CombinatorialOptimizationandGraphTheory ORCO …szigetiz/OCG/... · 2020. 11. 10. · Zoltan Szigeti Z. Szigeti OCG-ORCO 1/1. AlgorithmHungarianmethod(Kuhn) Example Execute the Hungarian

Combinatorial Optimization and Graph Theory

ORCOMatchings in bipartite graphs

Execution of the Hungarian Method

Zoltán Szigeti

Z. Szigeti OCG-ORCO 1 / 1

Algorithm Hungarian method (Kuhn)

Example

Execute the Hungarian method to find a minimum weight perfect

matching in the following bipartite graph.

U

V

u1 u2 u3 u4

v1 v2 v3 v4

v4

v3

v2

v1 3

u1 u2 u3 u4

2 1 1

1

3

4

4

5

58

8 8

2

7

9

Z. Szigeti OCG-ORCO 1 / 1

Execution of the Hungarian method

3

min{3, 9, 8, 7}

5 38 4 57 82

9311 2

4 18

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3

min{2, 4, 5, 8}

2

5 38 4 57 82

9311 2

4 18

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3

min{1, 1, 2, 4}

2 1

5 38 4 57 82

9311 2

4 18

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3

min{1, 5, 3, 8}

2 1 1

5 38 4 57 82

9311 2

4 18

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

y15 38 4 5

7 8293

11 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

3 + 0 = 3

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

3 + 0 6= 9

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

3 + 0 6= 8

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

3 + 0 6= 7

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

2 + 0 = 2

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

2 + 0 6= 4

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

2 + 0 6= 5

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

2 + 0 6= 8

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

1 + 0 = 1

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

1 + 0 = 1

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

1 + 0 6= 2

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

1 + 0 6= 4

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

1 + 0 = 1

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

1 + 0 6= 5

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

1 + 0 6= 3

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

1 + 0 6= 8

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

3 2 1 1

0 0 0 0

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1 5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

5 5

1 1 1 1

1 1 1 1

5 55

s

t

(D1, g1)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (0, 1)(0, 1)

s

t

(x0, g1)

(0, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (0, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (0, 1)(0, 1)

s

t

(x0, g1)

(0, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (0, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

s

t

D01

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (0, 1)(0, 1)

s

t

(x0, g1)

(0, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (0, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

1s

t

D01

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (0, 1)(0, 1)

s

t

(x0, g1)

(0, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (0, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

1s

t

D01

2

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (0, 1)(0, 1)

s

t

(x0, g1)

(0, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (0, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

3 2

1s

t

D01

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (0, 1)(0, 1)

s

t

(x0, g1)

(0, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (0, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

3 2

1s

t

D01

4

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (0, 1)(0, 1)

s

t

(x0, g1)

(0, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (0, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

4 3 2

1s

t

D01

5

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (0, 1)(0, 1)

s

t

(x0, g1)

(0, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (0, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

4 3 2

1s

t

D01

5

6

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (0, 1)(0, 1)

s

t

(x0, g1)

(0, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (0, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

6

4 3 2

1s

t

D01

5

7

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (0, 1)(0, 1)

s

t

(x0, g1)

(0, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (0, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

t

6

4 3 2

1s

D01

5

7

8

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (0, 1)(0, 1)

s

t

(x0, g1)

(0, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (0, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

s

D01

t

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (0, 1)(0, 1)

s

t

(x0, g1)

(0, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (0, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

s

D01

t

1

5

1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (0, 1)(0, 1)

s

t

(x0, g1)

(0, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (0, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

s

D01

t

1

5

1

ε1 = min{1, 5, 1}

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (1, 1)(0, 1)

s

t

(x1, g1)

(1, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (1, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

s

D01

t

1

5

1

ε1 = min{1, 5, 1}

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (1, 1)(0, 1)

s

t

(x1, g1)

(1, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (1, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

s

t

D11

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (1, 1)(0, 1)

s

t

(x1, g1)

(1, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (1, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

s

t

D11

1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (1, 1)(0, 1)

s

t

(x1, g1)

(1, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (1, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

s

t

D11

1

2

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (1, 1)(0, 1)

s

t

(x1, g1)

(1, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (1, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

2

s

t

D11

1

3

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (1, 1)(0, 1)

s

t

(x1, g1)

(1, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (1, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

3 2

s

t

D11

1

4

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (1, 1)(0, 1)

s

t

(x1, g1)

(1, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (1, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

3 2

s

t

D11

1

4

5

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (1, 1)(0, 1)

s

t

(x1, g1)

(1, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (1, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

5

3 2

s

t

D11

1

4

6

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (1, 1)(0, 1)

s

t

(x1, g1)

(1, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (1, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

t

5

3 2

s

D11

1

4

6

7

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (1, 1)(0, 1)

s

t

(x1, g1)

(1, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (1, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

t

5

3 2

s

D11

1

4

6

7

8

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (1, 1)(0, 1)

s

t

(x1, g1)

(1, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (1, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

t

s

D11

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (1, 1)(0, 1)

s

t

(x1, g1)

(1, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (1, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

t

s

D11

1

5

1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1) (0, 1) (1, 1)(0, 1)

s

t

(x1, g1)

(1, 1) (0, 1)(0, 1)(0, 1)

(0, 5) (0, 5)(0, 5) (1, 5)

(0, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

t

s

D11

1

5

1

ε2 = min{1, 5, 1}

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1)(0, 1)

(1, 5)

(0, 1) (0, 1) (1, 1)(1, 1)

s

t

(x2, g1)

(1, 1) (1, 1)

(0, 5) (0, 5)(0, 5)

(1, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

t

s

D11

1

5

1

ε2 = min{1, 5, 1}

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1)(0, 1)

(1, 5)

(0, 1) (0, 1) (1, 1)(1, 1)

s

t

(x2, g1)

(1, 1) (1, 1)

(0, 5) (0, 5)(0, 5)

(1, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

s

t

D21

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1)(0, 1)

(1, 5)

(0, 1) (0, 1) (1, 1)(1, 1)

s

t

(x2, g1)

(1, 1) (1, 1)

(0, 5) (0, 5)(0, 5)

(1, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

s

t

D21

1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1)(0, 1)

(1, 5)

(0, 1) (0, 1) (1, 1)(1, 1)

s

t

(x2, g1)

(1, 1) (1, 1)

(0, 5) (0, 5)(0, 5)

(1, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

s

t

D21

1

2

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1)(0, 1)

(1, 5)

(0, 1) (0, 1) (1, 1)(1, 1)

s

t

(x2, g1)

(1, 1) (1, 1)

(0, 5) (0, 5)(0, 5)

(1, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

2

s

t

D21

1

3

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1)(0, 1)

(1, 5)

(0, 1) (0, 1) (1, 1)(1, 1)

s

t

(x2, g1)

(1, 1) (1, 1)

(0, 5) (0, 5)(0, 5)

(1, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

2

s

t

D21

1

3

4

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1)(0, 1)

(1, 5)

(0, 1) (0, 1) (1, 1)(1, 1)

s

t

(x2, g1)

(1, 1) (1, 1)

(0, 5) (0, 5)(0, 5)

(1, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

2

s

t

Z1

1

3

4

5

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

(0, 1)(0, 1)

(1, 5)

(0, 1) (0, 1) (1, 1)(1, 1)

s

t

M1

(1, 1) (1, 1)

(0, 5) (0, 5)(0, 5)

(1, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

2

s

t

Z1

1

3

4

5

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G1

M1,T1 = (U \ Z1) ∪ (V ∩ Z1)

U

V

(0, 1)(0, 1)

(1, 5)

(0, 1) (0, 1) (1, 1)(1, 1)

s

t

M1

(1, 1) (1, 1)

(0, 5) (0, 5)(0, 5)

(1, 5)

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

2

s

t

Z1

1

3

4

5

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

U \ T1

G − T1

V \ T1

ε1 = min{c(uv) − y1(u) − y1(v) : uv ∈ E (G − T1)}= min{6, 5, 4, 2, 3, 6, 4, 2, 7} = 2

5 38 4 57 82

9311 2

4 18

0 0 0 0

3 2 1 1

y1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

U \ T1

G − T1

V \ T1

ε1 = min{c(uv) − y1(u) − y1(v) : uv ∈ E (G − T1)}= min{6, 5, 4, 2, 3, 6, 4, 2, 7} = 2

5 38 4 57 82

9311 2

4 18

-2 0 0 0

5 4 1 3

y2

y2(u) = y1(u) + ε1 : u ∈ U \ T1

y2(v) = y1(v) − ε1 : v ∈ V ∩ T1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G2 5 38 4 57 82

9311 2

4 18

-2 0 0 0

5 4 1 3

y2

y2(u) = y1(u) + ε1 : u ∈ U \ T1

y2(v) = y1(v) − ε1 : v ∈ V ∩ T1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G2

(0, 1)(0, 1)

(1, 5)

(0, 1) (0, 1) (1, 1)(1, 1)

s

t

(D2, x2, g2)

(1, 1) (1, 1)

(0, 5) (0, 5)(0, 5)

(1, 5) (0, 5)

5 38 4 57 82

9311 2

4 18

-2 0 0 0

5 4 1 3

y2

y2(u) = y1(u) + ε1 : u ∈ U \ T1

y2(v) = y1(v) − ε1 : v ∈ V ∩ T1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G2

(0, 1)(0, 1)

(1, 5)

(0, 1) (0, 1) (1, 1)(1, 1)

s

t

(D2, x2, g2)

(1, 1) (1, 1)

(0, 5) (0, 5)(0, 5)

(1, 5) (0, 5)

5 38 4 57 82

9311 2

4 18

-2 0 0 0

5 4 1 3

y2

y2(u) = y1(u) + ε1 : u ∈ U \ T1

y2(v) = y1(v) − ε1 : v ∈ V ∩ T1

s

t

D22

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G2

(0, 1)(0, 1)

(1, 5)

(0, 1) (0, 1) (1, 1)(1, 1)

s

t

(D2, x2, g2)

(1, 1) (1, 1)

(0, 5) (0, 5)(0, 5)

(1, 5) (0, 5)

5 38 4 57 82

9311 2

4 18

-2 0 0 0

5 4 1 3

y2

y2(u) = y1(u) + ε1 : u ∈ U \ T1

y2(v) = y1(v) − ε1 : v ∈ V ∩ T1

3 6

t

s

D22

1

2

45

7

8

9

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G2

(0, 1)(0, 1)

(1, 5)

(0, 1) (0, 1) (1, 1)(1, 1)

s

t

(D2, x2, g2)

(1, 1) (1, 1)

(0, 5) (0, 5)(0, 5)

(1, 5) (0, 5)

5 38 4 57 82

9311 2

4 18

-2 0 0 0

5 4 1 3

y2

y2(u) = y1(u) + ε1 : u ∈ U \ T1

y2(v) = y1(v) − ε1 : v ∈ V ∩ T1

t

s

D22

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G2

(0, 1)(0, 1)

(1, 5)

(0, 1) (0, 1) (1, 1)(1, 1)

s

t

(D2, x2, g2)

(1, 1) (1, 1)

(0, 5) (0, 5)(0, 5)

(1, 5) (0, 5)

5 38 4 57 82

9311 2

4 18

-2 0 0 0

5 4 1 3

y2

y2(u) = y1(u) + ε1 : u ∈ U \ T1

y2(v) = y1(v) − ε1 : v ∈ V ∩ T1

t

s

D22

1

15

1

5

ε2 = min{1, 5, 1, 5, 1}

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G2

(0, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(0, 5)

(0, 1) (1, 1)

s

t

(D2, x3, g2) (1, 5) (1, 5)

5 38 4 57 82

9311 2

4 18

-2 0 0 0

5 4 1 3

y2

y2(u) = y1(u) + ε1 : u ∈ U \ T1

y2(v) = y1(v) − ε1 : v ∈ V ∩ T1

t

s

D22

1

15

1

5

ε2 = min{1, 5, 1, 5, 1}

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G2

(0, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(0, 5)

(0, 1) (1, 1)

s

t

(D2, x3, g2) (1, 5) (1, 5)

5 38 4 57 82

9311 2

4 18

-2 0 0 0

5 4 1 3

y2

y2(u) = y1(u) + ε1 : u ∈ U \ T1

y2(v) = y1(v) − ε1 : v ∈ V ∩ T1

s

t

D32

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G2

(0, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(0, 5)

(0, 1) (1, 1)

s

t

(D2, x3, g2) (1, 5) (1, 5)

5 38 4 57 82

9311 2

4 18

-2 0 0 0

5 4 1 3

y2

y2(u) = y1(u) + ε1 : u ∈ U \ T1

y2(v) = y1(v) − ε1 : v ∈ V ∩ T1

s

t

Z2

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G2

(0, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(0, 5)

(0, 1) (1, 1)

s

t

(D2, x3, g2) (1, 5) (1, 5)

5 38 4 57 82

9311 2

4 18

-2 0 0 0

5 4 1 3

y2

y2(u) = y1(u) + ε1 : u ∈ U \ T1

y2(v) = y1(v) − ε1 : v ∈ V ∩ T1

s

t

Z2

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

U

G2

M2,T2 = (U \ Z2) ∪ (V ∩ Z2)

V

(0, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(0, 5)

(0, 1) (1, 1)

s

t

(D2, x3, g2) (1, 5) (1, 5)

5 38 4 57 82

9311 2

4 18

-2 0 0 0

5 4 1 3

y2

y2(u) = y1(u) + ε1 : u ∈ U \ T1

y2(v) = y1(v) − ε1 : v ∈ V ∩ T1

s

t

Z2

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

U \ T2

G − T2

V \ T2

ε2 = min{c(uv) − y2(u) − y2(v) : uv ∈ E (G − T2)}= min{3, 2, 1, 4, 1, 3} = 1

5 38 4 57 82

9311 2

4 18

-2 0 0 0

5 4 1 3

y2

y2(u) = y1(u) + ε1 : u ∈ U \ T1

y2(v) = y1(v) − ε1 : v ∈ V ∩ T1

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

U \ T2

G − T2

V \ T2

ε2 = min{c(uv) − y2(u) − y2(v) : uv ∈ E (G − T2)}= min{3, 2, 1, 4, 1, 3} = 1

26

0-3

5 38 4 57 82

9311 2

4 18

-1 0

5 3

y2

y3(u) = y2(u) + ε2 : u ∈ U \ T2

y3(v) = y2(v) − ε2 : v ∈ V ∩ T2

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G3

26

0-3

5 38 4 57 82

9311 2

4 18

-1 0

5 3

y2

y3(u) = y2(u) + ε2 : u ∈ U \ T2

y3(v) = y2(v) − ε2 : v ∈ V ∩ T2

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G3

(0, 5)

(0, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(0, 1) (1, 1)

s

t

(D3, x3, g3) (1, 5) (1, 5)

(0, 5)

26

0-3

5 38 4 57 82

9311 2

4 18

-1 0

5 3

y2

y3(u) = y2(u) + ε2 : u ∈ U \ T2

y3(v) = y2(v) − ε2 : v ∈ V ∩ T2

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G3

(0, 5)

(0, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(0, 1) (1, 1)

s

t

(D3, x3, g3) (1, 5) (1, 5)

(0, 5)

26

0-3

5 38 4 57 82

9311 2

4 18

-1 0

5 3

y2

y3(u) = y2(u) + ε2 : u ∈ U \ T2

y3(v) = y2(v) − ε2 : v ∈ V ∩ T2

s

t

D33

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G3

(0, 5)

(0, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(0, 1) (1, 1)

s

t

(D3, x3, g3) (1, 5) (1, 5)

(0, 5)

26

0-3

5 38 4 57 82

9311 2

4 18

-1 0

5 3

y2

y3(u) = y2(u) + ε2 : u ∈ U \ T2

y3(v) = y2(v) − ε2 : v ∈ V ∩ T2

s

t

Z3

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G3

(0, 5)

(0, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(0, 1) (1, 1)

s

t

(D3, x3, g3) (1, 5) (1, 5)

(0, 5)

26

0-3

5 38 4 57 82

9311 2

4 18

-1 0

5 3

y2

y3(u) = y2(u) + ε2 : u ∈ U \ T2

y3(v) = y2(v) − ε2 : v ∈ V ∩ T2

s

t

Z3

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

U

G3

M3,T3 = (U \ Z3) ∪ (V ∩ Z3)

V

(0, 5)

(0, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(0, 1) (1, 1)

s

t

(D3, x3, g3) (1, 5) (1, 5)

(0, 5)

26

0-3

5 38 4 57 82

9311 2

4 18

-1 0

5 3

y2

y3(u) = y2(u) + ε2 : u ∈ U \ T2

y3(v) = y2(v) − ε2 : v ∈ V ∩ T2

s

t

Z3

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

V \ T3

U \ T3

G − T3

ε3 = min{c(uv) − y3(u) − y3(v) : uv ∈ E (G − T3)}= min{1, 3, 2, 5} = 1

26

0-3

5 38 4 57 82

9311 2

4 18

-1 0

5 3

y2

y3(u) = y2(u) + ε2 : u ∈ U \ T2

y3(v) = y2(v) − ε2 : v ∈ V ∩ T2

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

V \ T3

U \ T3

G − T3

ε3 = min{c(uv) − y3(u) − y3(v) : uv ∈ E (G − T3)}= min{1, 3, 2, 5} = 1

0-2-4

467 3

-1

5 38 4 57 82

9311 2

4 18

y2

y4(u) = y3(u) + ε3 : u ∈ U \ T3

y4(v) = y3(v) − ε3 : v ∈ V ∩ T3

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G4

0-2-4

467 3

-1

5 38 4 57 82

9311 2

4 18

y2

y4(u) = y3(u) + ε3 : u ∈ U \ T3

y4(v) = y3(v) − ε3 : v ∈ V ∩ T3

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G4

(0, 5)

(0, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(0, 1) (1, 1)

s

t

(D4, x4, g4) (1, 5) (1, 5)

(0, 5)(0, 5)

0-2-4

467 3

-1

5 38 4 57 82

9311 2

4 18

y2

y4(u) = y3(u) + ε3 : u ∈ U \ T3

y4(v) = y3(v) − ε3 : v ∈ V ∩ T3

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G4

(0, 5)

(0, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(0, 1) (1, 1)

s

t

(D4, x4, g4) (1, 5) (1, 5)

(0, 5)(0, 5)

0-2-4

467 3

-1

5 38 4 57 82

9311 2

4 18

y2

y4(u) = y3(u) + ε3 : u ∈ U \ T3

y4(v) = y3(v) − ε3 : v ∈ V ∩ T3

s

t

D44

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G4

(0, 5)

(0, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(0, 1) (1, 1)

s

t

(D4, x4, g4) (1, 5) (1, 5)

(0, 5)(0, 5)

0-2-4

467 3

-1

5 38 4 57 82

9311 2

4 18

y2

y4(u) = y3(u) + ε3 : u ∈ U \ T3

y4(v) = y3(v) − ε3 : v ∈ V ∩ T3

t

s

D44

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G4

(0, 5)

(0, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(0, 1) (1, 1)

s

t

(D4, x4, g4) (1, 5) (1, 5)

(0, 5)(0, 5)

0-2-4

467 3

-1

5 38 4 57 82

9311 2

4 18

y2

y4(u) = y3(u) + ε3 : u ∈ U \ T3

y4(v) = y3(v) − ε3 : v ∈ V ∩ T3

t

s

D44

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G4

(0, 5)

(0, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(0, 1) (1, 1)

s

t

(D4, x4, g4) (1, 5) (1, 5)

(0, 5)(0, 5)

0-2-4

467 3

-1

5 38 4 57 82

9311 2

4 18

y2

y4(u) = y3(u) + ε3 : u ∈ U \ T3

y4(v) = y3(v) − ε3 : v ∈ V ∩ T3

t

s

D44

1

1

5

ǫ4 = min{1, 5, 1}

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G4

(0, 5)

(1, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(1, 1) (1, 1)

s

t

(D4, x5, g4) (1, 5) (1, 5)

(0, 5)(1, 5)

0-2-4

467 3

-1

5 38 4 57 82

9311 2

4 18

y2

y4(u) = y3(u) + ε3 : u ∈ U \ T3

y4(v) = y3(v) − ε3 : v ∈ V ∩ T3

t

s

D44

1

1

5

ǫ4 = min{1, 5, 1}

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G4

(0, 5)

(1, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(1, 1) (1, 1)

s

t

(D4, x5, g4) (1, 5) (1, 5)

(0, 5)(1, 5)

0-2-4

467 3

-1

5 38 4 57 82

9311 2

4 18

y2

y4(u) = y3(u) + ε3 : u ∈ U \ T3

y4(v) = y3(v) − ε3 : v ∈ V ∩ T3

s

t

D54

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G4

(0, 5)

(1, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(1, 1) (1, 1)

s

t

(D4, x5, g4) (1, 5) (1, 5)

(0, 5)(1, 5)

0-2-4

467 3

-1

5 38 4 57 82

9311 2

4 18

y2

y4(u) = y3(u) + ε3 : u ∈ U \ T3

y4(v) = y3(v) − ε3 : v ∈ V ∩ T3

s

t

Z4

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

G4

(0, 5)

(1, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(1, 1) (1, 1)

s

t

(D4, x5, g4) (1, 5) (1, 5)

(0, 5)(1, 5)

0-2-4

467 3

-1

5 38 4 57 82

9311 2

4 18

y2

y4(u) = y3(u) + ε3 : u ∈ U \ T3

y4(v) = y3(v) − ε3 : v ∈ V ∩ T3

s

t

Z4

Z. Szigeti OCG-ORCO 2 / 1

Execution of the Hungarian method

V

U

G4

M4,T4 = (U \ Z4) ∪ (V ∩ Z4)

perfect matching of minimum weight

(0, 5)

(1, 1)(1, 1) (1, 1)

(0, 5) (1, 5)(0, 5)

(1, 1)(1, 1)

(1, 1)

(1, 1) (1, 1)

s

t

(D4, x5, g4) (1, 5) (1, 5)

(0, 5)(1, 5)

0-2-4

467 3

-1

5 38 4 57 82

9311 2

4 18

y2

y4(u) = y3(u) + ε3 : u ∈ U \ T3

y4(v) = y3(v) − ε3 : v ∈ V ∩ T3

s

t

Z4

Z. Szigeti OCG-ORCO 2 / 1