2. Combinatorial Optimization 2012 1 3.5 Minimum Cuts in Undirected Graphs 3.5.1 Global Minimum Cuts

3. Combinatorial Optimization 2012 2

4. 3 Identifying nodes f, g 2 1 2 5 2 6 3 3 2 a 5 3 h f g e d b c 2 1 2 5 2 3 3 2 a 5 3 h x e d b c 4 4

5. Combinatorial Optimization 2012 4

6. 5 Node Identification Algorithm

7. Combinatorial Optimization 2012 6 2 1 2 5 2 6 3 3 2 a 5 3 h f g e d c  Legal ordering beginning with a is : a, b, c, d, e, h, g, f b 4

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9. 8

10. 9

11. 10  randomized algorithm for minimum cut problem Random Contraction Algorithm

12. Combinatorial Optimization 2012 11

13. Combinatorial Optimization 2012 12

14. Combinatorial Optimization 2012 13 3.5.2 Cut-Trees p q

15. Combinatorial Optimization 2012 14 E3 E2E1 D2 D1 C1 A B2B1 B3 General Procedure E3 E2E1 Y B2B1 B3 D2 D1 C1 Z f(y,z)

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17. Combinatorial Optimization 2012 16 S (1) r w v s X (2) r

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19. Combinatorial Optimization 2012 18 A B X Y h h x y a a B b Proof of Lemma 3.42

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21. Combinatorial Optimization 2012 20  A variant of Gomory-Hu procedure can be used to identify the violated odd set constraint for the matching problem (Ref: M. W. Padberg and M. R. Rao (1982), Odd Minimum Cut-Sets and b- Matchings, Mathematics of Operations Research 7, 67-80.)  More efficient implementation: D. Gusfield, "Very simple methods for all pairs network flow analysis," SIAM Journal on Computing 19 (1990) 143- 155