Analysis of a heuristic algorithm for planar assignment problem

This paper describes a theoretical analysis of the heuristic algorithm for the efficient solution of (the linear) assignment problem defined on a plane. The evaluation function, or assignment cost, in the assignment problem discussed in this paper is defined as the square‐sum of Euclidean distances between points included in the two point sets. A heuristic assignment algorithm, based on a vertical or horizontal partitioning of the given plane, is presented. Then the average‐case analysis is performed for the algorithm, assuming that the points are distributed uniformly on the unit square plane. Letting the size of the point set be n, the number of partitioned stages in the algorithm be hm and the average assignment cost for n → ∞ be Ǐ, it is seen as a result of analysis that the rate of increase of Ǐ in regard to n decreases monotonously with the increase of hm. It is also seen that when hm is maximum, Ǐ is (1/4) log2. n. The result of theoretical analysis is compared with the result of experiment.