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Concept
Integer programming
Summary
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear.
Integer programming is NP-complete. In particular, the special case of 0-1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems.
If some decision variables are not discrete, the problem is known as a mixed-integer programming problem.
In integer linear programming, the canonical form is distinct from the standard form. An integer linear program in canonical form is expressed thus (note that it is the vector which is to be decided):
and an ILP in standard form is expressed as
where are vectors and is a matrix. As with linear programs, ILPs not in standard form can be converted to standard form by eliminating inequalities, introducing slack variables () and replacing variables that are not sign-constrained with the difference of two sign-constrained variables.
The plot on the right shows the following problem.
The feasible integer points are shown in red, and the red dashed lines indicate their convex hull, which is the smallest convex polyhedron that contains all of these points. The blue lines together with the coordinate axes define the polyhedron of the LP relaxation, which is given by the inequalities without the integrality constraint. The goal of the optimization is to move the black dashed line as far upward while still touching the polyhedron. The optimal solutions of the integer problem are the points and which both have an objective value of 2. The unique optimum of the relaxation is with objective value of 2.8. If the solution of the relaxation is rounded to the nearest integers, it is not feasible for the ILP.
The following is a reduction from minimum vertex cover to integer programming that will serve as the proof of NP-hardness.
Official source
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