Editing Linear programming (section)

== Integer unknowns ==
If all of the unknown variables are required to be integers, then the problem is called an [[integer programming]] (IP) or '''integer linear programming''' (ILP) problem.  In contrast to linear programming, which can be solved efficiently in the worst case, integer programming problems are in many practical situations (those with bounded variables) [[NP-hard]]. '''0–1 integer programming''' or '''binary integer programming''' (BIP) is the special case of integer programming where variables are required to be 0 or 1 (rather than arbitrary integers). This problem is also classified as NP-hard, and in fact the decision version was one of [[Karp's 21 NP-complete problems]].

If only some of the unknown variables are required to be integers, then the problem is called a '''mixed integer (linear) programming''' (MIP or MILP) problem.  These are generally also NP-hard because they are even more general than ILP programs.

There are however some important subclasses of IP and MIP problems that are efficiently solvable, most notably problems where the constraint matrix is [[totally unimodular]] and the right-hand sides of the constraints are integers or – more general – where the system has the [[total dual integrality]] (TDI) property.

Advanced algorithms for solving integer linear programs include:
* [[cutting-plane method]]
* [[Branch and bound]]
* [[Branch and cut]]
* [[Branch and price]]
* if the problem has some extra structure, it may be possible to apply [[delayed column generation]].
Such integer-programming algorithms are discussed by [[Manfred W. Padberg|Padberg]] and in Beasley.