1. 1 Integrality constraints Integrality constraints are often crucial when modeling optimizayion problems as linear programs. We have seen that if our linear program expresses a network flow problem, we may rephrase it so that such integrality constraints are guaranteed to be satisfied by the solution found. What do we do if we want integrality constraints but our linear program does not express a network flow problem?

2. 2 Mixed Integer Linear Programs (MILP) Find x 2 R n minimizing or maximizing a linear form h x,c i =  i c i x i (the objective function) so that a given set of linear non-strict inequalities and integrality constraints x i 2 Z are satisfied. A feasible solution to the program is a point x satisfying the inequalities and integrality constraints.

3. 3 Integer Linear Programs (ILP) Find x 2 Z n minimizing or maximizing a linear form h x,c i =  i c i x i (the objective function) so that a given set of linear non-strict inequalities are satisfied. A feasible solution to the program is a point x satisfying the inequalities.

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5. 5 Power of ILP 0-1 variables may be interpreted as Boolean variables. Logical constraints on Boolean variables may be expressed by inequalities. Consequence (to be seen in the course “Combinatorial Search”): ILP is a universal language. It can express any “simple” search/optimization problem.

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8. 8 Traveling Salesman Problem (TSP) Given n cities on a map, find the shortest tour visiting all cities and ending up where it started.

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11. 11 Traveling Salesman Problem (TSP) Given n £ n distance matrix (d ij ) find permutation  of {0,1,2,..,n-1} minimizing The special case of d ij being actual distances on a map is called the Euclidean TSP.

12. 12 TSP as ILP, first attempt

13. 13 TSP as ILP, correct formulation

14. 14 TSP as ILP, compact formulation

15. 15 Maximum matching Max Flow Shortest paths Min Cost Flow Linear Programming Mixed Integer Linear Programming Polynomial time (easy) by Local Search Exponential time (hard). = reduction TSP NP-completeness … …

16. 16 Maximum matching Max Flow Shortest paths Min Cost Flow Linear Programming Mixed Integer Linear Programming Polynomial time (easy) by Local Search Exponential time (hard). = reduction TSP… NP-completeness

17. 17 …. This doesn’t mean that we should give up solving concrete ILP or TSP instances! There is a java program finding the solution below in less than a minute.