1. EMIS 8373: Integer Programming Combinatorial Optimization Problems updated 27 January 2005

2. slide 1 Combinatorial Optimization Problems Input –A finite set N = {1, 2, …, n} –Weights (costs) c j for all j  N –A set F of feasible subsets of N Optimization Problem –Find a minimum-weight feasible subset

3. slide 2 COP Example: Minimum Spanning Tree (MST) Input –A (simple) graph G = (V,E) –Edge cost c ij for each edge (i,j)  E Optimization Problem –Find a minimum-cost spanning tree Spanning tree: a set of |V|-1 edges S such that each vertex is incident to at least one edge in S and S contains no cycles.

4. slide 3 MST Example: Input 1 3 2 5 4 4 2 3 3 5 7 1

5. slide 4 MST Example: Some Feasible Spanning Trees 1 3 2 5 4 4 3 3 1 1 3 2 5 4 2 3 7 1 1 3 2 5 4 4 2 3 5 1 3 2 5 4 2 3 3 1 cost = 12 cost = 9 cost = 14 cost = 11

6. slide 5 Example MST as COP N = {(1, 2), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4), (4, 5)}. c 12 = 4, c 15 = 2, c 23 = 3, c 24 = 5, c 25 = 7, c 34 = 3, c 45 =1. F is the set of all spanning trees in G –{(1, 2), (2, 3), (3, 4), (4, 5)}  F. –{(1, 2), (1, 5), (2, 3), (2, 4)}  F. –{(1, 2), (1, 5), (2, 5), (3, 4)}  F. |F| can be very large relative to |E| Optimization Problem –Find a minimum-weight feasible subset

7. slide 6 The Traveling Salesman Problem (TSP) Input –N is a set of cities {1, 2, …, n} –Travel time t ij between cities i and j Optimization Problem –Find a tour of N that starts at city 1, visits each other city exactly once, and then returns to city 1 in the minimum possible time.

8. slide 7 Symmetric TSP Example 3 1 1 3 2 5 4 4 2 3 5 7 5 1 7 t ij Table Graph Representation

9. slide 8 STSP Example: Feasible Tours 1 3 2 5 4 5 5 1 7 1 3 1 3 2 5 4 4 2 3 7 Tour 1 cost = 13 Tour 2 cost = 25 1 3 2 5 4 4 2 3 5 5 Tour 3 cost = 19

10. slide 9 Example STSP as COP N = {(1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3,4), (3,5), (4,5)}. c 12 = 4, c 13 = 7, c 12 = 7, etc. F is the set of tours G where a tour is cycle that contains all the vertices of the graph. –{(1, 2), (1, 5), (2, 3), (3,4), (4,5)}  F. Optimization Problem –Find a minimum-weight feasible subset

11. slide 10 Solving TSP by Enumeration Each feasible tour can be represented by as a permutation  of the cites. –The salesman visits the cities in the order determined by the permutation. Tour 1 corresponds to  = (1, 2, 3, 4, 5) Tour 2 corresponds to  = (1, 3, 5, 2, 4) Tour 3 corresponds to  = (1, 2, 4, 3, 5) –For n cities there are n – 1 choices for the second city, n – 2 for the third, etc. Thus, there are (n-1)! permutations that need be evaluated. –Solving a TSP with 11 cities this way requires evaluating 3.6 million permutations!

12. slide 11 Formulation TSP as BIP Let x ij = 1 if the tour goes from city i to city j; and zero, otherwise. Constraints –The tour enters city j exactly once –The tour leaves city i exactly once Objective function

13. slide 12 3 BIP Solution for Example STSP x 12 = x 23 = x 31 = x 45 = x 54 =1 and all other variables = 0. Objective function value = 10 “Tour” 1 3 2 5 4 4 1 1 1

14. slide 13 Subtour Elimination Constraints To eliminate (prevent) the subtour (cycle) of the cities 1, 2, and 3, add the constraint x 12 + x 13 + x 21 + x 23 + x 31 + x 32  2. To eliminate (prevent) the subtour (cycle) of the cities 4 and 5, add the constraint x 45 + x 54  1.

15. slide 14 x 15 = x 54 = x 42 = x 23 = x 31 =1 and all other variables = 0. Objective function value = 12. Need to add 2 5 -6 = 26 subtour elimination constraints. –Solving BIP with branch-and-bound may be faster than enumeration, but the number of constraints need grows exponentially with number of cities. BIP with Subtour Elimination: Solution for Example STSP

16. slide 15 COP Example: Minimum-Cost Cycle Cover (MCCP) Input –A (simple) graph G = (V,E) –Edge cost c ij for each edge (i,j)  E Optimization Problem –Find a minimum-cost set of cycles C = {C 1, C 2, …, C j } in G such that each edge in E is covered by (contained in) at least one cycle in C. –The cost of a cycle is the total cost of the edges it covers. –The cost of a cycle cover is the total cost of its cycles.

17. slide 16 MCCP Example 1 2 5 4 2 7 Cycle 1 Cost = 13 1 2 5 4 5 7 Cycle 2 Cost = 13 3 3 2 4 3 5 Cycle 3 Cost = 11 Cost of cycle cover = 13+13+11 =37 1 3 2 5 4 4 2 3 3 5 7 1

18. slide 17 Cycles in MCCP Example 1 2 5 4 2 7 Cycle 1 Cost = 13 1 2 5 4 5 7 Cycle 2 Cost = 13 3 3 2 4 3 5 Cycle 3 Cost = 11 3 1 3 2 54 4 2 3 1 Cycle 4 Cost = 13 1 2 54 4 2 5 1 Cycle 5 Cost = 12 3 3 2 54 3 7 1 Cycle 6 Cost = 14

19. slide 18 Example MCCP as COP N is the set of all cycles in G N = {Cycle 1, Cycle 2, Cycle 3, Cycle 4, Cycle 5, Cycle 6} Cycle 1 = {(1, 2), (1,5), (2, 5)}, Cycle 2 = {(2,4),(2,5),(4,5)}, etc. In general, |N| >> |E| c[Cycle 1] = 13, c[Cycle 2] = 13, …, c[Cycle 6] = 14. F: a subset S of N is feasible if the union of all the cycles in S = E. –S = {Cycle 1, Cycle 2} is not feasible. –S = {Cycle 1, Cycle 2, Cycle 3} if feasible.

20. slide 19 Formulation of MCCP as BIP Sets –Let C ={C 1, C 2, …C |C| } be the set of all cycles in the graph G=(V,E). Constants –Let a ij = 1 if cycle j covers edge i; and zero, otherwise. Label the edges 1, 2, 3, …, |E| –Let w j = the cost of cycle j Decision variables –Let the binary variable x j = 1 iff cycle j is selected as part of the cycle cover.

21. slide 20 Formulation of MCCP as BIP Constraints –For each edge i, we must select at least one cycle j such where a ij = 1. Therefore, Objective Function

22. slide 21 BIP Formulation of MCCP

23. slide 22 The Set Covering Problem Input –An m-row, n-column, zero-one matrix A (a ij  {0,1}) –A cost c j associated with each column j Optimization Problem

24. slide 23 The Set Covering Problem The matrix A is know as an incidence matrix and the columns are know as incidence vectors. MCCP is a special case of set covering.