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

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Integrality constraints

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Integrality constraints

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?


Mixed integer linear programs milp

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.


Integer linear programs ilp

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.


Power of ilp

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.


Traveling salesman problem tsp

Traveling Salesman Problem (TSP)

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


Traveling salesman problem tsp1

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.


Tsp as ilp first attempt

TSP as ILP, first attempt


Tsp as ilp correct formulation

TSP as ILP, correct formulation


Tsp as ilp compact formulation

TSP as ILP, compact formulation


Integrality constraints

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


Integrality constraints

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


Integrality constraints

…. 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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