It is shown that the quadratic assignment problem defined on the r-dimensional grid can be formalized using polynomial equations. Consequently, the problem can be classified into subgroups of relaxed problems according to the degree of the polynomial equations. Since the behavior of the solution is almost determined by a few polynomial equations with rather low degree, and it is generally easier to solve the relaxed problem represented by those equations, it is possible to derive more efficient and effective methods than those based on the conventional formulations. Solutions for the relaxed problem in which the degree is no more than two are also discussed, and applications of the present formulation to the floor-planning problem of VLSI and the graph partitioning problem are presented.
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