Maximum Bipartite Matching

1
Maximum Bipartite Matching

• M-augmenting path A path that alternates between
  edges in M and edges not in M and has endpoints
  unsaturated by M.
• Theorem 3.10 A matching M is a maximum matching
  in G iff G has no M-augmenting path.
• We can enlarge M if M-augmenting path exists.

X
Y
2
The Idea

• Iteratively seek augmenting paths to enlarge the
  current matching till no M-augmenting path can be
  found.

3
How to seek M-augmenting path?
X
Y

• Let U be the set of M-unsaturated vertices in X.
  One endpoint of an M-augmenting path is in U.
• If node x is in U and xy?E(G) for some
  unsaturated node y, then an M-augmenting path is
  found.
• Let S?X be the set of vertices reached by
  exploring M-alternating paths from U. If node x
  is in S and xy?E(G) for some unsaturated node y,
  then an M-augmenting path is found.

4
How to seek M-augmenting path?
S
x
x
x
x
x
U
X
w
w
w
y
y
y
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Algorithm 3.2.1

• 1. SU and T?.
• 2. If S has no unmarked vertex, stop and report
  T?(X-S) as a minimum cover and M as a maximum
  matching.
• 3. Explore alternating path from an unmark x?S.
• 3.1. If there exists unsaturated y?N(x) such
  that xy?M, then stop and report an M-augmenting
  path from U to y.
• 3.2. For each xy?E(G), include y in T and
  include w in S, where yw?M. Mark x and Goto Step
  2.

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Theorem 3.2.2

• Repeatedly applying Algorithm 3.2.1 to a
  bipartite graph produces a matching and a vertex
  cover of equal size

7
How to show the current matching M has maximum
size?

• Theorem 3.1.16 If G is a bipartite graph, then
  the maximum size of a matching in G equals to the
  minimum size of a vertex cover in G
• Find a vertex cover with the same size as the
  current matching
• If S has no unmarked vertex (no M-augmenting path
  is found), stop and report T?(X-S) as a minimum
  cover and M as a maximum matching.

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T?(X-S) is a vertex cover

• No edge joins S to Y-T.

S
X-S
X
Y
T
Y-T
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M TX-S

• Mlt TX-S
• T?(X-S) is a vertex cover
• M gt TX-S edges.
• Each y?T is M-saturated.
• Each vertex of X-S is M-saturated.
• No edge in M joins T to X-S.

S
X-S
X
Y
T
Y-T
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Maximum Weighted Bipartite Matching

• A farming company owns n farms and n processing
  plants. The profit that results from sending the
  output of farm i to plant j is wi,j.
• The company wants to select edges forming a
  matching to maximize total profit.

v1
v2
v3
u1
u2
u3
u1
u2
v3
v2
v1
u3
Edge weight Wi,j
w(M)65819
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Minimum Weighted Cover

• The government will pay ui if the company agrees
  not to use farm i and vj if it agrees not to use
  plant j.
• The government must offer amounts such that
  uivjgtwi,j for all i, j. Otherwise, the company
  makes more by using edge xiyj than taking the
  government payments for those vertices.
• The government wants to minimize ?ui?vj.

v1
v2
v3
c(u)
4
5
7
u1
u2
u3
c(v)
0
2
1
Edge weight Wi,j
c(u,v)19
12
Lemma 3.2.7 Duality of minimum c(u,v) and
maximum w(M)

• For a perfect matching M and weighted cover (u,v)
  in a weighted bipartite graph G,
• (1) c(u,v)gtw(M)
• (2) c(u,v) w(M) iff M consists of edges xiyj
  such that uivjwi,j.
• In (2), M and c(u,v) are optimal.

since uivjgtwi,j for all i, j.
4
5
7
c(u)
u1
u2
u3
v3
v2
v1
c(v)
0
2
1
w(M)65819
c(u,v)19
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Equality Subgraph

• Equality Subgraph The equality subgraph Gu,v for
  a weighted cover (u,v) is the spanning subgraph
  of Kn,n whose edges are the pairs xiyj such that
  uivjwi,j.
• In the cover, the excess for i,j is uivj-wi,j.

Kn,n
v1
v2
v3
c(u)
4
5
7
u1
u2
u3
u1
u2
u3
v3
v2
v1
c(v)
0
2
1
Edge weight Wi,j
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Algorithm 3.2.9

• Input A matrix of weights on the edges of Kn,n
  with bipartition X,Y.
• Output A maximum weighted matching.
• Idea Iteratively adjust the cover (u,v) until
  the equality subgraph Gu,v has a perfect matching.

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Algorithm 3.2.9

• Initialization Let (u,v) be a cover, such that
  uimaxj wi,j and vj0.

? u1 max 4, 1, 6, 2, 3 6, v1 0 u2
max 5, 0, 3, 7, 6 7, v2 0 u3 max 2,
3, 4, 5, 8 8, v3 0 u4 max 3, 4, 6, 3,
4 6, v4 0 u5 max 4, 6, 5, 8, 6 8,
v5 0
A matrix of weight
0 0 0 0 0
4 1 6 2 3 5 0 3 7 6 2 3 4 5 8 3 4 6
3 4 4 6 5 8 6
6 7 8 6 8
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Algorithm 3.2.9
0 0 0 0 0
4 1 6 2 3 5 0 3 7 6 2 3 4 5 8 3 4 6
3 4 4 6 5 8 6
60-4 60-1 60-6 60-2 60-3 70-5 70-0
70-3 70-7 70-6 80-2 80-3 80-4 80-5
80-8 60-3 60-4 60-6 60-3 60-4 80-4
80-6 80-5 80-8 80-6
excess matrix
6 7 8 6 8
R
2 5 0 4 3 2 7 4 0 1 6 5 4 3 0 3 2 0
3 2 4 2 3 0 2
X
R
Y
T T
T T
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Algorithm 3.2.9

• How to adjust the cover (u,v) such that the
  equality subgraph Gu,v has a perfect matching?
• Add any edge incident to a vertex in QR?T into
  Gu,v does not work.
• Decrease the excess for each edge not incident to
  a vertex in QR?T.
• No edge has a negative excess after adjusting the
  cover (u,v).

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Algorithm 3.2.9
0 0 0 0 0
R
2 5 0 4 3 2 7 4 0 1 6 5 4 3 0 3 2 0
3 2 4 2 3 0 2
6 7 8 6 8
min
X
e
R
Y
T T
T T T
T T
e
0 0 11 11 01
min
0 0 0 0 0
1 1
0 3 0 4 2 0 5 4 0
0 5 4 5 4 0 1 0 0 3
1 2 0 3 0 1
5-1 6-1 8-1 5-1 7-1
6-1 7-1 8 6-1 8-1
1 4 0 4 2 1 6 4 0
0 6 5 5 4 0 2 1 0 3
1 5 1 3 0 1
TTT
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Algorithm 3.2.9

• 1. Find a maximum matching M in Gu,v. If M is
  perfect matching, stop and report M as a maximum
  weighted matching.
• 2. Let Q be a vertex cover of size M in Gu,v.
  Let RX?Q and TY?Q. Let eminuivj- wi,j
  xi?X-R, yj?Y-T.
• 3. Decreases ui by e for xi ?X-R, and increases
  vj by e for yj ?T.
• 4. Form the new equality graph and Goto Step 1.

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Theorem 3.2.11

• Algorithm 3.2.9 finds a maximum weight matching
  and a minimum cost cover
• Let (u,v) be the current cover and suppose that
  the equality subgraph has no perfect matching.
  Let (u,v) be the new lists of numbers assigned
  to the vertices.
• Show (u,v) is a cover by discuss 3 cases 1)
  edge from X-R to T or from R to Y-T, 2) edges
  from R to T, and 3) edge from X-R to Y-T.

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Theorem 3.2.11

• Case 1 uivj uivj for edges xiyj from X-R to
  T or from R to Y-T.
• ? (u,v) is a cover

0 0 0 0 0
0 0 01 01 0
2 5 0 4 3 2 7 4 0 1 6 5 4 3 0 3 2 0
3 2 4 2 3 0 2
6 7 8 6 8
6-1 7-1 8 6-1 8-1
1 4 0 4 2 1 6 4 0
0 6 5 5 4 0 2 1 0 3
1 5 1 3 0 1
R
R
T T
T T
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Theorem 3.2.11

• Case 2 uivj uivje for edges xiyj from R to
  T.
• ? (u,v) is a cover

0 0 0 0 0
0 0 01 01 0
2 5 0 4 3 2 7 4 0 1 6 5 4 3 0 3 2 0
3 2 4 2 3 0 2
6 7 8 6 8
6-1 7-1 8 6-1 8-1
1 4 0 4 2 1 6 4 0
0 6 5 5 4 0 2 1 0 3
1 5 1 3 0 1
R
R
T T
T T
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Theorem 3.2.11

• Case 3 uivj uivj-e for edges xiyj from X-R
  to Y-T.
• ? (u,v) is a cover

since eminuivj- wi,j xi?X-R, yj?Y-T.
0 0 0 0 0
0 0 01 01 0
2 5 0 4 3 2 7 4 0 1 6 5 4 3 0 3 2 0
3 2 4 2 3 0 2
6 7 8 6 8
6-1 7-1 8 6-1 8-1
1 4 0 4 2 1 6 4 0
0 6 5 5 4 0 2 1 0 3
1 5 1 3 0 1
R
R
T T
T T
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Theorem 3.2.11

• 3. It terminates only when the equality subgraph
  has a perfect matching. ? equal value for the
  current matching and cover.
• 4. We need to show Algorithm 3.2.9 always
  terminate.
• 5. It suffices to show at each iteration the cost
  of the cover is reduced at least e because the
  cost starts at some value and is bounded below by
  the weight of a perfect matching.

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Theorem 3.2.11

• M is not a perfect matching. ? QMltX. ?
  RTltX-RR.

? TltX-R.
-e
X-R
R
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e