Tucker,%20Applied%20Combinatorics,%20Sec.%204.3,%20prepared%20by%20Jo%20E-M

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Tucker, Applied Combinatorics, Sec. 4.3, prepared
by Jo E-M
Some Definitions

• Bipartite Graph Matching

Y
X
A set of independent edges
Edges only between X and Y
X-Matching Maximal Matching
All vertices in X are used
The largest possible number of edges
Note an X-matching is necessarily maximal.
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More Definitions
R(A) Range of a set A of vertices

• Edge Cover

R(A) is the set of vertices adjacent to at least
one vertex in A A red, R(A) blue
A set of vertices so that every edge is incident
to at least one of them
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Matching Network
All edges infinite capacity
a
z
All edges capacity 1
By converting the bipartite graph to a network,
we can use network flow techniques to find
matchings.
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Matching Network
All edges infinite capacity
a
z
All edges capacity 1
Matching a z flow
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Matching Network
All edges infinite capacity
a
z
All edges capacity 1
X matching saturation at a
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Matching Network
All edges infinite capacity
a
z
All edges capacity 1
Maximal matching Maximal flow
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Matching Network
All edges infinite capacity
S red vertices, an edge cover
a
z
All edges capacity 1
A red on left, B red on right
Edge cover means that all edges have at least one
red endpoint P to P,
to ,
to P. Thus only uncovered edges,
would go from P to .
Edge cover Finite a z cut
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Matching Network
All edges infinite capacity
S red vertices, NOT an edge cover
a
z
All edges capacity 1
A red on left, B red on right
An infinite cut would go through an edge with two
black endpoints, which corresponds to an
uncovered edge.
Not an edge cover Infinite a z cut
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Theorem 1

• Recall Corollary 3a from Section 4.2
• The size of a maximal flow is equal to the
  capacity of a minimal cut.
• Since.
• Then,
• The size of a maximal matching is equal
• to the size of a minimal edge cover.

Matching a z flow
and
Edge cover Finite a z cut
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Finding Matchings
All edges infinite capacity
a
z
All edges capacity 1
Take any matching, convert to a network, and then
use the augmenting flow algorithm to find a
maximal flow, hence a maximal matching.
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An easy way to think about it

1. Start with any matching.
2. From an unmatched vertex in X, alternate between
   nonmatching and matching edges until you hit an
   unmatched vertex in Y.
3. Then switch between the nonmatching and matching
   edges along to path to pick up one more matching
   edge.
4. Continue this until no unmatched vertex in X
   leads to an unmatched vertex in Y.

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Example
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Theorem 2Halls Matching Theorem

• A bipartite graph has an X-matching if and only
  if for every subset A of X, R(A) is greater
  than or equal to A.

X matching
True for all A in X
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Proof ofHalls Matching Theorem

• An X-matching implies that for
  every subset A of X, R(A) is greater than or
  equal to A.

For any A, the X matching gives at least one
vertex in R(A) for every vertex in A.
X matching
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Proof ofHalls Matching Theorem

• If R(A) is greater than or equal
  to A for every subset A of X, then there is an
  X-matching.

• First note that if M is a maximal matching, then
  .
• Taking A X, we have that
  since the range of X is contained in Y.
  Thus .
• Also, by Theorem 1, if S is a minimal edge cover,
  then .
• Note that if , then M must be
  an X matching.
• Thus, it suffices to show that
  for all edge covers S.

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Proof of Halls Matching Theorem

• Continued we need to show that
  for all edge covers S.

Since ,
. Now let A
be the vertices in X but not in S, so that
. Thus,

. Now, S is an edge-cover, so if it doesnt
contain on the X side, it must contain
all the vertices a goes to on the Y side in order
to cover the edges in between. Thus
, so . Thus,
But , hence
, so
as needed. ///
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Example To Try
Make up a random bipartite graph with about 20
vertices total. Swap with someone, and then both
of you find an X-matching