On 1-skeleton of the polytope of pyramidal tours with step-backs

TL;DR

This paper studies the structure of the polytope formed by pyramidal tours with step-backs, providing algorithms to verify adjacency, and estimates for its diameter and clique number, which are relevant for understanding its combinatorial complexity.

Contribution

It introduces a linear-time algorithm for verifying vertex adjacencies in the polytope's 1-skeleton and estimates key graph parameters like diameter and clique number.

Findings

Diameter of the 1-skeleton is at most 4.

Clique number grows quadratically with n.

Efficient adjacency verification algorithm provided.

Abstract

Pyramidal tours with step-backs are Hamiltonian tours of a special kind: the salesperson starts in city 1, then visits some cities in ascending order, reaches city $n$, and returns to city 1 visiting the remaining cities in descending order. However, in the ascending and descending direction, the order of neighboring cities can be inverted (a step-back). It is known that on pyramidal tours with step-backs the traveling salesperson problem can be solved by dynamic programming in polynomial time. We define the polytope of pyramidal tours with step-backs $\operatorname{PSB}(n)$ as the convex hull of the characteristic vectors of all possible pyramidal tours with step-backs in a complete directed graph. The 1-skeleton of $\operatorname{PSB}(n)$ is the graph whose vertex set is the vertex set of the polytope, and the edge set is the set of geometric edges or one-dimensional faces of the polytope. We present a linear-time algorithm to verify vertex adjacencies in 1-skeleton of the polytope $\operatorname{PSB}(n)$ and estimate the diameter and the clique number of 1-skeleton: the diameter is bounded above by 4 and the clique number grows quadratically in the parameter $n$.