A Randomized Fully Polynomial-time Approximation Scheme for Weighted Perfect Matching in the Plane

Abstract

In the approximate Euclidean min-weighted perfect matching problem, a set of points in the plane and a real number are given. Usually, a solution of this problem is a partition of points of into pairs such that the sum of the distances between the paired points is at most times the optimal solution. In this paper, the authors give a randomized algorithm which follows a Monte-Carlo method. This algorithm is a randomized fully polynomial-time approximation scheme for the given problem. Fortunately, the suggested algorithm is a one tackled the matching problem in both Euclidean nonbipartite and bipartite cases. The presented algorithm outlines as follows: With repeating times, we choose a point from to build the suitable pair satisfying the suggested condition on the distance. If this condition is achieved, then remove the points of the constructed pair from and put this pair in (the output set of the solution). Then, choose a point and the nearest point of it from the remaining points in to construct a pair and put it in . Remove the two points of the constructed pair from and repeat this process until becomes an empty set. Obviously, this method is very simple. Furthermore, our algorithm can be applied without any modification on complete weighted graphs and complete weighted bipartite graphs , where and m is an even.