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Integrality constraints

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- 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?

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.

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.

- 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.

- Given n cities on a map, find the shortest tour visiting all cities and ending up where it started.

- 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.

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

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

…. 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.