Hamiltonian decomposition and verifying vertex adjacency in 1-skeleton of the traveling salesperson polytope by variable neighborhood search

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 co…

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 …

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

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…

On the skeleton of the pyramidal tours polytope

We consider the skeleton of the pyramidal tours polytope. Hamiltonian tour is called pyramidal if the salesperson starts in city $1$, then visits some cities in increasing order, reaches city $n$ and returns to city $1$, visiting the remaining cities in decreasing order. The polytope $PYR(n)$ is defined as the convex hull of characteristic vectors of all pyramidal tours in the complete graph $K_{n}$. The skeleton of the polytope $PYR(n)$ is the graph whose vertex set is the vertex set of $PYR(n)$ and edge set is the set of geometric edges or one-dimensional faces of $PYR(n)$. We describe the n…

Graph realizations constrained by skeleton graphs

In 2008 Amanatidis, Green and Mihail introduced the Joint Degree Matrix (JDM) model to capture the fundamental difference in assortativity of networks in nature studied by the physical and life sciences and social networks studied in the social sciences. In 2014 Czabarka proposed a direct generalization of the JDM model, the Partition Adjacency Matrix (PAM) model. In the PAM model the vertices have specified degrees, and the vertex set itself is partitioned into classes. For each pair of vertex classes the number of edges between the classes in a graph realization is prescribed. In this paper …

通过整数线性规划寻找4-正则多重图的第二个Hamilton分解

Finding a second Hamiltonian decomposition of a 4-regular multigraph by integer linear programming

针对4-正则多重图的第二个Hamilton分解问题，提出基于DFJ与MTZ公式的整数线性规划模型，并结合变邻域下降和链式边固定，实验表明二者分别在有向与无向情形表现突出。