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Integrality constraints

Integrality constraints. Integrality constraints are often extremely desirable when modeling problems as linear programs.

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Integrality constraints

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  1. Integrality constraints • Integrality constraints are often extremely desirable when modeling 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. Mixed Integer Linear Programs (MILP) Find x2Rn minimizing or maximizing a linear formhx,ci = i ci xi (the objective function) so that a given set of linear non-strict inequalitiesand integrality constraintsxi2Z are satisfied. A feasible solution to the program is a point xsatisfying the inequalities and integrality constraints.

  3. Integer Linear Programs (ILP) Find x2Zn minimizing or maximizing a linear formhx,ci = i ci xi (the objective function) so that a given set of linear non-strict inequalitiesare satisfied. A feasible solution to the program is a point xsatisfying the inequalities.

  4. 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.

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

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

  7. TSP as ILP, first attempt

  8. TSP as ILP, correct formulation

  9. TSP as ILP, compact formulation

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

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

  12. …. 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.

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