The quadratic assignment problem (QAP) is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics, from the category of the facilities location problems first introduced by Koopmans and Beckmann.

The problem models the following real-life problem:

Intuitively, the cost function encourages facilities with high flows between each other to be placed close together.

The problem statement resembles that of the assignment problem, except that the cost function is expressed in terms of quadratic inequalities, hence the name.

The formal definition of the quadratic assignment problem is as follows:

Usually weight and distance functions are viewed as square real-valued matrices, so that the cost function is written down as:

In matrix notation:

where ${\displaystyle \Pi _{n}}$ is the set of ${\displaystyle n\times n}$ permutation matrices, ${\displaystyle W}$ is the weight matrix and ${\displaystyle D}$ is the distance matrix.