We present a linear time approximation algorithm with a performance ratio of 1/2 for nding a maximum weight matching in an arbitrary graph. Such a result is already known and is due to Preis [7].

Approximation algorithms have so far mainly been studied for problems that are not known to have polynomial time algorithms for solving them exactly. Here we propose an approximation algorithm for the weighted matching problem in graphs which can be solved in polynomial time. The weighted matching problem is to find a matching in an edge weighted graph that has maximum weight. The first polynomial time algorithm for this problem was given by Edmonds in 1965. The fastest known algorithm for the weighted matching problem has a running time of O(nm+n 2 log n). Many real world problems require graphs of such large size that this running time is too costly. Therefore there is considerable need for faster approximation algorithms for the weighted matching problem. We present a linear time approximation algorithm for the weighted matching problem with a performance ratio arbitrarily close to 2/3

Given a weighted graph, the weighted matching problem is to find a matching with maximum weight. The fastest known exact algorithm runs in O(nm + n2 log n) however for many real world applications this is too costly, and an approximate matching is sufficient. A c-approximation algorithm is one which always finds a weight of at least c times the optimal weight. Drake and Hougardy developed a linear time 2/3 - epsilon approximation algorithm which is the best known serial algorithm. They also developed a parallel 1- epsilon approximation algorithm for the PRAM model, however it requires a large number of processors which is not as useful in practice. Hoepman developed a distributed 1/2 approximation algorithm which is the best known distributed algorithm. We present a shared memory parallel version of the best 2/3 - epsilon algorithm, which is simple to understand and easy to implement.