  Let M be an n X n symmetric cost matrix. Assume that D is a derangement of
edges in M, i.e., a set of point-disjoint cycles containing all of the n points
of M.The modified Floyd-Warshall algorithm applied to ((D')^-1)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 algorithm to D^-1)M^-, we may possibly obtain a
smaller-valued derangement than D consisting of entries in M. A
minimally-valued derangement would be of great value as a good and natural
lower bound for an optimal tour in M.
