and integrating the flow backward in time [@Cit:TSP_sahai]. The solutions are shown in Fig. \[fig:polar\]. These computations are interesting and capture the “hardness” of the problem. In particular, one can see that the solutions of relaxed versions of the problem (such as the relaxations to the manifolds of orthogonal matrices) do not, in general, lie in the basin of attraction of the optimal solutions of the original problem. Other such instances of analysis of relaxed solutions of the TSP using dynamical systems theory are outlined in [@Cit:TSP_sahai].

Although, dynamical systems theory demonstrates that the Procrustes solutions do not typically lie in the basin of attraction of the optimal solutions of the TSP, a new biasing scheme for the Lin–Kernighan heuristic is constructed using the aforementioned relaxation [@Cit:TSP_sahai].

The Lin–Kernighan heuristic is a popular heuristic for the TSP [@Cit:LKH]. Starting from an initial tour, the approach progresses by extracting edges from the tour and replacing them with new edges, while maintaining the Hamiltonian cycle constraint. If $k$ edges in the tour are simultaneously replaced, this is known as the $k$-opt move [@Cit:LKH]. To prune the search space, the algorithm relies on minimum spanning trees to identify edges that are more likely 