Combinatorial Optimization Minimum Cuts in Undirected Graphs Global Minimum Cuts

Combinatorial Optimization

3 Identifying nodes f, g a 5 3 h f g e d b c a 5 3 h x e d b c 4 4

Combinatorial Optimization

5 Node Identification Algorithm

Combinatorial Optimization 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

Combinatorial Optimization

8

9

10  randomized algorithm for minimum cut problem Random Contraction Algorithm

Combinatorial Optimization

Combinatorial Optimization

Combinatorial Optimization Cut-Trees p q

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

Combinatorial Optimization

Combinatorial Optimization S (1) r w v s X (2) r

Combinatorial Optimization

Combinatorial Optimization A B X Y h h x y a a B b Proof of Lemma 3.42

Combinatorial Optimization

Combinatorial Optimization  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, )  More efficient implementation: D. Gusfield, "Very simple methods for all pairs network flow analysis," SIAM Journal on Computing 19 (1990)