3.2.1. augmenting path algorithm two theorems to recall: theorem 3.1.10 (berge). a matching m in a...
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3.2.1. Augmenting path algorithm
Two theorems to recall:
Theorem 3.1.10 (Berge). A matching M in a graph G is a maximum matching in G iff G has no M-augmenting path.
Theorem 3.1.16 (König,Egerváry) If G is bipartite, then a maximum matching and a minimum vertex cover of G have the same size (’(G)=(G)).
Augmenting path algorithm for maximum bipartite matching
Input: X,Y-bigraph. Output: M,Q with |M|=|Q|
Use modified breadth-first search to find augmenting paths.
Initialize with empty matching and iteratively increase by 1.
Produce a vertex cover of same size to certify the output.
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3.2.1. Augmenting path algorithm
Algorithm (Augmenting path algorithm)
Input An X,Y-bigraph G, a (partial matching) M, and
the set U of M-unsaturated vertices of X
Idea Explore augmenting paths to all possible vertices, marking explored vertices and their predecessors, and tracking reached vertices SµX and TµY
Initialization S=U and T=;Iteration If S is all marked, stop and output M and Q=T[(X-S).
Otherwise, explore from an unmarked x2S.
For any edge xy2E(G)-M: (1) mark y and put y in T. (2) For any edge yw2M, mark w and put w in S.
Stop if an unsaturated y is found; report an M-augmenting path.
Otherwise continue exploring in this fashion.
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Augmenting path algorithm example
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
Iteration 1 Input: M=; S=U=X, T=;, x=x1
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SSS S S
Augmenting path algorithm example
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
Iteration 1: y1 unsaturated. Halt and augment M.
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T
SSS S S
Augmenting path algorithm example
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
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Iteration 2 Input: M={x1y1} S={x2, x3, x4, x5}, T=;, x=x2
SSS S
Augmenting path algorithm example
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
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Iteration 2: From x2, explore y1 and its matched neighbors
T
SSS S S
Augmenting path algorithm example
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
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Iteration 2: From x1, explore y2 and its matched neighbors.y2 is unsaturated: halt and augment M.
T
SSS S S
T
Augmenting path algorithm example
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
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Iteration 3 Input: M={x1y2, x2y1} S={x3, x4, x5}, T=;, x=x3
SS S
Augmenting path algorithm example
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
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Iteration 3: From x3, explore y3 and its matched neighbors.y3 unsaturated, so terminate and report augmenting path.
SS S
T
Augmenting path algorithm example
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
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Iteration 4 Input: M={x1y2, x2y1, x3y3} S={x4, x5}, T=;, x=x4
S S
Augmenting path algorithm example
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
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Iteration 4: From x4, explore y2,y3 and their (distinct) matched neighbors.
SSS
T T
S
Augmenting path algorithm example
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
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Iteration 4: From x1, and x3, explore to find y1 via non-matching edges. Explore back up to x2 via a matching edge.
SSS
T T
S
T
S
Augmenting path algorithm example
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
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Iteration 4 (continued): x5 is still unexplored. Explore from x5 to find unsaturated vertex y4. Terminate and report an M-augmenting path.
SSS S
T TT
S
T
Augmenting path algorithm example
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
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Iteration 5 Input: M={x1y2,x2y1,x3y3,x5y4} S={x4}, T=;, x=x4
S
Augmenting path algorithm example
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
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Iteration 5: From x4, explore y2,y3 and their (distinct) matched neighbors.
SSS
T T
Augmenting path algorithm example
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
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Iteration 5: From x1, and x3, explore to find y1 via non-matching edges. Explore back up to x2 via a matching edge.
SSS S
T TT
Augmenting path algorithm example
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
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Termination: No vertex of S is unexplored.Output: maximum matching M={x1y2,x2y1,x3y3,x5y4}Minimum vertex cover Q=T[(X-S) ={x5,y1,y2,y3}.
SSS S
T TT
Q
QQQ
Maximum Weighted Transversal
0 0 0 0 0
6 4 4 4 3 6
4 1 1 4 3 4
5 1 4 5 3 5
9 5 6 4 7 9
8 5 3 6 8 3
Transversal: A set M of n entries of an n£n, matrix, no two in the same column or row. Its weight is the sum of the entries.Cover: A pair of vectors (u,v)=(u1,…,un;v1,…,vn) such that every entry wi,j of the matrix satisfies wi,j · ui+vj.
u
v
w(M)=23
w(u,v)=32
3.2.7. Maximum Transversal = Minimum Cover
3.2.7. Lemma. (Duality of weighted matching and weighted cover problems) For a perfect matching M and a weighted cover (u,v) in a weighted bipartite graph G, c(u,v)¸w(M). Also, c(u,v)=w(M) iff M consists of edges xiyj such that ui+vj=wi,j. In this case, M and (u,v) are optimal.
0 1 2
4 4 4 4
2 1 1 4
3 1 4 5
u
v
w(M)=12
w(u,v)=12
3.2.9. Hungarian Algorithm (Maximum Weighted Matching/Minimum Weighted Cover)
Algorithm (Hungarian Algorithm)
Input An n£n matrix of nonnegative edge weights of Kn,n
Idea Iteratively adjust the cover (u,v) until the equality subgraph Gu,v has a perfect matching
Initialization Any cover (u,v), such as ui=maxi wi,j and vj=0
Iteration Find a maximum matching M in Gu,v. If M is a perfect matching, stop and output M and (u,v) as a maximum weight matching and minimum weight cover.
Otherwise:
Let = smallest excess in Gu,v of an edge from X-R to Y-T.
Replace uià ui- for all xi2 X-R.
Replace vjà vj+ for all yj2 Y-T.
Form the new equality subgraph Gu,v and repeat.
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Hungarian Algorithm Example
0 0 0 0 06 4 1 6 2 3
7 5 0 3 7 6
8 2 3 4 5 8
6 3 4 6 3 4
8 4 6 5 8 6
uv
w(M)=21 w(u,v)=35
0 0 0 0 0
6 2 5 0 4 3
7 2 7 4 0 1
8 6 5 4 3 0
6 3 2 0 3 2
8 4 2 3 0 2
uv excess matrix
T T
R
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
T T
R
Equality subgraphvertex cover: Q=R[T
Add =1 to T, subtract from X-R.
Hungarian Algorithm Example
0 0 1 1 05 4 1 6 2 3
6 5 0 3 7 6
8 2 3 4 5 8
5 3 4 6 3 4
7 4 6 5 8 6
uv
w(M)=21 w(u,v)=33
0 0 1 1 0
5 1 4 0 4 2
6 1 6 4 0 0
8 6 5 5 4 0
5 2 1 0 3 1
7 3 1 3 0 1
uv excess matrix
T T
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
T T T
Equality subgraphvertex cover: Q=R[T
T
Add =1 to T, subtract from X-R.
Hungarian Algorithm Example
0 0 2 2 14 4 1 6 2 3
5 5 0 3 7 6
7 2 3 4 5 8
4 3 4 6 3 4
6 4 6 5 8 6
uv
w(M)=31 w(u,v)=31
0 0 2 2 1
4 0 3 0 4 2
5 0 5 4 0 0
7 5 4 5 4 0
4 1 0 0 3 1
6 2 0 3 0 1
uv excess matrix
T T
x1 x2 x3 x4 x5
y1 y2 y3 y4 y5
T
Matching weight=cover weight.Halt and output M, and (u,v).
equality subgraph