Using group theory and transition matrices to study a class of metaheuristic neighborhoods

The one-step conjugative rearrangement neighborhood of all possible incumbent tours in an n-city single-agent Traveling Salesperson Problem is represented by a transition matrix. Using these matrices and employing group theory and the symmetric group on n letters, we show that all such matrices will fall into three different types: (1) irreducible matrices with one set of tours, (2) irreducible cyclic matrices of period 2 with two distinct sets of tours, and (3) reducible matrices with two equal-sized distinct sets of tours. In addition to giving the required conditions that yield each neighborhood type, we briefly discuss how these results are easily extended to multi-agent traveling salesperson problems and suggest directions for future investigations.