In this paper, we give new polynomially testable sufficiency conditions for
a given instance of the traveling salesman problem (TSP) to have an optima
l tour that is pyramidal. This properly generalizes the Demidenko condition
and the conditions of Warren. We thus have new, nontrivial polynomially te
stable and polynomially solvable eases of TSP. (C) 1999 Elsevier Science Lt
d. All rights reserved.