The Kuhn-Munkres Algorithm step by step
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The Kuhn-Munkres
Algorithm
step by step
The Kuhn-Munkres Algorithm:
It’s algorithm to find the maximum weight perfect
matching in a weighted complete bipartite graph
Kuhn-Munkres algorithm consider assignment
problem in terms of bipartite graph
The Kuhn-Munkres algorithm
Key idea
find a good feasible labeling that remains enough edges
in equality graph.
to ensure perfect matching can be done
Algorithm describe
Start with any feasible labeling L
and some matching M in GL
While M is not perfect matching, repeat:
Find an augmenting path in GL to increase the size of M
or
if no path exists, improve L to L’ such that GL ⊂ GL’
The Kuhn-Munkres algorithm
Before the algorithm
Y1 Y2 Y3
X1 X2 X3
1 6 8
4
1
6
Original Graph
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
1 6 8
4
1
6
Initial feasiblelabeling
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
6 8
4
Equality Graph + Matching
1)()()( xywylxl
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
1 6 8
4
1
6
Updating feasible labeling:
Or how we compute:
Before the algorithm
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
)()()(min xywylxl
TySx
l
If we have the following labeling L. Assume S={X1,X2} T={y2}
Y1 0 Y2 2 Y3 0
X1 4 X2 6 X3 4
1 6 8
4
1
6
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
1 6 8
4
1
6
Compute only for x1,x2
With all Y except y2l
Find minimum of:
x1, y1 = l(x1)+l(y1)-w(x1,y2)
= 6 + 0 – 1 = 5
x2, y3 = l(x2)+l(y3)-w(x1,y2)
= 8 + 0 – 6 = 2
Subtract 2 from x1,x2Add 2 to y2
TySx &
Generate initial labeling L and matching M in GL
Step 1. If X is M-saturateda (perfect matching), stop
Otherwise, let u be an M-unsaturated vertex.
Set S = {u} and T = .
Step 2. If , go to step 3.
Otherwise, Compute
Replace L by L’ and G by Gl’
Algorithm
TSNlG )(
)()()(min xywylxl
TySx
l
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
Algorithm
TSNylG )(
M yz
)(PEMM
Apply Kuhn- Munkers algorithm to find an optimal matching:
Example :
Y1 Y2 Y3
X1 X2 X3
1 6 8
4
1
6
Original Graph
TSNlG )(
)()()(min xywylxl
TySx
l
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
)(PEMM
Y1 Y2 Y3
X1 X2 X3
1 6 8
4
1
6
Original Graph
Example :
TSNlG )(
)()()(min xywylxl
TySx
l
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
)(PEMM
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
1 6 8
4
1
6
Original Graph
Example :
TSNlG )(
)()()(min xywylxl
TySx
l
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
)(PEMM
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
Example :
TSNlG )(
)()()(min xywylxl
TySx
l
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
)(PEMM
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
Example :
)()()(min xywylxl
TySx
l
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
)(PEMM
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
Example :
TSNlG )(
TSNlG )(
)()()(min xywylxl
TySx
l
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
)(PEMM
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
Example :
)()()(min xywylxl
TySx
l
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
)(PEMM
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
Example :
S = {X1}
T = {}
)(SNlG {y2}
TSNlG )(
)()()(min xywylxl
TySx
l
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
)(PEMM
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
S = {X1}
T = {}
)(SNlG {y2}
Example :
TSNlG )(
)()()(min xywylxl
TySx
l
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
)(PEMM
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
S = {X1}
T = {}
)(SNlG {y2}
Example :
TSNlG )(
)()()(min xywylxl
TySx
l
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
)(PEMM
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
S = {X1}
T = {}
)(SNlG {y2}
Example :
TSNlG )(
)()()(min xywylxl
TySx
l
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
M yz
)(PEMM
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
S = {X1}
T = {}
)(SNlG {y2}
Example :
TSNlG )(
TSNylG )(
TSNlG )(
)()()(min xywylxl
TySx
l
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
M yz
)(PEMM
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
S = {X1}
T = {}
)(SNlG {y2}
Example :
TSNylG )(
TSNlG )(
)()()(min xywylxl
TySx
l
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
M yz
)(PEMM
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
S = {X1, X2}
T = {y2}
)(SNlG {y2}
Go to 2
Example :
TSNylG )(
TSNlG )(
)()()(min xywylxl
TySx
l
)(PEMM
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
S = {X1, X2}
T = {y2}
)(SNlG {y2}
Example :
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
TSNlG )(
)()()(min xywylxl
TySx
l
)(PEMM
Y1 0 Y2 0 Y3 0
X1 6 X2 8 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
S = {X1, X2}
T = {y2}
)(SNlG {y2}
=2
Example :
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
TSNlG )(
)()()(min xywylxl
TySx
l
)(PEMM
Y1 0 Y2 2 Y3 0
X1 4 X2 6 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
S = {X1, X2}
T = {y2}
)(SNlG {y2}
Example :
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
TSNlG )(
)()()(min xywylxl
TySx
l
)(PEMM
Y1 0 Y2 2 Y3 0
X1 4 X2 6 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
S = {X1, X2}
T = {y2}
)(SNlG {y2}
Example :
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
TSNlG )(
)()()(min xywylxl
TySx
l
)(PEMM
Y1 0 Y2 2 Y3 0
X1 4 X2 6 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
S = {X1, X2}
T = {y2}
)(SNlG {y2,y3}
Example :
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
TSNlG )(
)()()(min xywylxl
TySx
l
)(PEMM
Y1 0 Y2 2 Y3 0
X1 4 X2 6 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
S = {X1, X2}
T = {y2}
)(SNlG {y2,y3}
Example :
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
TSNlG )(
)()()(min xywylxl
TySx
l
)(PEMM
Y1 0 Y2 2 Y3 0
X1 4 X2 6 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
S = {X1, X2}
T = {y2}
)(SNlG {y2,y3}
Example :
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
TSNlG )(
)()()(min xywylxl
TySx
l
)(PEMM
Y1 0 Y2 2 Y3 0
X1 4 X2 6 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
S = {X1, X2}
T = {y2}
)(SNlG {y2,y3}
Example :
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
TSNlG )(
)()()(min xywylxl
TySx
l
)(PEMM
Y1 0 Y2 2 Y3 0
X1 4 X2 6 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
S = {X1, X2}
T = {y2}
)(SNlG {y2,y3}
Example :
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
TSNlG )(
)()()(min xywylxl
TySx
l
)(PEMM
Y1 0 Y2 2 Y3 0
X1 4 X2 6 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
S = {X1, X2}
T = {y2}
)(SNlG {y2,y3}
Go to 1
Example :
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
TSNlG )(
)()()(min xywylxl
TySx
l
)(PEMM
Y1 0 Y2 2 Y3 0
X1 4 X2 6 X3 4
1 6 8
4
1
6
Original Graph
Y1 Y2 Y3
X1 X2 X3
Equality Graph + Matching
S = {X1, X2}
T = {y2}
)(SNlG {y2,y3}
Go to 1
stop
Example :
otherwise,)(
if,)(
if,)(
)(^
vl
Tvvl
Svvl
vl l
l
TSNylG )(
M yz
Thanks
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