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

Simulated annealing approach to verify vertex adjacencies in the traveling salesperson polytope