  Let M be an nXn symetric matrix, n, even, T, an upper bound for T_OPT, an
optimal tour, sigma_T, the smaller-valued perfect matching obtained from
alternate edges of T expressed as a product of 2-cycles. Applying the modified
Floyd-Warshall algorithm to (sigma_T)^-1M^-, we construct acceptable and
2-circuit cycles some sets of which may yield circuits that can be patched into
tours. We obtain necessary and sufficient conditions for a set, S, of cycles to
yield circuits that may be patched into a tour.Assume that the following
(Condition A)is valid: If (sigma_T)s = T*, |T*|<T, then all cycles of s have
values less than |T| - |sigma_T|.Let SFWOPT),S(OPT)be the respective sets of
cycles yielding T_FWOPT, T_OPT. Given Condition(A), using F-W, we can always
obtain S(FWOPT). Using Condition A but not F-W, S_OPT is always obtainable from
a subset of the cycles obtained.
