Derangements in Symmetric Cost Matrices
  Let M be an n X n symmetric cost matrix. Assume that D is a derangement in M, i.e.,a set of disjoint cycles consisting of edges that contains all of the n points of M. The modified Floyd-Warshall algorithm applied to (D')^-1(M^-)A^- (where A is an asymmetric cost matrix containing D', a derangement)yielded a solution to the Assignment Problem in O((n^2)logn) running time. Here, applying a variation of the modified F-W to (D^-1)M^-1, we can obtain D = D_FWABS, the smallest-valued derangement obtainable using the modified F-W. Let T_TSPOPT be an optimal tour in M. We give conditions for obtaining D_ABSOLUTE, the smallest-valued derangement obtainable in M, where |D_ABSOLUTE| <= |T_TSPOPT|. 
Howard Kleiman