match.

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Suppose that that
edge was missing.

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We used that edge in the match.

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But let's suppose
it was not there.

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Let's get rid of it.

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And what we find now is that
this last girl no longer

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can be matched to
this second boy, which

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is what we previously had.

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So let's try to find
some other match.

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And there isn't any.

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And the reason is
that if you look

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at this group of three
girls on the left

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and you look at all of
the boys on the right

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that they are collectively
compatible with-- that

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is, one of these
three girls at least

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is willing to be paired with
one of the boys on the right--

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there are only
two boys that have

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to be shared among three girls.

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And that is one example of
what's called a bottleneck.

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So we have three girls.

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And collectively, they
only like two boys.

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There just are not enough boys
to go around for these girls.

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That proves that a match is
not going to be possible.

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So more generally,
if you have a set

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S of girls on the left and
you look at the image of S

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under the edge
relation-- that is

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E of S, which is collectively
the set of all of the boys

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that are compatible with one or
more of the girls in S-- then

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whenever you have--
So we previously just

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had an example where S
was 3 and E of S was 2.

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And because 3 was
greater than 2--

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because S was greater than E
of S-- we were bottlenecked.

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And we couldn't
possibly find a match.

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And more generally, the
definition of a bottleneck

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is that if you have a
set where the size of S

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is greater than the
size of the image of S,

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then that's called a bottleneck.

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And the first
observation we can make

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is the bottleneck lemma says
that a bottleneck is a set

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S of girls without enough boys.

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And if S is greater than E of
S, that's called a bottleneck.

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And when there is one, no
match is possible, obviously.

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So this is a reason
why there might not

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be a match, is that
there is a bottleneck.

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Now, a rather deep
theorem is conversely,

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if there are no bottlenecks,
then in fact there is a match.

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This is known as Hall's theorem.

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It's not obvious,
although we'll find

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an understandable proof of it.

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And that's what we're going
to do in the next segment.