Tour merging via branch-decomposition

Robertson and Seymour introduced branch-width as a new connectivity invariant of graphs in their proof of the Wagner conjecture. Decompositions based on this invariant provide a natural framework for implementing dynamic-programming algorithms to solve graph optimization problems. We describe a heuristic method for finding branch-decompositions; the method is based on the eigenvector technique for finding graph separators. We use this as a tool to obtain high-quality tours for the traveling salesman problem by merging collections of tours produced by standard traveling salesman heuristics.