1. 2. Valid Inequalities for the 0-1 Knapsack Polytope Integer Programming 2011 1

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4. 4 C E(C)\C N\E(C)

5. Integer Programming 2011 5

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7. Application to 0-1 IP Integer Programming 2011 7

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9.  For application of valid inequalities for 0-1 knapsack problem to 0-1 IP, see II.6.2.  For the separation of the violated cover inequality, we need to solve 0-1 knapsack problem which is NP-hard. May use heuristics for separation. However, some sophisticated algorithms for the knapsack problem works very well computationally. Hence exact separation may be worth doing.  We also need to solve the 0-1 knapsack problem if we do lifting. However, for 0  1 lifting, it can be solved in polynomial time since the coefficients in the objective function is bounded by n. We can reverse the roles of the objective function and the constraint. See pp. 462, pp. 440-441 Prop 1.6. (The results are for general knapsack problem, but can be modified for 0-1 knapsack.) Integer Programming 2011 9

10. 3. Valid Inequalities for the Symmetric Traveling Salesman Polytope Integer Programming 2009 10

11. Integer Programming 2009 11 1/2 1 1 1 1 1 1 Fractional solution that can’t be cut off by subtour elimination (cut set) constraints (called envelope)

12. Integer Programming 2009 12

13.  Multiply degree constraints for all v  H by ½ and sum them   e  E(H) x e + ½  e   (H) x e = |H|. (3.8) add -½ x e  0, for all e   (H)\  i=1 k E(W i ) to (3.8)   e  E(H) x e + ½  i=1 k  e   (H)  E(Wi ) x e  |H|. (3.9) Consider subtour elimination constraints for W i, H  W i, W i \H, respectively  e  E(Wi) x e  | W i | - 1,for i = 1, …, k  e  E(H  Wi) x e  |H  W i | - 1,for i = 1, …, k  e  E(Wi\H) x e  |W i \ H| - 1,for i = 1, …, k. multiply each of the above by ½, and add to (2.5) ( since E(W i ) = E(W i  H)  E(W i \H)  (E(W i )   (H)) )  e  E(H) x e +  i=1 k  e  E(Wi ) x e  |H| + ½  i=1 k [ (|W i | – 1) + (|H  W i | - 1) + (|W i \ H| - 1) ] = |H| +  i=1 k (|W i | – 1) – k/2 since k is odd  = |H| +  i=1 k (|W i | – 1) – (k+1)/2 Integer Programming 2009 13

14.  There are polynomial time algorithms for separation of subtour elimination constraints and 2-matching inequalities. However, no polynomial time algorithm is known for the separation of more general comb inequalities. (use heuristics)  The comb inequalities can be generalized further  generalized comb inequalities  Thm 3.7: The generalized comb inequalities give facets of conv(S). Integer Programming 2009 14