A criterion for the adjacency of vertices of polytopes generated by subsets of symmetric groups

We propose a characterization of the adjacency of vertices in the class of permutation polytopes generated by arbitrary subsets of symmetric groups. In particular, this class contains polytopes for the well-known classical problems, such as the assignment problem, $2$- and $3$-combinations, the traveling salesman problem and their various modifications. Up to now, the problem of vertex adjacency has been studied for a single polytope only. In the present paper, we obtain, for general permutation polytopes, necessary and sufficient conditions that guarantee that two given vertices are adjacent (or not) to each other. The conditions are formulated in terms of permutations and of the solvability of certain special systems of linear equations. The presently known adjacency criteria for vertices of polytopes for the assignment problem are simple corollaries of our conditions. The latter allow us to develop a general algorithmic scheme for recognizing vertex adjacency of a general permutation polytope and estimate its complexity.