Integer Unknowns
If some or 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 programming (MIP) problem. These are generally also NP-hard.
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.
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 Padberg and in Beasley.
Read more about this topic: Linear Programming
Famous quotes containing the word unknowns:
“Pregnancy represents one of the great unknowns in life. Everyone’s at least a little worried about how it will turn out.”
—Lawrence Kutner (20th century)