Approximating minimum weight perfect matchings for complete graphs satisfying the triangle inequality

Abstract

We describe an O(log3n) time NC approximation algorithm for the CREW P-RAM, using n3/ log n processors with a 2log3n performance ratio, for the problem of finding a minimum-weight perfect matching in complete graphs satisfying the triangle inequality. The algorithm is conceptually very simple and has a work measure within a factor of log2n of the best exact sequential algorithm. This is the first NC approximation algorithm for the problem with a sub-linear performance ratio. As was the case in the development of sequential complexity theory, matching problems are on the boundary of what problems might ultimately be described as tractable for parallel computation. Future work in this area is likely to decide whether these ought to be regarded as those problems in NC or those problems in RNC.