A O(n^8) X O(n^7) Linear Programming Model of the Quadratic Assignment Problem

TL;DR

This paper proposed a linear programming model for the quadratic assignment problem but was withdrawn due to errors in key theorems, highlighting the need for 9-dimensional variables instead of 8.

Contribution

It introduced a modeling approach for the quadratic assignment problem, which was later found to be incorrect due to dimensionality issues.

Findings

The initial model contained errors in theorems and corollaries.

Correct modeling requires 9-dimensional variables instead of 8.

Related correct models are cited from Diaby's work.

Abstract

This paper has been withdrawn because Theorem 21 and Corollary 22 are in error; The modeling idea is OK, but it needs 9-dimensional variables instead of the 8-dimensional variables defined in notations 6.9. Examples of the correct model (with 9-index variables) are: (1) Diaby, M., "Linear Programming Formulation of the Set Partitioning Problem," International Journal of Operational Research 8:4 (August 2010) pp. 399-427; (2) Diaby, M., "Linear Programming Formulation of the Vertex Coloring Problem," International Journal of Mathematics in Operational Research 2:3 (May 2010) pp. 259-289; (3) Diaby, M., "The Traveling Salesman Problem: A Linear Programming Formulation," WSEAS Transactions on Mathematics, 6:6 (June 2007) pp. 745-754.