An assignment problem is any mathematical problem whose solution consists of assigning members of one set to members of another set if each set is of equal size and each element is assigned to exactly one element from the other set.
It is one of a class of general scientific problems, that has a known approximate solution (one that can be calculated in a reasonable amount of time).
A problem may be classified as an 'assignment problem' if the nature of the problem is such that there exist 2 sets of data, each with identical numbers of elements. Then there exists some function C, that maps one of the 'A' elements, to one of the 'B' elements in a one-to-one manner.
As opposed to, for instance, a function D, which maps several elements of 'A' to a single element of 'B', or vice-versa; in which case one cannot say that this is an 'assignment problem'.
Mathematical Definition: An assignment problem (or "linear assignment") is any problem involving minimising the sum of C(a, b) over a set P of pairs (a, b) where a is an element of some set A and b is an element of set B, and C is some function, under constraints such as "each element of A must appear exactly once in P" or similarly for B, or both.
For example, the a's could be workers and the b's projects.
The problem is "linear" because the "cost function" C() depends only on the particular pairing (a, b) and is independent of all other pairings.
External Links
http://forum.swarthmore.edu/epigone/comp.soft-sys.matlab/bringhyclu
http://www.soci.swt.edu/capps/prob.htm
http://mat.gsia.cmu.edu/GROUP95/0577.html
http://www.informs.org/Conf/WA96/TALKS/SB24.3.html]
This article was originally based on material from FOLDOC, used with permission. Update as needed.