Continuous Relaxations for the Traveling Salesman Problem

In this work, we construct heuristic approaches for the traveling salesman problem (TSP) based on embedding the discrete optimization problem into continuous spaces. We explore multiple embedding techniques – namely, the construction of dynamical flows on the manifold of orthogonal matrices and associated Procrustes approximations of the TSP cost function. In particular, we find that the Procrustes approximation provides a competitive biasing approach for the Lin–Kernighan heuristic. The Lin–Kernighan heuristic is typically based on the computation of edges that have a “high probability” of being in the shortest tour, thereby effectively pruning the search space for the k-opt moves. This computation is traditionally based on generating 1-trees for a modified distance matrix obtained by a subgradient optimization method. Our novel approach, instead, relies on the solution of a two-sided orthogonal Procrustes problem, a natural relaxation of the combinatorial optimization problem to the manifold of orthogonal matrices and the subsequent use of this solution to bias the k-opt moves in the Lin–Kernighan heuristic. Although the initial cost of computing these edges using the Procrustes solution is higher than the 1-tree approach, we find that the Procrustes solution, when coupled with a homotopy computation, contains valuable information regarding the optimal edges. We explore the Procrustes based approach on several TSP instances and find that our approach often requires fewer k-opt moves.