The maximum weight perfect matching problem for complete weighted graphs is in PC

There are efficient sequential algorithms that use linear programming (LP) for computing maximum weight matchings. Finding a deterministic parallel algorithm for computing maximum weight matchings in complete graphs has been an open problem for some time. Since LP is known to be P-complete, then, by the parallel computation thesis, it is unlikely that there exists an NC algorithm that uses LP to solve the maximum weight matching problem. The authors present an LP-based parallel algorithm for maximum weight matching in a complete weighted graph. The algorithm is designed for the EREW PRAM model of parallel computation, and runs in O(n³/p+n²logn) time for p⩽n, where p is the number of processors and n is the number of vertices in the graph. This algorithm provides an optimal speedup with respect to the O(n ³) sequential LP-based solution of Gabow (1974) or Lawler (1976), for p⩽n/log n. This is the first deterministic optimal speedup parallel algorithm designed for the maximum weight matching problem on complete graphs