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

Yasser M., Salwa M., Hanaa A.E., Soheir M.
2012 International Journal of Advanced Computer Science and Applications
In the approximate Euclidean min-weighted perfect matching problem, a set V of n 2 points in the plane and a real number 0   are given. Usually, a solution of this problem is a partition of points of V into n pairs such that the sum of the distances between the paired points is at most ) 1 (   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
more » ... tion 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    / 1 times, we choose a point from V 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 V and put this pair in M (the output set of the solution). Then, choose a point and the nearest point of it from the remaining points in V to construct a pair and put it in M . Remove the two points of the constructed pair from V and repeat this process until V becomes an empty set. Obviously, this method is very simple. Furthermore, our algorithm can be applied without any modification on complete weighted graphs m K and complete weighted bipartite graphs n n K , , where 1 ,  m n and m is an even. Keywords-Perfect matching; approximation algorithm; Monte-Carlo technique; a randomized fully polynomial-time approximation scheme; and randomized algorithm. I.