Iterative
improvement
Bipartite matching
Bipartite graph
Definition: A graph that can be partitioned into
two disjoint sets (not necessarily of the same
size) so that all edges go between them.
Equivalent: does not have any odd length cycles;
or is 2-colorable
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G = V, U, E
Checking 2-colorability
a
b
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d
Using either DFS or BFS, alternately mark
vertices even or odd and check that no back or
cross edges connect ones of the same parity.
Matching
A subset M  E of edges such that no two share
a common vertex.
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Try to achieve the largest number |M| of edges.
Augmenting paths
Idea: augment current matching M by a path P
between two free vertices which alternates
between edges in and out of M to form M  P.
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Is perfect when all original vertices are included.
Example
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(c)
(b)
(a)
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Augmenting
Augmenting
Augmenting
path:
path:
3,
path:
8,2,4,6,
1,9,1,
65,710
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(c)
(b)
(d)
Maximum matching
Maximum matching
Theorem: A matching M is maximal if and only if there is
no augmenting path with respect to M.
Proof: One direction is obvious: if there is an augmenting
path, then apply it to increase M. For the other direction,
if M* is a larger matching, then |M*−M| > |M−M*|. Each
endpoint in M' = MM* is incident to no more than one
edge in M and one edge in M*. So with degree two or
less, every connected component of M' is either a path or
an even-length cycle of alternating edges from M−M* to
M*−M. Since the latter is bigger, there must be at least
one such path that starts and ends with an edge in
M*−M, and this is an augmenting path.