Pyramidal traveling salesman problem

In this paper, we give new polynomially testable sufficiency conditions for a given instance of the traveling salesman problem (TSP) to have an optimal tour that is pyramidal. This properly generalizes the Demidenko condition and the conditions of Warren. We thus have new, nontrivial polynomially testable and polynomially solvable cases of TSP. Scope and purpose The problem of identifying polynomially testable and polynomially solvable subclasses of the TSP is of significant theoretical and practical value. Pyramidal TSP is a well-studied subclass of the TSP that is polynomially solvable. However, it is an NP-hard problem to test if a given instance of TSP is pyramidal. Various polynomially testable subclasses of this have been identified in literature. In this paper, we give a general polynomially testable sufficiency condition for an instance of TSP to be pyramidal. This provides simpler proofs and also a proper generalization of the known classes.