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| The '''quadratic assignment problem''' ('''QAP''') is one of fundamental [[combinatorial optimization]] problems in the branch of [[Optimization (mathematics)|optimization]] or [[operations research]] in [[mathematics]], from the category of the [[facilities location]] problems.
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| The problem models the following real-life problem:
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| :There are a set of ''n'' facilities and a set of ''n'' locations. For each pair of locations, a ''distance'' is specified and for each pair of facilities a ''weight'' or ''flow'' is specified (e.g., the amount of supplies transported between the two facilities). The problem is to assign all facilities to different locations with the goal of minimizing the sum of the distances multiplied by the corresponding flows.
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| Intuitively, the cost function encourages factories with high flows between each other to be placed close together.
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| The problem statement resembles that of the [[assignment problem]], except that the [[Loss function|cost function]] is expressed in terms of quadratic inequalities, hence the name.
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| ==Formal mathematical definition==
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| The formal definition of the quadratic assignment problem is as follows:
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| :Given two sets, ''P'' ("facilities") and ''L'' ("locations"), of equal size, together with a [[weight function]] ''w'' : ''P'' × ''P'' → '''[[real number|R]]''' and a distance function ''d'' : ''L'' × ''L'' → '''[[real number|R]]'''. Find the [[bijection]] ''f'' : ''P'' → ''L'' ("assignment") such that the cost function:
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| ::<math>\sum_{a,b\in P}w(a,b)\cdot d(f(a), f(b))</math>
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| :is minimized.
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| Usually weight and distance functions are viewed as square real-valued [[matrix (mathematics)|matrices]], so that the cost function is written down as:
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| :<math>\sum_{a,b\in P}w_{a,b}d_{f(a),f(b)}</math>
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| In matrix notation:
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| :<math>min_{X\in\Pi_n} trace(WXDX^T)</math> where <math>\Pi_n</math> are the permutation matrices, "W" is the weight matrix and "D" is the distance matrix.
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| == Computational complexity ==
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| The problem is [[NP-hard]], so there is no known [[algorithm]] for solving this problem in polynomial time, and even small instances may require long computation time. The [[travelling salesman problem]] may be seen as a special case of QAP if one assumes that the flows connect all facilities only along a single ring, all flows have the same non-zero (constant) value. Many other problems of standard [[combinatorial optimization]] problems may be written in this form.
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| == Applications ==
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| In addition to the original plant location formulation, QAP is a mathematical model for the problem of placement of interconnected [[electronic component]]s onto a [[printed circuit board]] or on a [[integrated circuit|microchip]], which is part of the [[place and route]] stage of [[computer aided design]] in the electronics industry.
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| ==See also==
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| *[[Quadratic bottleneck assignment problem]]
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| == References ==
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| * {{cite book|author = [[Michael R. Garey]] and [[David S. Johnson]] | year = 1979 | title = [[Computers and Intractability: A Guide to the Theory of NP-Completeness]] | publisher = W.H. Freeman | isbn = 0-7167-1045-5}} A2.5: ND43, pg.218.
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| ==External links==
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| * http://www.seas.upenn.edu/qaplib/ QAPLIB - A Quadratic Assignment Problem Library
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| * http://issuu.com/spconguy/docs/ant-algorithm-applied-to-the-quadratic-assignment- - A QAP sample application
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| [[Category:NP-hard problems]]
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| [[Category:Combinatorial optimization]]
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Not much to tell about me at all.
Hurrey Im here and a part of wmflabs.org.
I just hope I am useful at all
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