Chapter 2. Optimal Trees and Paths Combinatorial Optimization 2014 1.

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Presentation transcript:

Chapter 2. Optimal Trees and Paths Combinatorial Optimization

2 2.1 Minimum Spanning Trees

Combinatorial Optimization

4 a b f g k d h

Combinatorial Optimization

6

7 a b f g k d h

Combinatorial Optimization a b f g k d h

Combinatorial Optimization

10

Combinatorial Optimization

Combinatorial Optimization MST and LP

Combinatorial Optimization

Combinatorial Optimization

15 Idea of Branch-and-Bound Algorithm (IP) (LPR) Combinatorial Optimization 2014

16 Combinatorial Optimization 2014

17  Results of solving LP relaxation. 1.unbounded  integer program unbounded 2.infeasible  integer program infeasible 3.optimal solution which is integer  it is optimal to integer program 4.optimal solution not integer  only obtain lower bound. Need to branch  How to divide the solution set in case of 4. Suppose x * optimal solution to LP relaxation and Then consider 2 sets Any feasible solution to integer program is contained in one of (a), (b). So we do not miss any feasible solution. Then we solve LP relaxation of (a), (b) again. (Search procedure with tree structure) Combinatorial Optimization 2014

18 Combinatorial Optimization 2014

 Procedure to solve the LP relaxation (with many constraints) Cutting plane approach 19 Solve LP relaxation with small number of constraints (e.g., w/o subtour elim. constr.)  violated constr? Solve LP after adding the violated constraint. Y N Stop Combinatorial Optimization 2014 * Dual form of column generation, e.g. cutting stock problem.

Combinatorial Optimization

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Combinatorial Optimization  The structure of the survivable network may depend on the facilities used (e.g. ADM (Add-Drop Multiplexor) needs to configure ring networks )