On vertex adjacencies in the polytope of pyramidal tours with step-backs

We consider the traveling salesperson problem in a directed graph. The pyramidal tours with step-backs are a special class of Hamiltonian cycles for which the traveling salesperson problem is solved by dynamic programming in polynomial time. The polytope of pyramidal tours with step-backs $PSB (n)$ is defined as the convex hull of the characteristic vectors of all possible pyramidal tours with step-backs in a complete directed graph. The skeleton of $PSB (n)$ is the graph whose vertex set is the vertex set of $PSB (n)$ and the edge set is the set of geometric edges or one-dimensional faces of $PSB (n)$. The main result of the paper is a necessary and sufficient condition for vertex adjacencies in the skeleton of the polytope $PSB (n)$ that can be verified in polynomial time.Comment: in Englis

Simulated Annealing Approach to Verify Vertex Adjacencies in the Traveling Salesperson Polytope

We consider 1-skeletons of the symmetric and asymmetric traveling salesperson polytopes whose vertices are all possible Hamiltonian tours in the complete directed or undirected graph, and the edges are geometric edges or one-dimensional faces of the polytope. It is known that the question whether two vertices of the symmetric or asymmetric traveling salesperson polytopes are nonadjacent is NP-complete. A sufficient condition for nonadjacency can be formulated as a combinatorial problem: if from the edges of two Hamiltonian tours we can construct two complementary Hamiltonian tours, then the corresponding vertices of the traveling salesperson polytope are not adjacent. We consider a heuristic simulated annealing approach to solve this problem. It is based on finding a vertex-disjoint cycle cover and a perfect matching. The algorithm has a one-sided error: the answer "not adjacent" is always correct, and was tested on random and pyramidal Hamiltonian tours.Comment: in Englis