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{{Short description|A mathematical optimization problem restricted to integers}}
An '''integer programming''' problem is a [[mathematical optimization]] or [[Constraint satisfaction problem|feasibility]] program in which some or all of the variables are restricted to be [[
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]].<ref>{{cite book
| first = Richard M. |last = Karp
|chapter = Reducibility among Combinatorial Problems
| author-link = Richard M. Karp
| chapter-url = http://cgi.di.uoa.gr/~sgk/teaching/grad/handouts/karp.pdf
| title = Complexity of Computer Computations
| editor = R. E. Miller |editor2=J. W. Thatcher |editor3=J.D. Bohlinger
| publisher = New York: Plenum
| pages = 85–103
| year = 1972
| doi = 10.1007/978-1-4684-2001-2_9
| isbn = 978-1-4684-2003-6
}}</ref>
If some decision variables are not discrete, the problem is known as a '''mixed-integer programming''' problem.<ref>{{cite web
▲<u>Canonical and standard form for ILP(Interger Linear Programming)</u>
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 <math>\mathbf{x}</math> vector which is to be decided):<ref name="optBook">{{cite book|last1=Papadimitriou|first1=C. H.|author1-link=Christos Papadimitriou|last2=Steiglitz|first2= K.|author2-link=Kenneth Steiglitz|title=Combinatorial optimization: algorithms and complexity|year=1998|publisher=Dover|___location=Mineola, NY|isbn=0486402584}}</ref>
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