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

3. 2 2.1 Minimum Spanning Trees

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5. 4 a b f g k d h 20 29 25 12 16 28 31 32 23 15 18 22 35

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8. 7 a b f g k d h 20 29 25 12 16 28 31 32 23 15 18 22 35

9. Combinatorial Optimization 2014 8 a b f g k d h 20 29 25 12 16 28 31 32 23 15 18 22 35

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13. Combinatorial Optimization 2014 12 MST and LP

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16. 15 Idea of Branch-and-Bound Algorithm (IP) (LPR) Combinatorial Optimization 2014

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18. 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

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20.  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.

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22. Combinatorial Optimization 2014 21  General questions for a problem: Extreme points of LP relaxation all integer vectors? If not, can we make the polyhedron have only integer extreme points by adding some inequalities to the LP relaxation? How much efforts? If not, can we approximate the integer polyhedron using only part of the inequalities needed to describe it? (obtain good lower bound) How much efforts?

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29. Steiner tree problem Combinatorial Optimization 2014 28

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31.  Example of a Steiner arborescence Combinatorial Optimization 2014 30 a b f g k {r} h : Terminal nodes

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33. Combinatorial Optimization 2014 32 Models for Connectivity of Graphs

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39. Combinatorial Optimization 2014 38  The structure of the survivable network may depend on the facilities used (e.g. ADM (Add-Drop Multiplexor) needs to configure ring networks )