1. EMIS 8373: Integer Programming Combinatorial Relaxations and Duals Updated 8 February 2005

2. slide 1 TSP Example t ij Table

3. slide 2 Assignment Problem 1 2 3 4 1’ 2’ 3’ 4’ 3 7 10 3 6 7 5 5 7 6 7 5 1 1 1 1

4. slide 3 Mapping Tour 1234 to an Assignment 1 2 3 4 1’ 2’ 3’ 4’ 3 6 7 5 1 1 1 1 assignment cost = 3 + 6 +7 + 5 = 21. cost of tour 1  2  3  4  1 = 21.

5. slide 4 Optimal Assignment 1 2 3 4 1’ 2’ 3’ 4’ 3 3 7 7 1 1 1 1

6. slide 5 Assignment Relaxation of TSP Every TSP tour solution corresponds to a feasible assignment with the same objective function value: –Tour 1  2  3  4  1  x 12 = x 23 = x 34 = x 41 = 1 –Tour 1  3  2  4  1  x 13 = x 32 = x 24 = x 41 = 1 Some assignments do not map back to tours –Assignment x 12 = x 21 = x 34 = x 41 = 1 gives sub-tours 1  2  1 and 3  4  3 Since every TSP tour is a feasible assignment and the cost of the tour = cost of the assignment, the assignment problem is a relaxation of TSP.

7. slide 6 Example 2: Symmetric TSP 10 1 3 2 5 4 4 t ij Table Graph Representation 2 6 9 3 1 2 6 5

8. slide 7 The 1-Tree Relaxation Given an graph G=(V, E) –Let E 1 be the set of edges adjacent to node 1. –Let E 2 = E \ E 1 be the set of edges not adjacent 1. –Let H =(V\{1}, E 2 ) be the subgraph induced by E 2. –A 1-tree is subgraph T=(V,F) of G where F consists of a spanning tree of H and two edges from E 1.

9. slide 8 1-Trees 10 1 3 2 5 4 4 2 6 9 3 1 2 6 5 1 3 2 5 4 6 9 2 6 G T1T1 1 3 2 5 4 2 2 6 T2T2 T1 is TSP tour of G T2 is not a TSP tour of G All tours of G are 1-trees The 1-tree and TSP have the same cost function 1-tree is a relaxation of TSP

10. slide 9 Finding a Minimum-Cost 1-Tree 9 2 10 1 3 2 5 4 4 2 6 3 1 6 5 1 3 2 5 4 4 2 2 1 1. Find MST in H=(V\{1}, E\E 1 ) 2. Add shortest two edges in E 1 Cost = 14 5

11. slide 10 Vertex Covers and Matchings, Let G=(V,E) be an undirected graph. –A vertex cover of G is a subset of the vertices C such that for every edge (i,j) in E vertex i is in C and/or vertex j is in C. –A matching in G is a subset of the edges M such that no two edges in M share the same vertex.

12. slide 11 Example Graph 1 2 4 3 5 6

13. slide 12 A Matching in the Example Graph 1 2 4 3 5 6 M = {(1,5), (2,3), (4,6)}

14. slide 13 A Vertex Cover in the Example 1 2 4 3 5 6 C = {1, 2, 3, 4}

15. slide 14 Duality Result for Matching and Vertex Covers Let G=(V,E) be an undirected graph. –The maximum cardinality matching problem and the minimum cardinality vertex covering problem form a weak-dual pair. –Proof Let M = {(i 1, j 1 ), …, (i k, j k )}, The 2k nodes {i 1, j 1, …, i k, j k } are distinct Any cover C of the edges by V must contain at least one node from (i 1, j 1 ), a least one node from (i 2, j 2 ), …, and a least one node from (i k, j k ). Thus, |C| ≥ k = |M|.